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We study the problems of uniqueness for Hardy-H´enon parabolic +equations, which are semilinear heat equations with the singular potential (Hardy +type) or the increasing potential (H´enon type) in the nonlinear term. To deal +with the Hardy-H´enon type nonlinearities, we employ weighted Lorentz spaces as +solution spaces. We prove unconditional uniqueness and non-uniqueness, and we +establish uniqueness criterion for Hardy-H´enon parabolic equations in the weighted +Lorentz spaces. The results extend the previous works on the Fujita equation and +Hardy equations in Lebesgue spaces. +1. Introduction and main results +1.1. Introduction and our setting. We consider the Cauchy problem of the +Hardy-H´enon parabolic equation +� +∂tu − ∆u = |x|γ|u|α−1u, +(t, x) ∈ (0, T) × Rd, +u(0) = u0 ∈ Lq,r +s (Rd), +(1.1) +where T > 0, d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞], r ∈ (0, ∞] and s ∈ R. Here, +∂t := +∂ +∂t is the time derivative, ∆ := �d +j=1 +∂2 +∂x2 +j is the Laplace operator on Rd, +u = u(t, x) is an unknown complex-valued function on (0, T) × Rd, u0 = u0(x) is +a prescribed complex-valued function on Rd, and Lq,r +s (Rd) is the weighted Lorentz +space (see Definition 2.3), which includes the Lebesgue space Lq(Rd) = Lq,q +0 (Rd) as +a special case r = q and s = 0. The equation (1.1) in the case γ = 0 is the Fujita +equation, which has been extensively studied in various directions. The equation +(1.1) with γ < 0 is known as a Hardy parabolic equation, while that with γ > 0 +is known as a H´enon parabolic equation. The corresponding stationary problem to +(1.1), that is, +− ∆U = |x|γ|U|α−1U, +(1.2) +was proposed by H´enon as a model to study the rotating stellar systems (see [19]), +and has also been extensively studied in the mathematical context, especially in the +fields of nonlinear analysis and variational methods (see [14] for example). +In this paper we study the problem on unconditional uniqueness and non-uniqueness +for (1.1) in weighted Lorentz spaces Lq,r +s (Rd). Here, unconditional uniqueness means +2020 Mathematics Subject Classification. Primary 35A02, 35K58; Secondary 35B33. +Date: January 3, 2023. +Key words and phrases. Hardy-H´enon parabolic equations, semilinear heat equations, uncon- +ditional uniqueness, non-uniqueness, uniqueness criterion, singular stationary solutions, weighted +Lorentz spaces. +1 +arXiv:2301.00506v1 [math.AP] 2 Jan 2023 + +2 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +uniqueness of the solution to (1.1) in the sense of the integral form +u(t) = et∆u0 + +� t +0 +e(t−τ)∆(| · |γ|u(τ)|α−1u(τ)) dτ +(1.3) +in L∞(0, T; Lq,r +s (Rd)) or C([0, T]; Lq,r +s (Rd)), where T > 0 and {et∆}t>0 is the heat +semigroup. We say that non-uniqueness holds for (1.1) if unconditional uniqueness +fails. +In contrast, we say that conditional uniqueness holds if uniqueness of the +solution to (1.1) holds in the entire space with some auxiliary function spaces. In +addition, we also study uniqueness criterion which is a necessary and sufficient con- +dition on the Duhamel term (i.e. the second term in the right-hand side of (1.3)) +for uniqueness to hold. +Let us here state previous works on uniqueness for (1.1). For (1.1) with γ ≤ 0, the +problem on uniqueness has been well studied (see [3, 4, 7–9, 18, 27, 29, 36, 38, 42, 43] +for example). In the study of unconditional uniqueness for (1.1) in Lebesgue spaces +Lq(Rd) or Lorentz spaces Lq,r(Rd), the following two critical exponents are known +to be important. The first one is the so-called scale-critical exponent qc given by +qc = qc(d, γ, α) := d(α − 1) +2 + γ +, +(1.4) +and we say that the problem (1.1) is scale-critical if q = qc, scale-subcritical if +q > qc, and scale-supercritical if q < qc. The second one is the critical exponent Qc +given by +Qc = Qc(d, γ, α) := +dα +d + γ , +(1.5) +which is related to well-definedness of the Duhamel term in (1.3) in Lq,r(Rd). In +fact, the nonlinear term |x|γ|u|α−1u ∈ L1 +loc(Rd) for any u ∈ Lq,r(Rd) if and only if +“q > Qc” or “q = Qc and r ≤ α”. In the case γ = 0, unconditional uniqueness for +(1.1) in C([0, T]; Lq(Rd)) was proved in the double subcritical case q > max{qc, Qc} +by Weissler [42] and in the single critical cases q = Qc > qc and q = qc > Qc by +Brezis and Cazenave [7]. In the double critical case q = qc = Qc, non-uniqueness +was proved for some initial data u0 ∈ Lq(Rd) by Terraneo [38], and then, for any +initial data u0 ∈ Lq(Rd) by Matos and Terraneo [27]. In [38], uniqueness criterion +was also obtained in the double critical case. In the scale-supercritical case q < qc, +non-uniqueness for (1.1) was proved for initial data u0 = 0 by Haraux and Weissler +[18]. +Uniqueness and non-uniqueness have also been studied for heat equations +with exponential nonlinearities (see [21, 23] and references therein). In the Hardy +case − min{2, d} < γ < 0, similar results were obtained by [4,36], where the Lorentz +spaces Lq,r(Rd) is used to study unconditional uniqueness in the critical case q = Qc +in [36]. In contrast, the H´enon case γ > 0 has not been well studied. This is due +to the difficulty of treating the increasing potential |x|γ in the nonlinear term at +infinity. To overcome this difficulty, the weighted spaces are effective, and recently, +conditional uniqueness was obtained in Lq +s(Rd) = Lq,q +s (Rd) in [10]; however, uncon- +ditional uniqueness and non-uniqueness are completely open. The main purpose +of this paper is to prove unconditional uniqueness, non-uniqueness and uniqueness +criterion for (1.1) with all γ > − min{2, d}, including the H´enon case, in Lq,r +s (Rd). + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +3 +Figure 1. The figure shows the domain of (α, q) for d ≥ 3 and +γ ≤ 0, where α0 := 1 + γ +d , αF := 1 + 2+γ +d +is the Fujita exponent, +α∗ := d+γ +d−2 is the Serrin exponent and αHS := d+2+2γ +d−2 +is the Hardy- +Sobolev exponent. Table 1 and Table 2 summarize the previous results +on uniqueness for (1.1) with γ ≤ 0. + +q= qc +q=Qc +b +d-2 +1 +0 +αo +1 αF +Q* +SHo +α +Table l: Unconditional Uniqueness +q > min[qc, Qc}] + >= b ----- +>= +== +YES +YES +YES +NO +[42, Thm 1] ( = 0) +[7, Thm 4] ( = 0) +[7, Thm 4] (= 0) +[27, Thm 1] (= 0) +[4, Thm 1.1] (< 0) +[36, Thm 1.1] (< 0) +[36, Thm 1.1] (< 0) +[36, Thm 1.3] (< 0) +Table 2: Conditional Uniqueness + q 0 and X = Lq,r +s (Rd) or Lq,r +s (Rd). We say that a function +u = u(t, x) on (0, T) × Rd is a mild solution to (1.1) with initial data u0 ∈ X in +C([0, T]; X) (L∞(0, T; X) resp.) if u belongs to C([0, T]; X) (L∞(0, T; X) resp.) +and satisfies the integral equation (1.3) for almost everywhere (t, x) ∈ (0, T) × Rd. +We define two critical cases in the framework of Lq,r +s (Rd) in a similar manner to +qc and Qc, respectively. The equation (1.1) is invariant under the following scale +transformation: +uλ(t, x) := λ +2+γ +α−1u(λ2t, λx), +λ > 0. +More precisely, if u is a solution to (1.1), then so is uλ with the rescaled initial data +λ +2+γ +α−1u0(λx). Moreover, we calculate +∥uλ(0)∥Lq,r +s += λ−s+ 2+γ +α−1 − d +q ∥u0∥Lq,r +s += λ−d( s +d + 1 +q − 1 +qc )∥u0∥Lq,r +s , +λ > 0. +Hence, if q and s satisfy +s +d + 1 +q = 1 +qc +, +(1.6) +then ∥uλ(0)∥Lq,r +s += ∥u0∥Lq,r +s +for any λ > 0, i.e., the norm ∥uλ(0)∥Lq,r +s +is invariant +with respect to λ. +Therefore, we say that the problem (1.1) is scale-critical if +s +d + 1 +q = +1 +qc , scale-subcritical if +s +d + 1 +q < +1 +qc , and scale-supercritical if +s +d + 1 +q > +1 +qc . +Another critical case is when the following holds: +s +d + 1 +q = 1 +Qc +. +(1.7) +This is related to local integrability of the nonlinear term |x|γ|u|α−1u. +In fact, +|x|γ|u|α−1u ∈ L1 +loc(Rd) for any u ∈ Lq,r +s (Rd) if and only if +s +d + 1 +q < 1 +Qc +or +s +d + 1 +q = 1 +Qc +and r ≤ α. +(1.8) +Then, it is ensured for the Duhamel term in (1.3) to be well-defined in Lq,r +s (Rd). +In terms of the two critical cases, we divide the problem into the following four +cases: Double subcritical case ( s +d + 1 +q < min{ 1 +qc, 1 +Qc}), single critical case I ( s +d + +1 +q = +1 +Qc < +1 +qc ), single critical case II ( s +d + 1 +q = +1 +qc < +1 +Qc ), and double critical case +( s +d + 1 +q = 1 +qc = +1 +Qc ). Moreover, we define the exponent α∗ by +α∗ = α∗(d, γ) := +� +� +� +d + γ +d − 2 +if d ≥ 3, +∞ +if d = 1, 2, +(1.9) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +5 +which is often referred to the Serrin exponent (see [34, 35] and also [16]). +The +exponents α∗, qc and Qc are related as follows: +α ⋚ α∗ +if and only if +qc ⋚ Qc. +In our results on unconditional uniqueness below, we assume that +� +� +� +� +� +d ∈ N, +γ > − min{2, d}, +α > max +� +1, 1 + γ +d +� +, +α ≤ q ≤ ∞, +γ +α − 1 ≤ s < d, +0 < r ≤ ∞. +(1.10) +Our results on unconditional uniqueness are the following: +Theorem 1.2 (Scale-subcritical case). Let T > 0, and let d, γ, α, q, r, s be as in +(1.10). Assume either (1) or (2): +(1) (Double subcritical case) r ≤ α if q = α, and 0 < s +d + 1 +q < min{ 1 +qc, 1 +Qc}. +(2) (Single critical case I) α < α∗, q ̸= ∞, r ≤ α and +s +d + 1 +q = +1 +Qc < 1 +qc . +Then unconditional uniqueness holds for (1.1) in L∞(0, T; Lq,r +s (Rd)). +Theorem 1.3 (Scale-critical case). Let T > 0, and let d, γ, α, q, r, s be as in (1.10). +Assume d ≥ 3, q ̸= ∞, and either (1) or (2): +(1) (Single critical case II) α > α∗ and +s +d + 1 +q = +1 +qc < +1 +Qc (replace Lq,∞ +s +(Rd) by +Lq,∞ +s +(Rd) if r = ∞). +(2) (Double critical case) α = α∗, r ≤ α∗ − 1 and +s +d + 1 +q = 1 +qc = +1 +Qc . +Then unconditional uniqueness holds for (1.1) in C([0, T]; Lq,r +s (Rd)). +Remark 1.4. In Theorem 1.2 (1), the condition “r ≤ α if q = α” comes from the +restriction on parameters in linear estimates. More precisely, the condition is due +to the restriction r1 = 1 for linear estimates with q1 = 1 in Proposition 3.1 (see +(3.4) and also Lemma 4.1 (ii)). +Next, we consider the following two cases where the unconditional uniqueness is +not obtained in the above theorems: r > α in the single critical case I; r > α∗ − 1 +in the double critical case. +In the single critical case I, the condition r ≤ α naturally appears from the +viewpoint of well-definedness of mild solutions to (1.1) as seen in (1.8). On the +other hand, when r > α, we can define mild solutions to (1.1) with the auxiliary +condition and we know that conditional uniqueness holds (see [10, Theorem 1.13]). +We are interested in the questions whether unconditional uniqueness holds in this +case or whether the conditional uniqueness can be improved. Unfortunately, we do +not know if unconditional uniqueness holds in this case. However, we can give the +following sufficient condition for uniqueness to hold which improves the conditional +uniqueness [10, Theorem 1.13]. +Proposition 1.5. Let T > 0, and let d, γ, α, q, r, s be as in (1.10). Assume that +α < α∗, q ̸= ∞, α < r ≤ ∞, and +s +d + 1 +q = +1 +Qc < +1 +qc . Let u0 ∈ Lq,r +s (Rd). Then, if + +6 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +u1, u2 ∈ L∞(0, T; Lq,r +s (Rd)) are mild solutions to (1.1) with u1(0) = u2(0) = u0 such +that +ui(t) − et∆u0 ∈ L∞(0, T; Lq,r′(α−1) +s +(Rd)) +for i = 1, 2, +(1.11) +then u1 = u2 on [0, T]. Here, r′ is the H¨older conjugate of r, i.e., 1 = 1 +r + 1 +r′ . +In the double critical case, we prove the result on non-uniqueness for (1.1) if +α∗ − 1 < r ≤ ∞. More precisely, we have the following: +Theorem 1.6 (Double critical case). Let d ≥ 3, γ > −2, α = α∗, α∗ ≤ q < ∞, +α∗ − 1 < r ≤ ∞, and +s +d + 1 +q = +1 +qc = +1 +Qc . Then, for any initial data u0 ∈ Lq,r +s (Rd), +there exists T = T(u0) > 0 such that the problem (1.1) has at least two different +solutions in C([0, T]; Lq,r +s (Rd)) (replace Lq,r +s (Rd) by Lq,∞ +s +(Rd) if r = ∞). +By Theorem 1.3 (2) and Theorem 1.6, we reveal that the exponent r = α∗ −1 is a +threshold of dividing unconditional uniqueness and non-uniqueness for (1.1) in the +double critical case. In Theorem 1.6, one of two different solutions is regular and +the other is singular at x = 0 (see Section 5), and this threshold comes essentially +from the logarithmic rate of the singularity at x = 0 of the singular solution (see +Theorem 5.5). In addition, we give the following uniqueness criterion. +Theorem 1.7. Let T > 0, and let d, γ, α, q, r, s be as in (1.10). +Assume that +d ≥ 3, γ > −2, α = α∗, α∗ ≤ q < ∞, α∗ − 1 < r ≤ ∞, and +s +d + 1 +q = 1 +qc = +1 +Qc . Let +u0 ∈ Lq,r +s (Rd). Then, if u1, u2 ∈ C([0, T]; Lq,r +s (Rd)) are mild solutions to (1.1) with +u1(0) = u2(0) = u0 such that +ui(t) − et∆u0 ∈ C([0, T]; Lq,α∗−1 +s +(Rd)) +for i = 1, 2, +(1.12) +then u1 = u2 on [0, T] (replace Lq,r +s (Rd) by Lq,∞ +s +(Rd) if r = ∞). +Remark 1.8. The exponent r = α∗ − 1 in (1.12) of Theorem 1.7 is optimal for the +same reason as above (see Theorem 5.4). +In the scale-supercritical case, we have the following result on non-uniqueness for +(1.1). Here, we define the exponents αF and αHS by +αF = αF(d, γ) := 1 + 2 + γ +d +and +αHS = αHS(d, γ) := d + 2 + 2γ +d − 2 +, +which are often referred to as the Fujita exponent (see [32, 33]) and the critical +Hardy-Sobolev exponent (see [26]). +Proposition 1.9 (Scale-supercritical case). Let d ≥ 3, γ > −2, α > 1, 1 < q ≤ ∞, +0 < r ≤ ∞ and s ∈ R be such that +γ ≤ +�√ +3 − 1 +if d = 3, +0 +if d ≥ 4, +αF < α < αHS +and +1 +qc +< s +d + 1 +q < 1. +(1.13) +Then the equation (1.1) has a global positive solution in C([0, ∞); Lq,r +s (Rd)) with +initial data 0. + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +7 +Figure 2. +The figure shows the domain of ( s +d, 1 +q) in the case γ < 0 +and min{ 1 +qc, 1 +Qc} < 1 +qc . (U.U.) and (N.U.) mean unconditional unique- +ness and non-uniqueness, respectively. The cases γ = 0 and γ > 0 +are deduced by moving the line s +d = +γ +d(α−1) to the right. +To visually understand our above results, we give Figure 2 for the case γ < 0 and +min{ 1 +qc, 1 +Qc} < 1 +qc . +Herein, we compare our results with previous ones. Our results generalize the +previous works [4, 7, 18, 27, 36, 38, 42], since s can be taken as s = 0 if γ ≤ 0 in +our results. More precisely, our results on unconditional uniqueness (Theorem 1.2 +and Theorem 1.3 (1)) include the results in [42, Theorem 4] and [7, Theorem 4] +(γ = 0 and s = 0) and [4, Theorem 1.1] and [36, Theorem 1.1] (γ < 0 and s = 0), +and our result on non-uniqueness (Theorem 1.6) generalizes the previous works [27, +Theorem 1] (γ = 0 and s = 0) and [36, Theorem 1.3] (γ < 0 and s = 0). Moreover, +our results on the double critical case (Theorem 1.3 (2) and Theorem 1.6) also reveal +the threshold r = α∗ − 1 of dividing unconditional uniqueness and non-uniqueness, +which is not mentioned in the previous ones even if γ = 0. Regarding the uniqueness +criterion, Theorem 1.7 generalizes the previous works [38, Theorem 0.10] (γ = 0 and +s = 0) and [36, Theorem 1.4] (γ < 0 and s = 0), and Proposition 1.5 has not been +mentioned in the previous works. In the scale-supercritical case, Proposition 1.9 + +1-9 ++ +1 +1 += 1 +d +q +qc +b +1 +1-α +q +1 +qc +(N.U.) ++=0 +9 +min(l, +(U.U.) +0 +1 +1 +sid +d(α-1) +元 +qc8 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +Figure 3. The figure shows the domain of (α, s) for d ≥ 3 and +q > 1. Here, α0 := min{1, 1 + γ +d}, αF, α∗, αHS are given in Figure 1, +sc, Sc are given in (1.14), and s∗ := d − 2 − d +q . Table 3 and Table 4 +summarize our results on uniqueness for (1.1). + +S +s=Sc +s= Sc +d(1- 1) +S=Sc +α +0 +αo +αF +α* +SHo +Table 3: Unconditional Uniqueness +s < min{sc, Sc] +-- s= Sc< Sc +'S>'s= += Sc = Sc +YES +YES if r ≤α +YES +YES if r< α* - 1 +Thm 1.2 (1) +OPEN if r > α +Thm 1.3 (1) +NO if r >α*- 1 +Thm 1.2 (2) +Thm 1.3 (2), Thm 1.6 +Table 4: Conditional Uniqueness +s'S +>Sc, α<αHS +YES +NO +[10, Thm 1.4, Thm 1.13] +Prop 1.9UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +9 +corresponds to [18, Theorem 1] (γ = 0 and s = 0) and [36, Proposition B.1] (γ < 0 +and s = 0). +To easily compare our results with the previous work [10] which includes the +H´enon case γ > 0, we can rewrite our results by using the two critical exponents on +s: +sc = sc(d, γ, α, q) := 2 + γ +α − 1 − d +q +and +Sc = Sc(d, γ, α, q) := d + γ +α +− d +q . +(1.14) +The exponents sc and Sc correspond to qc and Qc in the case without weights, +respectively. In fact, we can see that +sc = 0 if and only if q = qc +and +Sc = 0 if and only if q = Qc. +Hence, we can also say that the problem (1.1) is scale-critical if s = sc, scale- +subcritical if s < sc, and scale-supercritical if s > sc. Moreover, the four cases can +be rewritten as follows: Double subcritical case (s < min{sc, Sc}), single critical +case I (s = Sc < sc), single critical case II (s = sc < Sc), and double critical case +(s = sc = Sc). The results in [10] show local well-posedness, including the condi- +tional uniqueness, for (1.1) if s ≤ sc and non-existence of positive mild solution to +(1.1) for some initial data u0 ≥ 0 if s > sc. However, unconditional uniqueness and +non-uniqueness are not mentioned in [10]. Our results are summarized in Figure 3. +This paper is organized as follows. In Section 2, we summarize the definitions and +fundamental lemmas on Lorentz spaces and weighted Lorentz spaces. In Section 3, +we establish the two kinds of weighted linear estimates. In Subsection 3.1, we extend +the usual Lq1 -Lq2 estimates to the weighted Lorentz spaces, which are fundamental +tools in this paper. In Subsection 3.2, we prove a certain space-time estimate in the +weighted Lorentz spaces. We call it the weighted Meyer inequality. This inequality +corresponds to a certain endpoint case of the weighted Strichartz estimates, and it +is an important tool in studying the scale-critical case. In Section 4, we prove our +results on unconditional uniqueness and uniqueness criterion (Theorem 1.2, Theo- +rem 1.3, Proposition 1.5 and Theorem 1.7), based on the weighted linear estimates. +In Section 5, we prove our result on non-uniqueness (Theorem 1.6). In Section 6, we +discuss the non-uniqueness in the scale-supercritical case and prove Proposition 1.9. +In Section 7, we give a remark on the number of solutions in the double critical case, +and additional results on the critical singular case γ = − min{2, d} and the exterior +problem on domains not containing the origin. +2. Weighted Lorentz spaces +In this paper we use the symbols a ≲ b and b ≳ a for a, b ≥ 0 which mean that +there exists a constant C > 0 such that a ≤ Cb. The symbol a ∼ b means that +a ≲ b and b ≲ a happen simultaneously. Let Ω be a domain in Rd. We denote by +C∞ +0 (Ω) the set of all C∞-functions having compact support in Ω, and by L0(Ω) the +set of all Lebesgue measurable functions on Ω. We define the distribution function +df of a function f by +df(λ) := |{x ∈ Ω ; |f(x)| > λ}| , + +10 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +where |A| denotes the Lebesgue measure of a set A. +Definition 2.1. For 0 < q, r ≤ ∞, the Lorentz space Lq,r(Ω) is defined by +Lq,r(Ω) := +� +f ∈ L0(Ω) ; ∥f∥Lq,r(Ω) < ∞ +� +endowed with a quasi-norm +∥f∥Lq,r(Ω) := +� +� +� +� +� +� +� +�� ∞ +0 +(t +1 +q f ∗(t))r dt +t +� 1 +r +if r < ∞, +sup +t>0 t +1 +q f ∗(t) +if r = ∞, +where f ∗ is the decreasing rearrangement of f given by +f ∗(t) := inf{λ > 0 ; df(λ) ≤ t}. +We refer to [15] for the properties of the distribution function, the decreasing +rearrangement and the Lorentz space. +Remark 2.2. For 0 < q, r < ∞, the quasi-norm of Lq,r(Ω) is equivalent to +∥f∥Lq,r(Ω) ∼ q +1 +r +�� ∞ +0 +(df(λ) +1 +q λ)r dλ +λ +� 1 +r +. +For 0 < q < ∞ and r = ∞, +∥f∥Lq,∞(Ω) ∼ sup +� +λdf(λ) +1 +q ; λ > 0 +� += inf +� +C > 0 ; λdf(λ) +1 +q ≤ C +for all λ > 0 +� +. +Definition 2.3. Let 0 < q, r ≤ ∞ and s ∈ R. +(i) The weighted Lebesgue space Lq +s(Ω) is defined by +Lq +s(Ω) := +� +f ∈ L0(Ω) ; ∥f∥Lq +s < ∞ +� +endowed with a quasi-norm +∥f∥Lq +s(Ω) := +� +� +� +� +� +� +� +�� +Ω +(|x|s|f(x)|)q dx +� 1 +q +if q < ∞, +ess sup +x∈Ω +|x|s|f(x)| +if q = ∞. +The space Lq +s(Ω) is defined as the completion of C∞ +0 (Ω) with respect to ∥ · +∥Lq +s(Ω). +(ii) The weighted Lorentz space Lq,r +s (Ω) is defined by +Lq,r +s (Ω) := +� +f ∈ L0(Ω) ; ∥f∥Lq,r +s +(Ω) < ∞ +� +endowed with a quasi-norm +∥f∥Lq,r +s +(Ω) := ∥| · |sf∥Lq,r(Ω). +The space Lq,r +s (Ω) is defined as the completion of C∞ +0 (Ω) with respect to +∥ · ∥Lq,r +s +(Ω). +Only when Ω = Rd, we omit Ω and we write ∥ · ∥Lq,r +s += ∥ · ∥Lq,r +s +(Rd) for simplicity. + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +11 +Remark 2.4. There are several ways to define weighted Lorentz spaces. For exam- +ple, the definitions in [11,13,24] are different from ours. +Remark 2.5. Let us give several remarks on Lq,r +s (Ω). +(a) Lq,q +s (Ω) = Lq +s(Ω) and Lq,q +s (Ω) = Lq +s(Ω) for any 0 < q ≤ ∞ and s ∈ R. +(b) L∞,r +s +(Ω) = {0} for any r < ∞. Hence, in this paper, we always take r = ∞ +when q = ∞ in Lq,r +s (Ω) even if it is not explicitly stated. +(c) Lq,r +s (Ω) is a quasi-Banach space for any 0 < q, r ≤ ∞ and s ∈ R, and it is +normable if 1 < q < ∞ and 1 ≤ r ≤ ∞. +(d) Lq,r +s (Ω) = Lq,r +s (Ω) if q < ∞ and r < ∞, and Lq,r +s (Ω) ⊊ Lq,r +s (Ω) if q = ∞ +or r = ∞. +(e) Let 0 ∈ Ω. Then Lq,r +s (Ω) ⊂ L1 +loc(Ω) if and only if either of (e-1)–(e-3) holds: +(e-1) q > 1 and +s +d + 1 +q < 1; +(e-2) q > 1, +s +d + 1 +q = 1 and r ≤ 1; +(e-3) q = 1, +s +d + 1 +q ≤ 1 and r ≤ 1. +We have the H¨older and Young inequalities in Lorentz spaces. +Lemma 2.6 (Generalized H¨older’s inequality). Let 0 < q, q1, q2 < ∞ and 0 < +r, r1, r2 ≤ ∞. Then the following assertions hold: +(i) If +1 +q = 1 +q1 ++ 1 +q2 +and +1 +r ≤ 1 +r1 ++ 1 +r2 +, +then there exists a constant C > 0 such that +∥fg∥Lq,r ≤ C∥f∥Lq1,r1∥g∥Lq2,r2 +for any f ∈ Lq1,r1(Rd) and g ∈ Lq2,r2(Rd). +(ii) There exists a constant C > 0 such that +∥fg∥Lq,r ≤ C∥f∥Lq,r∥g∥L∞ +for any f ∈ Lq,r(Rd) and g ∈ L∞(Rd). +Lemma 2.7 (Generalized Young’s inequality). Let 1 < q, q1, q2 < ∞ and 0 < +r, r1, r2 ≤ ∞. Then the following assertions hold: +(i) If +1 +q = 1 +q1 ++ 1 +q2 +− 1 +and +1 +r ≤ 1 +r1 ++ 1 +r2 +, +then there exists a constant C > 0 such that +∥f ∗ g∥Lq,r ≤ C∥f∥Lq1,r1∥g∥Lq2,r2 +for any f ∈ Lq1,r1(Rd) and g ∈ Lq2,r2(Rd). +(ii) If +1 = 1 +q1 ++ 1 +q2 +and +1 ≤ 1 +r1 ++ 1 +r2 +, +then there exists a constant C > 0 such that +∥f ∗ g∥L∞ ≤ C∥f∥Lq1,r1∥g∥Lq2,r2 + +12 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +for any f ∈ Lq1,r1(Rd) and g ∈ Lq2,r2(Rd). +(iii) There exists a constant C > 0 such that +∥f ∗ g∥Lq,r ≤ C∥f∥Lq,r∥g∥L1 +for any f ∈ Lq,r(Rd) and g ∈ L1(Rd). +Lemmas 2.6 and 2.7 are originally proved by O’Neil [31] (see also Yap [44] for +Lorentz spaces with second exponents less than one). Lemma 2.7 (iii) is known in +the abstract setting (cf. Lemari´e-Rieusset [25, Chapter 4, Proposition 4.1]). It is +also recently proved by Wang, Wei and Ye [41, Lemma 2.2]. +3. Linear estimates +In this section, we summarize linear estimates for the heat semigroup in the +weighted Lorentz spaces. +3.1. Smoothing and time decay estimates in weighted spaces. Let {et∆}t>0 +be the heat semigroup whose element is defined by +et∆f := Gt ∗ f, +f ∈ S′(Rd) +with the Gaussian kernel +Gt(x) := (4πt)− d +2e− |x|2 +4t , +t > 0, x ∈ Rd. +Here, ∗ denotes the convolution operator and S′(Rd) denotes the space of tempered +distributions on Rd. We use the notation ∥ · ∥X→Y for the operator norm from a +quasi-normed space X to another one Y , i.e., +∥T∥X→Y := +sup +∥f∥X=1 +∥Tf∥Y +for an operator T from X into Y . In this subsection, we prove the following: +Proposition 3.1. Let d ∈ N, 1 ≤ q1 ≤ ∞, 1 < q2 ≤ ∞, 0 < r1, r2 ≤ ∞ and +s1, s2 ∈ R. Then there exists a constant C > 0 such that +∥et∆∥Lq1,r1 +s1 +→Lq2,r2 +s2 += Ct− d +2 ( 1 +q1 − 1 +q2 )− s1−s2 +2 +(3.1) +for any t > 0 if and only if q1, q2, r1, r2, s1, s2 satisfy +� +� +� +0 ≤ s2 +d + 1 +q2 +≤ s1 +d + 1 +q1 +≤ 1, +s2 ≤ s1, +(3.2) +(3.3) +and +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +r1 ≤ 1 +if s1 +d + 1 +q1 += 1 or q1 = 1, +r2 = ∞ +if s2 +d + 1 +q2 += 0, +r1 ≤ r2 +if s1 +d + 1 +q1 += s2 +d + 1 +q2 +, +ri = ∞ +if qi = ∞ +(i = 1, 2). +(3.4) +(3.5) +(3.6) +(3.7) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +13 +Remark 3.2. The estimate (3.1) can be also obtained for 0 < q2 ≤ 1. +More +precisely, let d ∈ N, 1 ≤ q1 ≤ ∞, 0 < q2 ≤ 1, 0 < r1, r2 ≤ ∞ and s1, s2 ∈ R, and +assume (3.2)–(3.7) with the additional condition +r2 ≥ 1 +if s2 +d + 1 +q2 += s1 +d + 1 +q1 += 1. +(3.8) +Then we have (3.1) for any t > 0. The additional condition (3.8) is required due +to use of the embedding L1(Rd) �→ L1,r2(Rd) for r2 ≥ 1 and Young’s inequality +∥f ∗ g∥L1 ≤ ∥f∥L1∥g∥L1 in the case +s2 +d + 1 +q2 = s1 +d + 1 +q1 = 1. On the other hand, we +can also prove the necessity of (3.2)–(3.7), but we do not know if (3.8) is necessary. +The proof is similar to that of Proposition 3.1, and we omit it. In the proofs of +the nonlinear estimates (Lemmas 4.1, 4.2, 5.2 and 5.9), we do not use the case +0 < q2 ≤ 1. +Remark 3.3. The estimates (3.1) are known in some particular cases, for example, +the case s2 = 0 in Lebesgue spaces in [4], the case s2 ≥ 0 in Lorentz spaces in [37], +and the case q1 ≤ q2 in Lebesgue spaces in [30,39] (see also [10]). Similar estimates +are proved in Herz spaces and weak Herz spaces in [30,39]. +Remark 3.4. Proposition 3.1 gives a precision of [36, Proposition 3.3] in the end- +point case (3.6) with s2 = 0. This implies that [36, Remark 3.4, (2)] does not hold. +However, this does not change the results in [36] as the endpoint case is not used in +[36]. +To reduce (3.1) for et∆ into that for e∆, we give the following lemma. +Lemma 3.5. Let d ∈ N, 1 ≤ q1, q2 ≤ ∞, 0 < r1, r2 ≤ ∞ and s1, s2 ∈ R. Then +e∆ is bounded from Lq1,r1 +s1 +(Rd) into Lq2,r2 +s2 +(Rd) if and only if et∆ is bounded from +Lq1,r1 +s1 +(Rd) into Lq2,r2 +s2 +(Rd) with +∥et∆∥Lq1,r1 +s1 +→Lq2,r2 +s2 += t− d +2 ( 1 +q1 − 1 +q2 )− s1−s2 +2 +∥e∆∥Lq1,r1 +s1 +→Lq2,r2 +s2 +(3.9) +for any t > 0. +Proof. It is enough to show (3.9) if e∆ is bounded from Lq1,r1 +s1 +(Rd) into Lq2,r2 +s2 +(Rd), +since the converse is trivial. +The proof is based on the scaling argument. +Let +f ∈ Lq1,r1 +s1 +(Rd). Since +(et∆f)(x) = +� +e∆(f(t +1 +2·)) +� +(t− 1 +2x), +(e∆f)(x) = +� +et∆(f(t− 1 +2·)) +� +(t +1 +2x), +for t > 0 and x ∈ Rd, we have +∥et∆f∥Lq2,r2 +s2 +≤ t− d +2 ( 1 +q1 − 1 +q2 )− s1−s2 +2 +∥e∆∥Lq1,r1 +s1 +→Lq2,r2 +s2 +∥f∥Lq1,r1 +s1 +, +∥e∆f∥Lq2,r2 +s2 +≤ t +d +2 ( 1 +q1 − 1 +q2 )+ s1−s2 +2 +∥et∆∥Lq1,r1 +s1 +→Lq2,r2 +s2 +∥f∥Lq1,r1 +s1 +. +Hence, (3.9) is proved. +□ + +14 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +Proof of the necessity part of Proposition 3.1. For the condition (3.7), see Remark +2.5 (b). +Step 1: Conditions +s1 +d + +1 +q1 ≤ 1 in (3.2) and (3.4). If either of these fails, then +Lq1,r1 +s1 +(Rd) is not included in L1 +loc(Rd) (see Remark 2.5 (e)), which implies that +et∆ : Lq1,r1 +s1 +(Rd) → Lq2,r2 +s2 +(Rd) is not well-defined. +Step 2: Conditions s2 +d + 1 +q2 ≥ 0 in (3.2) and (3.5). Suppose either of these fails, i.e., +s2 +d + 1 +q2 +< 0 +or +s2 +d + 1 +q2 += 0 and r2 < ∞. +When s2 +d + 1 +q2 = 0 and r2 < ∞, +lim inf +|x|→0 |g(x)| ≤ lim inf +|x|→0 |x|s2+ d +q2 | log |x|| +1 +r2 |g(x)| = 0 +holds for any g ∈ Lq2,r2 +s2 +(Rd) by Lemma A.1. However, there exists an f ∈ Lq1,r1 +s1 +(Rd) +such that +lim inf +|x|→0 |e∆f(x)| ̸= 0, +which implies e∆f ̸∈ Lq2,r2 +s2 +(Rd). Hence, it is impossible to obtain (3.1). The case +s2 +d + 1 +q2 < 1 is similarly proved. +Step 3: Condition +s1 +d + 1 +q1 ≤ s2 +d + 1 +q2 in (3.2). Suppose that s1 +d + 1 +q1 ≤ s2 +d + 1 +q2 does +not hold. Let f ∈ C∞ +0 (Rd) with f ̸= 0. Then we have +∥et∆f∥Lq2,r2 +s2 +≤ Ctδ∥f∥Lq1,r1 +s1 +, +t > 0, +where +δ := −d +2 +� 1 +q1 +− 1 +q2 +� +− s1 − s2 +2 +> 0. +Hence, et∆f → 0 in S′(Rd) as t → 0. Combining this with the continuity et∆f → f +in S′(Rd) as t → 0, we have f = 0 by uniqueness of the limit. However, this is a +contradiction to f ̸= 0. Thus, s1 +d + 1 +q1 ≤ s2 +d + 1 +q2 is necessary. +Step 4: Condition (3.3). +The proof is based on the translation argument as in +[11, 37]. In fact, take a non-negative function f ∈ C∞ +0 (Rd) with supp f ⊂ {x = +(x1, x′) ∈ R × Rd−1 ; x1 ≥ 0}, and let x0 = (1, 0, . . . , 0) ∈ Rd and τ > 0. Since +(e∆f(· − τx0))(x) = (e∆f)(x − τx0), it follows from (3.1) that +��| · |s2(e∆f)(· − τx0) +�� +Lq2,r2 ≤ C ∥| · |s1f(· − τx0)∥Lq1,r1 . +By making the changes of variables, we have +τ −(s1−s2) ��� +��� · +τ + x0 +��� +s2 e∆f +��� +Lq2,r2 ≤ C +��� +��� · +τ + x0 +��� +s1 f +��� +Lq1,r1 . +(3.10) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +15 +The weight | · +τ + x0|s2 has the uniform lower bounds with respect to sufficient large +τ : +���x +τ + x0 +��� +s2 ≥ +� +� +� +� +� +� +� +� +1 − |x| +τ +�s2 +≥ 2−s2 +for |x| ≤ 1, τ ≥ 2 +if s2 ≥ 0, +�|x| +τ + 1 +�s2 +≥ (|x| + 1)s2 +for τ ≥ 1 +if s2 ≤ 0. +Hence, once +lim sup +τ→∞ +��� +��� · +τ + x0 +��� +s1 f +��� +Lq1,r1 < ∞ +(3.11) +is obtained, we deduce from (3.10), (3.11) and positivity of e∆f that s2 ≤ s1. +Therefore, it is enough to show (3.11). In the case s1 ≥ 0, we have the uniform +upper bound +���x +τ + x0 +��� +s1 ≤ +�|x| +τ + 1 +�s1 +≤ (|x| + 1)s1 +for τ ≥ 1, +which implies (3.11). In the other case s1 < 0, the weight +���x +τ + x0 +��� +s1 = +��x1 +τ + 1 +�2 ++ |x′|2 +τ 2 +� s1 +2 +has a singularity only at x = x∗(τ) = (−τ, 0, . . . , 0), and is increasing with respect +to τ for each x ∈ {x1 ≥ 0}. Here, we note that | · +τ + x0|s1f ∈ Lq1,r1 for any τ > 0, +since the singular points x∗(τ) are not included in supp f for any τ > 0. Hence, +��� · +τ + x0 +��� +s1 f ↗ f +a.e. x ∈ {x1 ≥ 0} +as τ → ∞, +and we can use the monotone convergence theorem (Lemma A.2) to obtain +lim +τ→∞ +��� +��� · +τ + x0 +��� +s1 f +��� +Lq1,r1 = ∥f∥Lq1,r1 < ∞. +(3.12) +This implies (3.11). Thus, the necessity of s2 ≤ s1 is proved. +Final step: Condition (3.6). Let +f(x) = (1 + |x|)− d +q1 −s1(log(2 + |x|))− a +r1 +with a > 1. +Then f ∈ Lq1,r1 +s1 +(Rd). +Since f is a positive, radially symmetric and decreasing +function, we have +e∆f(x) ≥ +� +|y|≤1 +G1(y)f(x − y) dy ≥ Cf(x) +(3.13) +for |x| ≥ 1 sufficiently large. +If (3.6) fails, then f ̸∈ Lq2,r2 +s2 +(Rd), which implies +e∆f ̸∈ Lq2,r2 +s2 +(Rd) by (3.13). The proof of the necessity part is finished. +□ +Proof of the sufficiency part of Proposition 3.1. By Lemma 3.5, it is enough to prove +(3.1) with t = 1: +∥e∆f∥Lq2,r2 +s2 +≤ C∥f∥Lq1,r1 +s1 +. +(3.14) + +16 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +We start the proof with the case 1 < q1, q2 < ∞. We first prove (3.14) with the +non-endpoint case: +0 < s2 +d + 1 +q2 +< s1 +d + 1 +q1 +< 1 +and +s2 ≤ s1. +(3.15) +From Lemma 3.5 and the embedding Lq1,r1(Rd) �→ Lq1,∞(Rd) for any 0 < r1 ≤ ∞, +it is sufficient to show that e∆ is bounded from Lq1,∞ +s1 +(Rd) into Lq2,r2 +s2 +(Rd). We divide +the proof into three cases: +s2 ≥ 0, +s2 < 0 ≤ s1 +and +s1 < 0. +In the case s2 ≥ 0, we use the inequality |x|s2 ≤ C(|x − y|s2 + |y|s2) to obtain +||x|s2e∆f(x)| = |x|s2 |(G1 ∗ f)(x)| +≤ C {(G1 ∗ (| · |s2|f|))(x) + ((| · |s2G1) ∗ |f|))(x)} . +(3.16) +Then we use Lemma 2.6 (i) and Lemma 2.7 (i) to estimate +∥G1 ∗ (| · |s2|f|)∥Lq2,r2 ≤ C∥G1∥Lp1,r2∥| · |s2|f|∥Lp2,∞ +≤ C∥G1∥Lp1,r2∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥| · |s1f∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.17) +where 1 < p1 < ( s2 +d + 1 +q2)−1 and (1 − s2 +d )−1 < p2 < q2 satisfy +1 +q2 = +1 +p1 + 1 +p2 − 1 and +1 +p2 = s1−s2 +d ++ 1 +q1 , and +∥|(| · |s2G1) ∗ |f|∥Lq2,r2 ≤ C∥| · |s2G1∥Lp3,r2∥f∥Lp4,∞ +≤ C∥| · |s2G1∥Lp3,r2∥| · |−s1∥ +L +d +s1 ,∞∥| · |s1f∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.18) +where (1 − s2 +d )−1 < p3 < q2 and 1 < p4 < ( s2 +d + 1 +q2)−1 satisfy +1 +q2 = +1 +p3 + 1 +p4 − 1 and +1 +p4 = s1 +d + 1 +q1 . Here, we note that such p1, p2, p3 and p4 exist if (3.15) and s2 ≥ 0 +hold. Hence, (3.14) is proved in this case. +In the case s2 < 0 ≤ s1, we use Lemma 2.6 (i) to obtain +∥e∆f∥Lq2,r2 +s2 +≤ C∥| · |s2∥ +L +− d +s2 ,∞∥e∆f∥Lp5,r2, +(3.19) +where ( s1 +d + 1 +q1)−1 < p5 < ∞ satisfies +1 +q2 = − s2 +d + 1 +p5 . Then it has been proved above +that +∥e∆f∥Lp5,r2 ≤ C∥f∥Lq1,∞ +s1 +. +(3.20) +Thus, the case s2 < 0 ≤ s1 is also proved. +In the case s1 < 0, setting g := |x|s1|f|, and using the inequality |y|−s1 ≤ +C(|x − y|−s1 + |x|−s1), we have +��|x|s2e∆f(x) +�� ≤ |x|s2 +� +Rd G1(x − y)|y|−s1g(y) dy +≤ C +� +|x|s2−s1e∆g(x) + |x|s2 � +(| · |−s1G1) ∗ g +� +(x) +� +. +(3.21) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +17 +Then we use Lemma 2.6 (i) and Lemma 2.7 (i) to estimate +∥| · |s2−s1e∆g∥Lq2,r2 ≤ C∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥e∆g∥Lp6,r2 +≤ C∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥G1∥Lp7,r2∥g∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.22) +where q1 < p6 < − d +s1 and 1 < p7 < ( s2 +d + +1 +q2)−1 satisfy +1 +q2 = +s1−s2 +d ++ +1 +p6 and +1 +p6 = +1 +p7 + 1 +q1 − 1, and +��| · |s2 � +(| · |−s1G1) ∗ g +��� +Lq2,r2 ≤ C∥| · |s2∥ +L +− d +s2 ,∞∥(| · |−s1G1) ∗ g∥Lp8,r2 +≤ C∥| · |s2∥ +L +− d +s2 ,∞∥| · |−s1G1∥Lp9,r2∥g∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.23) +where ( s1 +d + 1 +q1)−1 < p8 < ∞ and (s1 +d + 1)−1 < p9 < (1 − 1 +q1)−1 satisfy +1 +q2 = − s2 +d + 1 +p8 +and +1 +p8 = +1 +p9 + 1 +q1 − 1. Here, we note that such p6, p7, p8 and p9 exist if (3.14) and +s1 < 0 hold. Thus, the case s1 < 0 is also proved. +Next, we consider the endpoint cases (3.4), (3.5) or (3.6) with 1 < q1, q2 < ∞. +Here, we give only sketch of proofs of single endpoint cases. If two or more endpoints +overlap, simply combine them. +As to the case (3.4), i.e., +s1 +d + 1 +q1 = 1 and r1 ≤ 1, we note that s1 ≥ 0, and the +proof is almost the same as the non-endpoint case (3.15) with s1 ≥ 0. In fact, we +can take p1 = ( s2 +d + 1 +q2)−1, p2 = (1 − s2 +d )−1, p3 = q2 and p4 = 1, and use Lemma 2.7 +(iii) (instead of Lemma 2.7 (i)) in (3.18), where ∥f∥Lp4,∞ is replaced by ∥f∥L1 and +the restriction r1 ≤ 1 appears. +As to the case (3.5), i.e., s2 +d + 1 +q2 = 0 and r2 = ∞, we note that s2 < 0, and the +proof is similar to the non-endpoint case (3.15) with s2 < 0 ≤ s1 or s2 ≤ s1 < 0. +For s2 < 0 ≤ s1, we use Lemma 2.6 (ii) to obtain +∥e∆f∥Lq2,∞ +s2 +≤ C∥| · |s2∥ +L +− d +s2 ,∞∥e∆f∥L∞ ≤ C∥e∆f∥L∞ +(this corresponds to taking p5 = ∞ in (3.19)). The estimate ∥e∆f∥L∞ ≤ C∥f∥Lq1,r1 +s1 +will be given later (see the proof of the case q2 = ∞ below). For s2 ≤ s1 < 0, we +also have +∥e∆f∥Lq2,∞ +s2 +≤ ∥| · |s2−s1e∆g∥Lq2,∞ + +��| · |s2 � +(| · |−s1G1) ∗ g +��� +Lq2,∞ +≤ C +� +∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥e∆g∥ +L +− d +s1 ,∞ + ∥| · |s2∥ +L +− d +s2 ,∞∥(| · |−s1G1) ∗ g∥L∞ +� +≤ C +� +∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥G1∥Lp7,∞∥g∥Lq1,∞ ++ ∥| · |s2∥ +L +− d +s2 ,∞∥| · |−s1G1∥Lp9,1∥g∥Lq1,∞ +� +≤ C∥f∥Lq1,∞ +s1 +, +where we take p6 = − d +s1 , p7 = [1 − ( s1 +d + 1 +q1)]−1, p8 = ∞ and p9 = (1 − 1 +q1)−1. + +18 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +As to the case (3.6), i.e., s1 +d + 1 +q1 = s2 +d + 1 +q2 and r1 ≤ r2, we can use Lemma 2.7 (iii) +to make a similar argument to the non-endpoint case. In fact, when s2 ≥ 0, this +case corresponds to taking p1 = 1, p2 = q2, p3 = (1 − s2 +d )−1 and p4 = ( s2 +d + 1 +q2)−1 in +(3.17) and (3.18). In particular, in (3.17), Lemma 2.7 (iii) is used and the restriction +r1 ≤ r2 is required: +∥G1 ∗ (| · |s2|f|)∥Lq2,r2 ≤ C∥G1∥L1∥| · |s2|f|∥Lq2,r2 +≤ C∥G1∥L1∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥| · |s1f∥Lq1,r2 +≤ C∥f∥Lq1,r2 +s1 +≤ C∥f∥Lq1,r1 +s1 +. +The case s2 < 0 is similar, and we may omit it. +In the rest of the proof, we consider the cases q1 = 1, q1 = ∞ or q2 = ∞. The +case q1 = 1 and q2 = ∞ is just L1-L∞ estimate. The case q1 = q2 = ∞ has been +already proved (see, e.g., [10, Lemma 2.1]). +The case 1 < q1 < ∞ and q2 = ∞ is the estimate (3.14) with +0 ≤ s2 ≤ s1, +s1 +d + 1 +q1 +≤ 1 +and +r1 ≤ 1 if s1 +d + 1 +q1 += 1. +Since s2 ≥ 0, this case is proved in a similar way to (3.17) and (3.18). In fact, we +deduce from Lemma 2.7 (ii) and Lemma 2.6 (i) that +∥G1 ∗ (| · |s2|f|)∥L∞ ≤ C∥G1∥Lp10,1∥| · |s2|f|∥Lp11,∞ +≤ C∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥| · |s1f∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.24) +where 1 ≤ p10 < +d +s2 and +d +d−s2 < p11 ≤ ∞ satisfy 1 = +1 +p10 + +1 +p11 and +1 +p11 = s1−s2 +d ++ 1 +q1 , +and +∥|(| · |s2G1) ∗ |f|∥L∞ ≤ C∥| · |s2G1∥Lp12,1∥f∥Lp13,∞ +≤ C∥| · |−s1∥ +L +d +s1 ,∞∥| · |s1f∥Lq1,∞ +≤ C∥f∥Lq1,∞ +s1 +, +(3.25) +where +d +d−s2 ≤ p12 < ∞ and 1 < p13 ≤ +d +s2 satisfy 1 = +1 +p12 + +1 +p13 and +1 +p12 = s1 +d + 1 +q1 . +Here, we note that such p10, p11, p12 and p13 exist if 0 ≤ s2 ≤ s1 and s1 +d + 1 +q1 < 1. +For the case s1 +d + 1 +q1 = 1, the first term can be estimated in the same way as (3.17) +(where we take p8 = +d +s2 and p9 = +d +d−s2 ). For the second term, we take p10 = ∞ and +p11 = 1 and we use Lemma 2.7 (ii) to obtain +∥|(| · |s2G1) ∗ |f|∥L∞ ≤ C∥| · |s2G1∥L∞∥f∥L1 +≤ C∥| · |−s1∥ +L +d +s1 ,∞∥| · |s1f∥Lq1,1 +≤ C∥f∥Lq1,1 +s1 . +(3.26) +Thus, the estimate (3.14) is proved in the case q2 = ∞. + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +19 +The case q1 = 1 and 1 < q2 < ∞ is the estimate (3.14) with +s2 ≤ s1 ≤ 0, +0 ≤ s2 +d + 1 +q2 +< s1 +d + 1, +r1 ≤ 1 +and +r2 = ∞ if s2 +d + 1 +q2 += 0. +The proof is similar to (3.22) and (3.23). Let +s2 +d + 1 +q2 > 0. As to the first term, it +follows from Lemma 2.6 (i) and Lemma 2.7 (ii) that +∥| · |s2−s1e∆g∥Lq2,r2 ≤ C∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥e∆g∥Lp14,r2 +≤ C∥| · |s2−s1∥ +L +d +s1−s2 ,∞∥G1∥Lp14,r2∥g∥L1 +≤ C∥f∥L1s1, +(3.27) +where 1 < p14 < − d +s1 satisfies +1 +q2 = s1−s2 +d ++ +1 +p14 . The second term can be estimated +as +��| · |s2 � +(| · |−s1G1) ∗ g +��� +Lq2,r2 ≤ C∥| · |s2∥ +L +− d +s2 ,∞∥(| · |−s1G1) ∗ g∥Lp15,r2 +≤ C∥| · |s2∥ +L +− d +s2 ,∞∥| · |−s1G1∥Lp15,r2∥g∥L1 +≤ C∥f∥L1s1, +(3.28) +where ( s1 +d + 1)−1 < p15 < ∞ satisfies +1 +q2 = − s2 +d + +1 +p15 . Here, we note that such p14 +and p15 exist if s2 ≤ s1 ≤ 0 and 0 < s2 +d + 1 +q2 < s1 +d + 1. For the case s2 +d + 1 +q2 = 0, the +first term can be estimated in the same way as (3.27) (where we take p14 = − d +s1 ). +For the second term, we we have only to take p15 = ∞ and r2 = ∞ and use Young’s +inequality ∥f ∗g∥L∞ ≤ ∥f∥L1∥g∥L∞ . Thus, the estimate (3.14) is proved in the case +q1 = 1 and 1 < q2 < ∞. The proof of Proposition 3.1 is finished. +□ +3.2. Weighted Meyer inequality. In this subsection, we shall prove the following +proposition, which is a key tool to study unconditional uniqueness and uniqueness +criterion in the scale-critical case and the construction of a singular solution in the +double critical case. +Proposition 3.6. Let T ∈ (0, ∞], and let d ≥ 3, 1 ≤ q1 ≤ ∞, 0 < q2 < ∞, +0 < r1 ≤ ∞ and s1, s2 ∈ R satisfy +� +� +� +� +� +� +� +� +� +� +� +� +� +0 < s2 +d + 1 +q2 +< s1 +d + 1 +q1 +≤ 1, +s2 ≤ s1, +d +2 +� 1 +q1 +− 1 +q2 +� ++ s1 − s2 +2 += 1, +(3.29) +(3.30) +(3.31) +and +� +� +� +r1 ≤ 1 +if s1 +d + 1 +q1 += 1 or q1 = 1, +r1 = ∞ +if q1 = ∞. +(3.32) +(3.33) +Then there exists a constant C > 0 such that +���� +� t +0 +e(t−τ)∆f(τ) dτ +���� +Lq2,∞ +s2 +≤ C sup +0<τ 1 and s1 +d + 1 +q1 < 1, since the proofs of the +other cases are similar. By the argument in [28], it suffices to prove that +∥g∥Lq2,∞ +s2 +≤ C, +(3.35) +where we define +g(x) := +� ∞ +0 +et∆f(t, x) dt +and we may assume that +sup +t≥0 +∥f(t, ·)∥Lq1,r1 +s1 +≤ 1 +without loss of generality. Let λ ∈ (0, ∞) be arbitrarily fixed. For τ ∈ (0, ∞), +which is to be determined later, we divide g into two parts: +g(x) = +� τ +0 +et∆f(t, x) dt + +� ∞ +τ +et∆f(t, x) dt =: h(x) + ℓ(x). +Let p0 and p1 be such that +0 < p1 < q2 < p0 ≤ ∞ +(3.36) +and +0 ≤ s2 +d + 1 +pi +≤ s1 +d + 1 +q1 +(i = 0, 1). +(3.37) +Then, by Proposition 3.1 and Remark 3.2, we have +∥ℓ∥Lp0,∞ +s2 +≤ +� ∞ +τ +∥et∆f(t)∥Lp0,∞ +s2 +dt +≤ C +� ∞ +τ +t− d +2 ( 1 +q1 − 1 +p0 )− s1−s2 +2 +∥f(t)∥Lq1,∞ +s1 +dt +≤ Cτ − d +2 ( 1 +q2 − 1 +p0 ) +and +∥h∥Lp1,∞ +s2 +≤ +� τ +0 +��et∆f(t) +�� +Lp1,∞ +s2 +dt +≤ +� τ +0 +t− d +2 ( 1 +q1 − 1 +p1 )− s1−s2 +2 +∥f(t)∥Lq1,∞ +s1 +dt +≤ Cτ +d +2 ( 1 +p1 − 1 +q2 ). +Now, the definition of the Lorentz norms yields +d|·|s2ℓ +�λ +2 +� +≤ +� +∥ℓ∥Lp0,∞ +s2 +λ/2 +�p0 +≤ +� +Cτ − d +2 ( 1 +q2 − 1 +p0 ) +λ +�p0 + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +21 +and similarly, +d|·|s2h +�λ +2 +� +≤ +� +∥h∥Lp1,∞ +s2 +λ/2 +�p1 +≤ +� +Cτ +d +2 ( 1 +p1 − 1 +q2 ) +λ +�p1 +. +Thus, choosing τ such that τ = λ− 2q2 +d , we deduce +d|·|s2g(λ) ≤ d|·|s2h +�λ +2 +� ++ d|·|s2ℓ +�λ +2 +� +≤ C +λq2 , +which implies (3.35). Thus, we conclude Proposition 3.6. +□ +4. Unconditional uniqueness and uniqueness criterion +In this section, we prove Theorem 1.2, Theorem 1.3, Proposition 1.5 and Theo- +rem 1.7. +4.1. Nonlinear estimates. We define the Duhamel term N(u) by +N(u)(t) := +� t +0 +e(t−τ)∆(| · |γ|u(τ)|α−1u(τ)) dτ. +Then we have the following nonlinear estimates, which are used to prove uncondi- +tional uniqueness in the double subcritical case and in the single critical case I. +Lemma 4.1. Let d, γ, α, q, r, s be as in (1.10). Let T ∈ (0, ∞] and δ be given by +δ := d(α − 1) +2 +� 1 +qc +− +�s +d + 1 +q +�� +. +(4.1) +Then the following assertions hold: +(i) If 0 < s +d + 1 +q < min{ 1 +qc, 1 +Qc} and q > α, then there exists a constant C > 0 +such that +∥N(u1)(t) − N(u2)(t)∥Lq,r +s +≤ Ctδ max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,∞ +s +)∥u1 − u2∥L∞(0,t;Lq,∞ +s +) +(4.2) +for any t ∈ (0, T) and u1, u2 ∈ L∞(0, T; Lq,∞ +s +(Rd)). +(ii) If either “0 < s +d + 1 +q < min{ 1 +qc, 1 +Qc} and q = α” or “ s +d + 1 +q = +1 +Qc < 1 +qc ”, then +there exists a constant C > 0 such that +∥N(u1)(t) − N(u2)(t)∥Lq,α +s +≤ Ctδ max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,α +s +)∥u1 − u2∥L∞(0,t;Lq,α +s +) +(4.3) +for any t ∈ (0, T) and u1, u2 ∈ L∞(0, T; Lq,α +s (Rd)), provided that q ̸= ∞. +Proof. We define σ := αs − γ. First, we prove the assertion (i). Let T ∈ (0, ∞] and +u1, u2 ∈ L∞(0, T; Lq,∞ +s +(Rd)). We assume (1.10) and 0 < s +d + 1 +q < min{ 1 +qc, 1 +Qc}. Then +the parameters q, s, σ satisfy +1 ≤ q +α, q ≤ ∞, +0 < s +d + 1 +q < σ +d + α +q < 1, +s ≤ σ +and +d +�α +q − 1 +q +� ++ σ − s < 2. + +22 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +Hence, we use Proposition 3.1 with (q1, r1, s1) = ( q +α, ∞, σ) and (q2, r2, s2) = (q, ∞, s), +and then, Lemma 2.6 with (q, r) = ( q +α, ∞), (q1, r1) = ( +q +α−1, ∞) and (q2, r2) = (q, ∞) +to obtain +∥N(u1)(t) − N(u2)(t)∥Lq,∞ +s +≤ C +� t +0 +(t − τ)− d +2 ( α +q − 1 +q )− σ−s +2 ∥| · |γ(|u1(τ)|α−1u1(τ) − |u2(τ)|α−1u2(τ))∥ +L +q +α ,∞ +σ +dτ +≤ C +� t +0 +(t − τ)− d +2 ( α +q − 1 +q )− σ−s +2 ∥| · |γ(|u1(τ)|α−1 + |u2(τ)|α−1)|u1(τ) − u2(τ)∥ +L +q +α ,∞ +σ +dτ +≤ C +� t +0 +(t − τ)− d +2 ( α +q − 1 +q )− σ−s +2 dτ × max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,∞ +s +)∥u1 − u2∥L∞(0,t;Lq,∞ +s +) +≤ Ctδ max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,∞ +s +)∥u1 − u2∥L∞(0,t;Lq,∞ +s +). +(4.4) +Therefore, the assertion (i) is proved. +The assertion (ii) is also proved in the same way. In fact, when s +d + 1 +q = +1 +Qc < 1 +qc , +we use Proposition 3.1 with the endpoint case (3.4) to obtain +∥N(u1)(t) − N(u2)(t)∥Lq,α +s +≤ C +� t +0 +(t − τ)− d +2 ( α +q − 1 +q )− σ−s +2 ∥| · |γ(|u1(τ)|α−1 + |u2(τ)|α−1)|u1(τ) − u2(τ)∥ +L +q +α ,1 +σ +dτ +≤ Ctδ max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,α +s +)∥u1 − u2∥L∞(0,t;Lq,α +s +). +Note that this case corresponds to taking the endpoint +σ +d + α +q = 1 in the above, +which causes the restriction r ≤ α. Here, the exponent q = ∞ is excluded (see +Remark 2.5 (b)). The proof in the case 0 < +s +d + 1 +q < min{ 1 +qc, 1 +Qc} and q = α is +similar using also (3.4). Thus, the proof of Lemma 4.1 is finished. +□ +In addition, we prepare the nonlinear estimates of the following type. +These +estimates are used to prove uniqueness criterion in the single critical case I, and +unconditional uniqueness and uniqueness criterion in the scale-critical case. +Lemma 4.2. Let d, γ, α, q, r, s be as in (1.10). Assume that �q ∈ (q, ∞) satisfies +s +d + 1 +q − +2 +d(α − 1) < s +d + 1 +�q < min +� 1 +qc +, 1 +Qc +− +1 +α − 1 +� 1 +Qc +− +�s +d + 1 +q +��� +. +(4.5) +Let T ∈ (0, ∞] and β be defined by +β = β(d, q, �q) := d +2 +�1 +q − 1 +�q +� +. +(4.6) +Then the following assertions hold: + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +23 +(i) If +s +d + 1 +q = +1 +Qc < 1 +qc and r > α, then there exists a constant C > 0 such that +∥N(u1)(t) − N(u2)(t)∥Lq,r +s +≤ Ctδ� +max +i=1,2 ∥ui − eτ∆u0∥L∞(0,t;Lq,r′(α−1) +s +) ++ sup +0<τ 0 such that +∥N(u1)(t) − N(u2)(t)∥Lq,∞ +s +≤ C +� +max +i=1,2 ∥ui − eτ∆u0∥L∞(0,t;Lq,∞ +s +) ++ sup +0<τ 0 such that +∥N(u1)(t) − N(u2)(t)∥Lq,∞ +s +≤ C +� +max +i=1,2 ∥ui − eτ∆u0∥L∞(0,t;Lq,α∗−1 +s +) ++ sup +0<τ 0 is given in (4.1). Similarly, we have +∥II(t)∥Lq,r +s +≤ Ctδ ∥u2(τ) − eτ∆u0∥α−1 +L∞(0,t;Lq,r′(α−1) +s +)∥u1 − u2∥L∞(0,t;Lq,r +s +) +(4.12) +for any t ∈ (0, T). To estimate III(t), we take auxiliary parameters p, �q and σ +satisfying +1 < p < ∞, +q < �q < ∞, +0 < s +d + 1 +q < σ +d + 1 +p < 1, +s ≤ σ +(4.13) +1 +p = α − 1 +�q ++ 1 +q, +(4.14) +− d +2 +�1 +p − 1 +q +� +− σ − s +2 +> −1, +−(α − 1)β > −1. +(4.15) +Here, the above p, �q and σ exist if (1.10) and (4.5) hold. We use Proposition 3.1 +with (q1, r1, s1) = (p, ∞, σ) and (q2, r2, s2) = (q, r, s) to obtain +∥III(t)∥Lq,r +s +≤ C +� t +0 +(t − τ)− d +2 ( 1 +p − 1 +q )− σ−s +2 ∥| · |γ|eτ∆u0|α−1|u1(τ) − u2(τ)|∥Lp,∞ +σ +dτ, +where (4.13) is required. Moreover, it follows from Lemma 2.6 with (q, r) = (p, ∞), +(q1, r1) = ( +�q +α−1, ∞) and (q2, r2) = (q, ∞) that +∥| · |γ|eτ∆u0|α−1|u1(τ) − u2(τ)|∥Lp,∞ +σ +≤ C +��eτ∆u0 +��α−1 +L�q,∞ +s +∥u1(τ) − u2(τ)∥Lq,∞ +s +, +where (4.14) is required. Combining the above two estimates, and using the equality +−d +2 +�1 +p − 1 +q +� +− σ − s +2 +− (α − 1)β + 1 = δ, +we have +∥III(t)∥Lq,∞ +s +≤ Ct− d +2 ( 1 +p − 1 +q )− σ−s +2 −(α−1)β+1 +�� 1 +0 +(1 − τ)− d +2 ( 1 +p − 1 +q )− σ−s +2 τ −(α−1)βdτ +� +× +� +sup +0<τ 0 and u1, u2 ∈ L∞(0, T; Lq,α +s (Rd)) be mild solutions to +(1.1) with initial data u1(0) = u2(0). By Lemma 4.1 (ii), we have +∥u1(t) − u2(t)∥Lq,α +s +≤ C0tδ max +i=1,2 ∥ui∥α−1 +L∞(0,t;Lq,α +s +)∥u1 − u2∥L∞(0,t;Lq,α +s +) +for any t ∈ (0, T), where δ > 0 is given in (4.1). If we choose t0 ∈ (0, T] such that +C0tδ +0 max +i=1,2 ∥ui∥α−1 +L∞(0,T;Lq,α +s +) < 1, +then we can derive that u1 = u2 on [0, t0]. We can repeat this argument until +we reach t = T , and hence, we arrive at u1 = u2 on [0, T]. Thus, we conclude +Theorem 1.2. +□ +The proof of Proposition 1.5 is similar to that of Theorem 1.2, and we have only +to use Lemma 4.2 (i) instead of Lemma 4.1 (ii). +Proof of Theorem 1.3. We give the proof only for the case (2), since the proof of the +case (1) is similar. Let T > 0 and u1, u2 ∈ C([0, T]; Lq,α∗−1 +s +(Rd)) be mild solutions +to (1.1) with initial data u1(0) = u2(0) = u0 ∈ Lq,α∗−1 +s +(Rd). By Lemma 4.2 (iii), we +have +∥u1(t) − u2(t)∥Lq,∞ +s +≤ C +� +max +i=1,2 ∥ui − eτ∆u0∥L∞(0,t;Lq,α∗−1 +s +) ++ sup +0<τ q, where β is given in (4.6). The goal of this subsection is to prove the +following: +Proposition 5.1. Let d ≥ 3, γ > −2, α = α∗, α∗ ≤ q < ∞, 0 < r ≤ ∞, and +s +d + 1 +q = 1 +qc = +1 +Qc . Assume that �q satisfies +max +� +0, 1 +q − +1 +q∗(0), 1 +q − +2 +dα∗ +� +< 1 +�q < 1 +q. +(5.2) +Then, for any u0 ∈ Lq,r +s (Rd), there exist a time T = T(u0) > 0 and a unique mild +solution u ∈ C([0, T]; Lq,r +s (Rd)) to (1.1) with u(0) = u0 satisfying (5.1) (replace +Lq,r +s (Rd) by Lq,∞ +s +(Rd) if r = ∞). +Once the following nonlinear estimates are established, the proof can be done by +the standard fixed point argument (see, e.g., [10, Subsection 3.1]). Hence, we only +give a proof of the following. +Lemma 5.2. Let d ≥ 3, γ > −2, α = α∗, α∗ ≤ q < ∞, 0 < r ≤ ∞, and +s +d + 1 +q = 1 +qc = +1 +Qc . +(i) Assume that �q satisfies (5.2). Then there exists a constant C > 0 such that +∥N(u1) − N(u2)∥K�q(T) ≤ C max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T) +(5.3) +for any t ∈ (0, T) and any functions u1, u2 satisfying +∥ui∥K�q(T) < ∞, +i = 1, 2. +(5.4) +(ii) Assume that +max +� +0, 1 +q − 2 +α∗ +� +< 1 +�q < 1 +q. +(5.5) +Then there exists a constant C > 0 such that +∥N(u1) − N(u2)∥L∞(0,T;Lq,r +s +) ≤ C max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T) +(5.6) + +28 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +for any t ∈ (0, T) and any functions u1, u2 satisfying (5.4). +Remark 5.3. Note that (5.2) implies (5.5). +Proof. We first prove the assertion (i). We set σ := αs − γ and take +1 < �q, �q +α < ∞, +0 < s +d + 1 +�q < σ +d + α +�q < 1, +s ≤ σ, +(5.7) +− d +2 +�α +�q − 1 +�q +� +− σ − s +2 +> −1, +−βα > −1. +(5.8) +Here, there exists a �q as above if (5.2) holds. In a similar way to (4.4), we estimate +∥N(u1)(t) − N(u2)(t)∥L�q,∞ +s +≤ C +�� t +0 +(t − τ)− d +2( α +�q − 1 +�q)− σ−s +2 τ −βα dτ +� +max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T) +≤ Ct−β max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T), +where (5.7) and (5.8) are required in the first and second steps, respectively. +Similarly, we can prove the assertion (ii). In fact, taking +1 < q, �q +α < ∞, +0 < s +d + 1 +q < σ +d + α +�q < 1, +s ≤ σ, +(5.9) +− d +2 +�α +�q − 1 +q +� +− σ − s +2 +> −1, +−βα > −1, +(5.10) +we estimate +∥N(u1)(t) − N(u2)(t)∥Lq,r +s +≤ C +�� t +0 +(t − τ)− d +2( α +�q − 1 +q)− σ−s +2 τ −βα dτ +� +max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T) +≤ C max +i=1,2 ∥ui∥α−1 +K�q(T)∥u1 − u2∥K�q(T), +where (5.9) and (5.10) are required in the first and second steps, respectively. Here, +there exists a �q as above if (5.5) holds. Thus, the proof is complete. +□ +5.2. Existence of singular solution. The mild solution u obtained in Subsec- +tion 5.1 is a bounded solution (see [4, Remark 1.1 and Proposition 3.2] and also +[40, the remark after Definition 2.1]). In this subsection, we find a singular mild so- +lution v to (1.1) for any initial data u0 ∈ Lq,r +s (Rd). Here, the singular mild solution +means that v ̸∈ L�q,∞ +s +(Rd) for any �q satisfying (5.2) (in particular, this solution has +a singularity at x = 0). The goal of this subsection is to prove the following: +Theorem 5.4. Let d ≥ 3, γ > −2, α = α∗, α∗ ≤ q < ∞, α∗ − 1 < r ≤ ∞, and +s +d + 1 +q = +1 +qc = +1 +Qc . Then, for any u0 ∈ Lq,r +s (Rd), there exist T = T(u0) > 0 and a +mild solution v ∈ C([0, T]; Lq,r +s (Rd)) to (1.1) with v(0) = u0 such that v ̸∈ L�q,∞ +s +(Rd) +for any �q satisfying (5.2) and +v(t) − et∆u0 ̸∈ Lq,α∗−1 +s +(Rd) +and +v(t) − et∆u0 ∈ Lq,r +s (Rd) for any r > α∗ − 1 +(5.11) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +29 +for any t ∈ (0, T] (replace Lq,r +s (Rd) by Lq,∞ +s +(Rd) if r = ∞). +The proof is based on the argument in [27,38]. In order to construct the singular +solution v, we use a positive, radially symmetric and singular stationary solution of +∆U + |x|γU +d+γ +d−2 = 0 +in B \ {0}, +U > 0, +(5.12) +where d ≥ 3, γ > −2 and B := {x ∈ Rd ; |x| < 1}. We have the results on the +existence of the singular stationary solution and the sharp bound of its behavior at +x = 0. +Theorem 5.5. Let d ≥ 3 and γ > −2. The the following assertions hold: +(i) The equation (5.12) has a positive, radial, and singular solution at x = 0, +where the singular solution means that it is unbounded near x = 0. +(ii) Let U ∈ C2(B \ {0}) be a positive radial solution to (5.12). Then, U has +either a removable singularity at |x| = 0 or a singularity at |x| = 0 as +lim +x→0 |x|d−2| log |x|| +d−2 +γ+2U(x) = +�(d − 2)2 +2 + γ +� d−2 +2+γ +. +(5.13) +Remark 5.6. The constant (5.13) appears in [6, Theorem 2.1] for −2 < γ < 2, +giving a precise value to that in [2, Theorem A] and hence to that in [36, Remark 6.2]. +The proofs of (i) and (ii) can be found in [1, Example 1] and [12, Theorem 1.1 (ii)], +respectively. For completeness, we give the proof of (ii) in Appendix B. Therefore, +we denote by U0 the singular stationary solution with +U0(x) ∼ |x|−(d−2)| log |x||− d−2 +γ+2 = |x|−(d−2)| log |x||− +1 +α∗−1 +near x = 0. We extend U0 to a function V0 on Rd as follows. +Proposition 5.7. Let d, γ, α, q, s be as in Theorem 5.4. Then there exists a function +V0 ≥ 0 on Rd \ {0} with compact support such that +V0(x) ∼ |x|−(d−2)| log |x||− +1 +α∗−1 +(5.14) +in a neighborhood of x = 0, and +R := ∆V0 + |x|γV α∗ +0 +is of C1 with compact support. +(5.15) +Moreover, +� +� +� +� +� +� +� +V0 ∈ Lq,r +s (Rd) +for any r > α∗ − 1, +V0 ̸∈ Lq,α∗−1 +s +(Rd), +V0 ̸∈ L�q,∞ +s +(Rd) +for any �q > q. +(5.16) +(5.17) +(5.18) +The proof of Proposition 5.7 is the same as in [38, Theorem 0.7] (see also [36, +Proposition 6.1]). +To prove Theorem 5.4, we find a singular mild solution v to (1.1) of the form +v(t) = w(t) + V0. + +30 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +Here, w = w(t) is a (regular) solution to the perturbed problem +� +� +� +w(t) = et∆w0 + N(w)(t) + +� t +0 +e(t−τ)∆R dτ, +w(0) = w0 := u0 − V0, +(5.19) +where +N(w)(t) := +� t +0 +e(t−τ)∆ � +|x|γ|w(τ) + V0|α∗−1(w(τ) + V0) − |x|γV α∗ +0 +� +dτ. +More precisely, we have the following: +Lemma 5.8. Let d, γ, α, q, r, s be as in Theorem 5.4. Then, for w0 ∈ Lq,r +s (Rd), there +exist T > 0 and a unique solution w ∈ C([0, T]; Lq,r +s (Rd)) to (5.19) with w(0) = w0 +such that it satisfies (5.1) for any �q satisfying (5.2). +The proof of this lemma is based on the fixed point argument as in [27]. Hence, +it is sufficient to show some estimates for the term N(w). To prove the nonlinear +estimates, we use the following decomposition of V0. By density and (5.16), for any +ε > 0, there exist functions h ∈ C∞ +0 (Rd) and V 0 ∈ Lq,∞ +s +(Rd) such that +V0 = h + V 0, +∥V 0∥Lq,∞ +s +< ε. +(5.20) +Then we have the following estimates for N(w). +Lemma 5.9. Let d, γ, α, q, r, s be as in Theorem 5.4, γ+ := max{0, γ} and γ− := +− min{0, γ}. Assume �q satisfies (5.2). Then there exists a constant C > 0 such +that +∥N(u1) − N(u2)∥K�q(t) +≤ C +� +max +i=1,2 ∥wi∥α∗−1 +K�q(t) + ∥V 0∥α∗−1 +Lq,∞ +s ++ t1− +γ− +2 ∥| · |γ+|h|α∗−1∥L∞ +� +∥w1 − w2∥K�q(t) +(5.21) +and +∥N(u1) − N(u2)∥L∞(0,t;Lq,r +s +) +≤ C +� +max +i=1,2 ∥wi∥α∗−1 +K�q(t) + ∥V 0∥α∗−1 +Lq,∞ +s ++ ∥h∥α∗−1 +L�q,∞ +s +� +∥w1 − w2∥K�q(t) +(5.22) +for any two functions w1, w2 satisfying (5.1) and for any t > 0. +Proof. We write +N(w1)(t) − N(w2)(t) = +� t +0 +e(t−τ)∆� +| · |γ |w1(τ) + V0|α∗−1 (w1(τ) + V0) +− | · |γ |w2(τ) + V0|α∗−1 (w2(τ) + V0) +� +dτ. +By the decomposition (5.20) together with the inequality +��|x + y|α∗−1(x + y) − |x′ + y|α∗−1(x′ + y) +�� ≤ C|x − x′| +� +|x|α∗−1 + |x′|α∗−1 + |y|α∗−1� + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +31 +for x, x′, y ∈ R, we have +|N(w1)(t) − N(w2)(t)| ≤ C +� t +0 +e(t−τ)∆[| · |γ|w1(τ)|α∗−1|w1(τ) − w2(τ)|] dτ ++ C +� t +0 +e(t−τ)∆[| · |γ|w2(τ)|α∗−1|w1(τ) − w2(τ)|] dτ ++ C +� t +0 +e(t−τ)∆[| · |γ|h|α∗−1|w1(τ) − w2(τ)|] dτ ++ C +� t +0 +e(t−τ)∆[| · |γ|V 0|α∗−1|w1(τ) − w2(τ)|] dτ +=: I(t) + II(t) + III(t) + IV (t). +(5.23) +First, we prove the estimate (5.21). In the same way as in the proof of Lemma 4.2, +the norms of the terms I(t) and II(t) can be estimated as +∥I∥K�q(t) + ∥II∥K�q(t) ≤ C max +i=1,2 ∥wi∥α∗−1 +K�q(t)∥w1 − w2∥K�q(t). +(5.24) +As to the term III(t), we use Proposition 3.1 with (q1, r1, s1) = (�q, ∞, s + γ−) +and (q2, r2, s2) = (�q, ∞, s) to obtain +∥III(t)∥L�q,∞ +s +≤ C +� t +0 +(t − τ)− +γ− +2 ∥| · |γ|h|α∗−1|w1(τ) − w2(τ)|∥L�q +s+γ− dτ += C +� t +0 +(t − τ)− +γ− +2 ∥| · |γ+|h|α∗−1|w1(τ) − w2(τ)|∥L�q,∞ +s +dτ +≤ C∥| · |γ+|h|α∗−1∥L∞ +� t +0 +(t − τ)− +γ− +2 ∥w1(τ) − w2(τ)∥L�q,∞ +s +dτ +≤ C∥| · |γ+|h|α∗−1∥L∞ +�� t +0 +(t − τ)− +γ− +2 τ −β dτ +� +∥w1 − w2∥K�q(t) +≤ Ct1− +γ− +2 −β∥| · |γ+|h|α∗−1∥L∞∥w1 − w2∥K�q(t), +(5.25) +where we required that +0 < s +d + 1 +�q ≤ s + γ− +d ++ 1 +�q < 1 +and +s ≤ s + γ−. +Here, thanks to (5.2) and γ− ∈ [0, 2), the above conditions are satisfied. +As to the term IV (t), thanks to (5.2), we can take σ := α∗s − γ and +0 < s +d + 1 +�q ≤ σ +d + 1 +p < 1, +s ≤ σ, +1 +p = α∗ − 1 +q ++ 1 +�q. + +32 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +Then we use Proposition 3.6 with (q1, r1, s1) = (p, ∞, σ) and (q2, r2, s2) = (�q, ∞, s) +to obtain +∥IV (t)∥L�q,∞ +s +≤ C sup +0<τ 0 and any t ∈ (0, T], and hence, w(t) − +et∆w0 also belongs to Lq,�r +s (Rd). As to the second term et∆V0, we estimate +∥et∆V0∥Lq,�r +s +≤ +� +Ct− d +2 (1− 1 +q )+ s +2∥V0∥L1 = Ct−1∥V0∥L1 +if s ≤ 0, +Ct− d +2 (1− 1 +q )∥V0∥L1s = CV0t− d +2 (1− 1 +q )∥V0∥L1 +if s > 0, +where we used Propositions 3.1 and V0 ∈ L1(Rd) with compact support in Propo- +sition 5.7. Hence, et∆V0 ∈ Lq,�r +s (Rd) is also shown for any �r > 0 and t ∈ (0, T]. In +contrast, the third term V0 satisfies V0 ̸∈ Lq,α∗−1 +s +(Rd) and V0 ∈ Lq,r +s (Rd) for any +r > α∗ − 1 by Proposition 5.7. Therefore, (5.11) is proved for any t ∈ (0, T]. Thus, +Theorem 5.4 is proved. +Proof of Theorem 1.6. The proof is a combination of Proposition 5.1 and Theo- +rem 5.4. In fact, by these results, there exist a regular mild solution u and singular +mild solution v to (1.1) with the same initial data u0. When r = ∞, the above +arguments are also valid if Lq,r +s (Rd) is replaced by Lq,∞ +s +(Rd). +□ +6. Scale-supercritical case +In this section we discuss the scale-supercritical case. +We use the self-similar +solution of (1.1) to show the existence of a non-trivial mild solution of (1.1) with +initial data 0. More precisely, we have the following: + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +33 +Proposition 6.1. Let d ≥ 3, γ > −2, α > αF , 1 < q ≤ ∞, 0 < r ≤ ∞ and +s ∈ R be such that +1 +qc +< s +d + 1 +q < 1. +Assume that there exists a solution W of +∆W + 1 +2x · ∇W + +2 + γ +2(α − 1)W + |x|γ|W|α−1W = 0, +x ∈ Rd \ {0} +(6.1) +such that +(i) W > 0 and W ∈ C(Rd) ∩ C2(Rd \ {0}), +(ii) lim +|x|→0 |x||∇W| = 0, +(iii) +lim +|x|→∞ |x|mW(x) = 0 and +lim +|x|→∞ |x|m|∇W(x)| = 0 for all m > 0. +Let Ψ(t, x) = t− +2+γ +2(α−1)W(x/ +√ +t) be the positive self-similar solution of (1.1). Then +Ψ ∈ C([0, ∞); Lq,r +s (Rd)) satisfies the equation +Ψ(t) = +� t +0 +e(t−τ)∆ (| · |γ|Ψ(τ)|αΨ(τ)) dτ +(6.2) +for any t ∈ (0, ∞). In particular, Ψ is a non-trivial mild solution to (1.1) with +initial data 0 in C([0, ∞); Lq,r +s (Rd)). +Proof. By the assumptions (i)–(iii) on W, it follows that +1 +2x · ∇W + +2 + γ +2(α − 1)W + |x|γ|W|α−1W ∈ L1(Rd). +Then W satisfies the equation (6.1) in D′(Rd) and Ψ(t, x) = t− +2+γ +2(α−1)W(x/ +√ +t) satis- +fies the equation (1.1) in D′((0, ∞)×Rd). Here, D′(X) is the space of distributions +on an open set X . Hence, +Ψ(t) = e(t−ε)∆Ψ(ε) + +� t +ε +e(t−τ)∆ � +| · |γ|Ψ(τ)|α−1Ψ(τ) +� +dτ +for 0 < ε < t in the sense of distributions. It is clear that +∥Ψ(t)∥Lq,r +s += t +d +2 (( s +d + 1 +q )− 1 +qc )∥W∥Lq,r +s +< ∞, +t > 0, +where 0 < s +d + 1 +q < 1. Then +lim +t→0 ∥Ψ(t)∥Lq,r +s += 0 +for 1 +qc +< s +d + 1 +q < 1. +(6.3) +Finally, we prove that the integral +� t +0 +e(t−τ)∆(| · |γ|Ψ(τ)|αΨ(τ)) dτ +(6.4) +converges absolutely in Lq,r +s (Rd). By Proposition 3.1, we have +∥e(t−τ)∆ � +| · |γ|Ψ(τ)|α−1Ψ(τ) +� +∥Lq,r +s +≤ C(t − τ)− d +2 ( α +�q − 1 +q )− α�s−γ−s +2 +∥Ψ(τ)∥α +L�q,r +�s += C(t − τ)− d +2 ( α +�q − 1 +q )− α�s−γ−s +2 +τ α( d +2�q + �s +2 − +2+γ +2(α−1) )∥W∥α +L�q,r +�s , + +34 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +where we require that +α < �q < ∞, +0 < s +d + 1 +q < α�s − γ +d ++ α +�q < 1, +s ≤ α�s − γ. +(6.5) +If α, q, s, �q and �s satisfy +d +2 +�α +�q − 1 +q +� ++ α�s − γ − s +2 +< 1, +α +� 2 + γ +2(α − 1) − d +2 +�1 +�q + �s +d +�� +< 1, +(6.6) +then (6.4) converges absolutely in Lq,r +s (Rd). To check these conditions, let us choose +�q and �s such that +s + γ +α +≤ �s, +0 < α +�q < 1, +α +�q + α�s +d < γ + d +d +, +(6.7) +1 +q + γ + s +d +< α +�q + α�s +d < 2 +d + 1 +q + γ + s +d +. +(6.8) +It is obvious that under the assumptions in Proposition 6.1, it is possible to take �q, +�s satisfying (6.7) and (6.8). We now show that (6.5) and (6.6) hold if (6.3), (6.7) +and (6.8) are satisfied. Indeed, (6.5) is already in (6.7) and the first inequality in +(6.8). For (6.6), we have +d +2 +�α +�q − 1 +q +� ++ α�s − γ − s +2 += dα +2�q − d +2q + α�s − γ − s +2 += dα +2�q + α�s +2 − d +2q − γ + s +2 +< 1 + d +2q + γ + s +2 +− d +2q − γ + s +2 += 1, +α +� 2 + γ +2(α − 1) − d +2 +�1 +�q + �s +d +�� += α(2 + γ) +2(α − 1) − d +2 +�α +�q + α�s +d +� +< α(2 + γ) +2(α − 1) − d +2 +�1 +q + γ + s +d +� += 1 + γ +2 + +2 + γ +2(α − 1) − d +2q − γ + s +2 +, +< 1. +Thus, we conclude Proposition 6.1. +□ +The existence of positive self-similar solutions Ψ of (1.1) with (i)–(iii) in Propo- +sition 6.1 is proved for any α satisfying +αF < α < αHS +(6.9) +with γ = 0 by [18, Propositions 3.1, 3.4 and 3.5] and with γ satisfying +− 2 < γ ≤ +�√ +3 − 1 +if d = 3, +0 +if d ≥ 4 +(6.10) + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +35 +by Hirose [20, Theorem 1.2 (ii)]. Furthermore, +W(x) = C|x| +2+γ +α−1 −de− |x|2 +4 � +1 + O +� +|x|−2�� +as |x| → ∞. +From Proposition 6.1 and this result, it immediately follows that the equation (1.1) +has three different solutions 0 and ±Ψ with initial data 0 in C([0, ∞); Lq,r +s (Rd)) +under the assumptions (6.9) and (6.10) for d, γ, α, q, r, s as in Proposition 6.1. Thus, +Proposition 1.9 is proved. +Remark 6.2. When γ does not satisfy (6.10), the existence of self-similar solutions +with (i)–(iii) in Proposition 6.1 under the condition (6.9) is an open problem. +The situation of the case α > αHS is different from the case (6.9). In this case, the +nonexistence of positive self-similar solution Ψ satisfying (i)–(iii) in Proposition 6.1 +is proved by the following result on uniqueness in the Sobolev space H1(Rd): +Lemma 6.3. Let T > 0 and u = u(t, x) be a mild solution to (1.1) satisfying +u ∈ C1((0, T); L2(Rd)) ∩ C1((0, T); Lα+1 +γ +α+1(Rd)) ∩ C((0, T); H2(Rd)). +Assume that u(t) → 0 in H1(Rd) as t → 0. Then u ≡ 0 on [0, T]. +The proof of Lemma 6.3 is almost the same as that of [18, Theorem 2], and so +we omit the proof. If α > αHS and there exists a positive self-similar solution Ψ +satisfying (i)–(iii) in Proposition 6.1, then Ψ satisfies all assumptions in Lemma 6.3, +and hence, Ψ ≡ 0. This contradicts Ψ > 0. Thus, we see the nonexistence of such +a Ψ. +7. Additional results and remarks +7.1. Double critical case. We give a remark on the number of solutions in the +double critical case. Theorem 1.6 shows that the problem (1.1) has two different +solutions, where one is regular and the other is singular (see Section 5). In fact, +however, (1.1) has an uncountable infinite number of different mild solutions in +C([0, T]; Lq,r +s (Rd)) for any initial data u0 ∈ Lq,r +s (Rd). This can be confirmed by +constructing the family {ut0}t0∈(0,T) of solutions to (1.1) such that ut0 is a singular +solution for 0 < t ≤ t0 and a regular solution for t0 < t < T . +7.2. Case γ = − min{2, d}. The problem on well-posedness for (1.1) in the critical +singular case γ = − min{2, d} has not been studied. +Establishing the weighted +linear estimates (3.1) with the double endpoint s1 +d + 1 +q1 = 1 and s2 +d + 1 +q2 = 0, we can +present the following results on uniqueness for the case d = 1 and γ = −1. +Theorem 7.1. Let T > 0, and let d = 1, γ = −1, α > 1, α ≤ q < ∞, and +s +d + 1 +q = 0. Then the following assertions hold: +(i) Let 0 < r ≤ α − 1. +Then unconditional uniqueness holds for (1.1) in +L∞(0, T; Lq,r +s (R)). + +36 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +(ii) Let r > α − 1 and u0 ∈ Lq,r +s (R). Then, if u1, u2 ∈ L∞(0, T; Lq,r +s (R)) are +mild solutions to (1.1) with u1(0) = u2(0) = u0 such that +ui(t) − et∆u0 ∈ L∞(0, T; Lq,α−1 +s +(R)) +for i = 1, 2, +then u1 = u2 on [0, T]. +Proof. The proofs of (i) and (ii) are similar to those of Theorem 1.2 (2) and Propo- +sition 1.5, respectively. The only difference is use of Proposition 3.1 with the double +endpoint case (3.4) and (3.5), where the restriction on r is required. +□ +In the case d = 1 and γ = −1, the existence of a solution has not been proved, but +Theorem 7.1 implies that only one solution exists at most. It remains open whether +unconditional uniqueness holds in the critical singular case d ≥ 2 and γ = −2. +Once the weighted Meyer inequality (3.34) with the endpoint case +s2 +d + 1 +q2 = 0 is +proved, this problem is solved, but we do not know if the endpoint inequality holds. +7.3. Case of the exterior problem. It is also interesting to analyze in more detail +the influence of the potential |x|γ at the origin or at infinity. For this, we discuss +unconditional uniqueness for the initial-boundary value problem of the Hardy-H´enon +parabolic equation on the exterior domain Ω := {x ∈ Rd ; |x| > 1}. +� +� +� +� +� +∂tu − ∆u = |x|γ|u|α−1u, +(t, x) ∈ (0, T) × Ω, +u = 0, +(t, x) ∈ (0, T) × ∂Ω, +u(0) = u0 ∈ Lq,r +s (Ω), +(7.1) +where T > 0, d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞], r ∈ (0, ∞] and s ∈ R. Here, ∂Ω +denotes the boundary of Ω. In conclusion, the critical exponents (1.4) and (1.5) with +γ = 0 (i.e. qc(0) = d(α−1) +2 +and Qc(0) = α) appear in the results on unconditional +uniqueness for (7.1), since the effect near the origin x = 0 has been eliminated. +The results of this subsection can be extended to more general situations such as +the initial-boundary value problem on general domains Ω not containing the origin +with the Robin boundary condition (cf. [22, Section 6]). +In the following, we shall prove the result on unconditional uniqueness. +Proposition 7.2. Let d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞] and s ∈ R be such that +α ≤ q ≤ ∞, +−d +q < s < d +� +1 − α +q +� +and +γ +α − 1 ≤ s. +(7.2) +Then the following assertions hold: +(i) Assume either +q > min {qc(0), Qc(0)} +and +r = ∞ +(7.3) +or +q = Qc(0) > qc(0) +and +r = α. +Then unconditional uniqueness holds for (7.1) in L∞(0, T; Lq,r +s (Ω)). + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +37 +(ii) Assume either +q = qc(0) > Qc(0) +and +r = ∞. +or +q = qc(0) = Qc(0) +and +r = α − 1. +Then unconditional uniqueness holds for (7.1) in C([0, T]; Lq,r +s (Ω)). +Remark 7.3. Since Lq,r +s1 (Ω) ⊂ Lq,r +s2 (Ω) if s2 ≤ s1, the exponent s should be taken +as close to max{− d +q, +γ +α−1} as possible in the above proposition from the point of view +of unconditional uniqueness. +We denote by −∆D the Laplace operator with the homogeneous Dirichlet bound- +ary condition on Ω and by {et∆D}t>0 the semigroup generated by −∆D. +The +integral kernel GD(t, x, y) of et∆D satisfies the Gaussian upper bound +0 ≤ GD(t, x, y) ≤ Gt(x − y) +(7.4) +for any t > 0 and almost everywhere x, y ∈ Ω. Then, we have the following linear +estimates. +Lemma 7.4. Let d ∈ N, 1 ≤ q1 ≤ ∞, 1 < q2 ≤ ∞, 0 < r1, r2 ≤ ∞ and s1, s2 ∈ R. +(i) Assume (3.2)–(3.7). Then there exists a constant C > 0 such that +∥et∆Df∥Lq2,r2 +s2 +(Ω) ≤ Ct− d +2 ( 1 +q1 − 1 +q2 )− s1−s2 +2 +∥f∥Lq1,r1 +s1 +(Ω) +for any t > 0 and f ∈ Lq1,r1 +s1 +(Ω). +(ii) Assume (3.7) and +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +− d +q2 +≤ s2 ≤ min +� +s1, d +� +1 − 1 +q1 +�� +, +q1 ≤ q2, +r1 ≤ 1 +if s2 = d +� +1 − 1 +q1 +� +or q1 = 1, +r2 = ∞ +if s2 = − d +q2 +, +r1 ≤ r2 +if q1 = q2. +(7.5) +(7.6) +(7.7) +(7.8) +(7.9) +Then there exists a constant C > 0 such that +∥et∆Df∥Lq2,r2 +s2 +(Ω) ≤ Ct− d +2 ( 1 +q1 − 1 +q2 )∥f∥Lq1,r1 +s1 +(Ω) +for any t > 0 and f ∈ Lq1,r1 +s1 +(Ω). +Proof. The assertion (i) is obtained by combining the upper bound (7.4) with the +argument of proof of Propositions 3.1. The assertion (ii) is proved by combining the +assertion (i) with s1 = s2 and the inclusion Lq,r +s1 (Ω) ⊂ Lq,r +s2 (Ω) if s2 ≤ s1. +□ +Similarly, we also have the following: +Lemma 7.5. Let T ∈ (0, ∞], and let d ∈ N, q1 ∈ [1, ∞], q2 ∈ (1, ∞), r1 ∈ (0, ∞] +and s1, s2 ∈ R. + +38 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +(i) Assume (3.29)–(3.33). Then there exists a constant C > 0 such that +���� +� t +0 +e(t−τ)∆Df(τ) dτ +���� +Lq2,∞ +s2 +(Ω) +≤ C sup +0<τ 0. +Then there exists a positive constant δ such that c/2 ≤ |x| +1 +q | log |x|| +1 +r |f(x)| for +|x| ≤ δ. Using [15, Proposition 1.4.5 (4) and (5)], we have +(fχ|x|≤δ)∗(t) ≥ c +2(|x|− 1 +q | log |x||− 1 +r χ{|x|≤δ})∗(t) = c +2t− 1 +q | log t|− 1 +r χ{t≤δ′} +for some δ′ > 0. Hence, +∥fχ|x|≤δ∥Lq,r = +�� ∞ +0 +(t +1 +q (fχ|x|≤δ)∗(t))r dt +t +� 1 +r +≥ c +2 +�� δ′ +0 +| log t|−1dt +t +� 1 +r += +∞, +which implies f ̸∈ Lq,r(R). The second equality is similarly proved. +□ +Lemma A.2. Let q, r ∈ (0, ∞], and let {fn}n∈N ⊂ Lq,r(Rd) and f be a measurable +function on Rd. Assume that |fn| ↗ |f| a.e. as n → ∞ (i.e. |fn(x)| ≤ |fn+1(x)| +for any n ∈ N and fn(x) → f(x) as n → ∞ for almost everywhere x ∈ Rd). Then +f ∈ Lq,r(Rd) and +∥f∥Lq,r = lim +n→∞ ∥fn∥Lq,r. +Appendix B. +In this appendix, we give a proof of Theorem 5.5 (ii) for completeness. The proof +is based on the argument of the proof of [17, Theorem 4.1]. For simplicity, we write +U = U(r), where r = |x|. Then U satisfies the problem +− (rd−1U ′)′ = rd−1+γU +d+γ +d−2 , +r ∈ (0, 1). +(B.1) +Then the upper bound of U near x = 0 was already obtained. +Theorem B.1 (Theorem 1.1 (iv) in [5]). There exists a constant C > 0 such that +U(r) ≤ Cr−(d−2)| log r|− d−2 +γ+2, +r ∈ (0, 1). +We make the change of variable +U(r) = r−(d−2)u(t), +t = − log r. +(B.2) +The properties of u are as follows. +Lemma B.2. The function u is of C2 on (0, ∞) and is a positive and strictly +decreasing solution of the nonlinear ordinary differential equation +d +dt +�du +dt (t) + (d − 2)u(t) +� ++ u(t) +d+γ +d−2 = 0, +t ∈ (0, ∞) +(B.3) + +40 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +with +u(0) = lim +r→1 U(r) +and +du +dt (0) = − lim +r→1 +�dU +dr (r) − (d − 2)U(r) +� +(and hence, u is a C1-diffeomorphism from (0, ∞) to (0, u(0))). Moreover, u +d+γ +d−2 ∈ +L1((0, ∞)) and +du +dt (t) + (d − 2)u(t) > 0, +t ∈ (0, ∞). +(B.4) +Proof. It is obvious that u is positive and of C2, and a straightforward calculation +gives that u satisfies the nonlinear ordinary differential equation (B.3). It is shown +by Theorem B.1 that u +d+γ +d−2 ∈ L1((0, ∞)). +We shall prove that u is strictly decreasing on (0, ∞) by contradiction. Suppose +that u is not strictly decreasing on (0, ∞). Then there exist t0, t1 such that 0 < +t0 < t1 and +ut(t0) = ut(t1) = 0 +and +ut ≥ 0 on (t0, t1). +(B.5) +Since u is positive, we find from (B.5) that +(d − 2) {u(t1) − u(t0)} = [ut(τ) + (d − 2)u(τ)]τ=t1 +τ=t0 += − +� t1 +t0 +u(τ) +d+γ +d−2 dτ < 0, +which implies that u(t0) > u(t1). This is a contradiction to ut ≥ 0 on (t0, t1). +Therefore, u is strictly decreasing on (0, ∞). In addition, it is also shown by the +inverse function theorem that u is a C1-diffeomorphism from (0, ∞) to (0, u(0)). +Lastly, since u ∈ C2((0, ∞)), the fundamental theorem of calculus gives +u(t′) − u(t) = +� t′ +t +ut(τ) dτ +for t′ ≥ t > 0, and as t′ → ∞, +u(t) = − +� ∞ +t +ut(τ) dτ +for t > 0. Since ut < 0, the convergence ut(t) → 0 as t → ∞ must hold. Noting +u(t), ut(t) → 0 as t → ∞, and integrating (B.3) over [t, ∞), we have +ut(t) + (d − 2)u(t) = +� ∞ +t +u(τ) +d+γ +d−2 dτ +(B.6) +for any t > 0, which implies (B.4). The proof of Lemma B.2 is finished. +□ +Lemma B.3. Let d ≥ 3 and γ > −2. Assume that +lim +t→∞ +ut(t) +u(t) = 0 or − (d − 2). +(B.7) +Then the assertion (ii) in Theorem 5.5 holds. + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +41 +Proof. In the case where +lim +t→∞ +ut(t) +u(t) = −(d − 2) +� +i.e. lim +t→∞(log u(t))t = −(d − 2) +� +, +(B.8) +then for any ε ∈ (0, d − 2), there exists T = T(ε) > 0 such that +− (d − 2) − ε < (log u(t))t < −(d − 2) + ε +(B.9) +for any t ≥ T . By integrating (B.9) over [T, t], we estimate +u(t) < u(T)e(−(d−2)+ε)(t−T) < u(0)e(−(d−2)+ε)(t−T), +and by recalling (B.2), we find that +U(r) ≤ Ce((d−2)−ε)Tr−ε, +r ∈ (0, 1). +Hence, U can be extended as a C1 function on B (see [35, Theorem 1] and also +[12, Lemma 2.1 and Section 3]). +Next, we consider the other case: +lim +t→∞ +ut(t) +u(t) = 0. +(B.10) +Set +ψ(t) := +� ∞ +t +u(τ) +d+γ +d−2 d τ. +Then the following hold: +lim +t→∞ +ψ(t) +d+γ +d−2 +ψt(t) += −(d − 2) +d+γ +d−2 , +(B.11) +and +lim +t→∞ t +d−2 +2+γ ψ(t) = (2 + γ)− d−2 +2+γ (d − 2) +2d+γ−2 +2+γ . +(B.12) +In fact, noting that limt→∞ u(t) = limt→∞ ut(t) = 0, we see from (B.3) that +ψ(t) = ut(t) + (d − 2)u(t) +and +ψt(t) = −u(t) +d+γ +d−2 . +Hence, +ψ(t) +d+γ +d−2 +ψt(t) += (ut(t) + (d − 2)u(t)) +d+γ +d−2 +−u(t) +d+γ +d−2 += − +�ut(t) +u(t) + (d − 2) +� d+γ +d−2 +. +This and (B.10) imply (B.11). Moreover, we see from (B.11) that +lim +t→∞(ψ− 2+γ +d−2 (t))t = −2 + γ +d − 2 lim +t→∞ +ψt(t) +ψ(t) +d+γ +d−2 += (2 + γ)(d − 2)− d+γ +d−2 −1. +Integrating the above yields +lim +t→∞ t−1ψ− 2+γ +d−2 (t) = (2 + γ)(d − 2)− d+γ +d−2 −1, + +42 +N. CHIKAMI, M. IKEDA, K. TANIGUCHI AND S. TAYACHI +which implies (B.12). By using (B.11) and (B.12) and noting that u(t) = (−ψt(t)) +d−2 +d+γ , +we obtain +lim +t→∞ t +d−2 +2+γ u(t) = lim +t→∞ t +d−2 +2+γ (−ψt(t)) +d−2 +d+γ = lim +t→∞ t +d−2 +2+γ +� ψ(t) +d − 2 +� += +�(d − 2)2 +2 + γ +� d−2 +2+γ +. +Thus, we conclude Lemma B.3. +□ +Finally, we conclude the proof of (ii) of Theorem 5.5 by showing the following. +Lemma B.4. Let d ≥ 3 and γ > −2. Then (B.7) holds. +Proof. Since u is a C1-diffeomorphism from (0, ∞) to (0, u(0)) by Lemma B.2, we +can define +ρ = u(t) +and +v(ρ) = ut(t) +� +i.e. v(ρ) = ut(u−1(ρ)) +� +. +For convenience, we set +w(ρ) := v(ρ) +ρ . +Then our goal is to prove that +lim +ρ→+0 w(ρ) = 0 or − (d − 2). +(B.13) +By Lemma B.2, w satisfies −(d − 2) < w < 0 and +0 = ρw(ρ) d +dρ (ρw(ρ) + (d − 2)ρ) + ρ +d+γ +d−2 += ρw(ρ) {(w(ρ) + (d − 2)) + ρwρ(ρ)} + ρ +d+γ +d−2 , +that is, +wρ = −1 +ρ +� +ρ +γ+2 +d−2 +w ++ (w + (d − 2)) +� += −w2 + (d − 2)w + ρ +γ+2 +d−2 +ρw +=: F(ρ, w). +Now, we can prove that +wρρ(a) > 0 +if a ∈ (0, u(0)] is such that wρ(a) = 0. +(B.14) +Indeed, wρ(a) = 0 implies that +wρρ(a) = Fρ(a, w(a)) = −γ + 2 +d − 2 +a +γ+2 +d−2 +a2w(a) > 0. +Since (B.14) implies that the sign of wρ is constant near ρ = +0, w is monotone +near ρ = +0. Hence, since −(d − 2) < w < 0, there exists a limit of w as ρ → +0: +lim +ρ→+0 w(ρ) = m ∈ [−(d − 2), 0]. +Suppose that −(d − 2) < m < 0 for contradiction. We calculate +m = lim +ρ→+0 w(ρ) = lim +ρ→+0 +v(ρ) +ρ += lim +t→+∞ +ut(t) +u(t) = lim +t→+∞(log u(t))t. + +UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION +43 +Then, for any ε ∈ (0, −m), there exists T > 0 such that +(m − ε)u(t) < ut(t) < (m + ε)u(t) +(B.15) +and +m − ε < (log u(t))t < m + ε +(B.16) +for any t > T . Integrating (B.16) over [t, T] gives +u(T)e(m−ε)(t−T) < u(t) < u(T)e(m+ε)(t−T) < u(0)e(m+ε)(t−T) +(B.17) +for any t > T , and hence, +u(t) +d+γ +d−2 < u(0) +d+γ +d−2 e +(m+ε)(d+γ) +d−2 +(t−T) +(B.18) +for any t > T . By (B.15) and (B.17), we also have +ut(t) + (d − 2)u(t) > {(d − 2) + m − ε}u(t) +> {(d − 2) + m − ε}u(T)e(m−ε)(t−T). +(B.19) +By combining (B.6), (B.18) and (B.19), we have +{(d − 2) + m − ε}u(T)e(m−ε)(t−T) < ut(t) + (d − 2)u(t) += +� ∞ +t +u(τ) +d+γ +d−2 d τ +< +� ∞ +t +u(0) +d+γ +d−2 e +(m+ε)(d+γ) +d−2 +(τ−T)d τ += Ce +(m+ε)(d+γ) +d−2 +(t−T) +for any t > T . Then, if we further assume (d − 2) + m − ε > 0, we have +u(T) < Ce{ (m+ε)(d+γ) +d−2 +−(m−ε)}(t−T) +for any t > T . However, as t → ∞, this contradicts that +u(T) ≥ Cε +for some constant Cε > 0, +if we fix ε sufficiently small so that +(m + ε)(d + γ) +d − 2 +− (m − ε) < 0 +i.e. +0 < ε < (2 + γ)|m| +2d + γ − 2. +Therefore, m must be 0 or −(d − 2). Thus, we prove Lemma B.4. +□ +Acknowledgement +The first author is supported by Grant-in-Aid for Young Scientists (B) (No. +17K14216) and Challenging Research (Pioneering) (No.17H06199), Japan Society for +the Promotion of Science. The second author is supported by JST CREST (No. JP- +MJCR1913), Japan and the Grant-in-Aid for Scientific Research (B) (No.18H01132) +and Young Scientists Research (No.19K14581), JSPS. The third author is sup- +ported by Grant for Basic Science Research Projects from The Sumitomo Foun- +dation (No.210788). The fourth author is supported by the Laboratoire ´Equations +aux D´eriv´ees Partielles LR03ES04. + +44 +N. 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H., Some remarks on convolution operators and L(p, q) spaces, Duke +Math. J. 36 (1969), no. 4, 647–658. +(N. Chikami) Graduate School of Engineering, Nagoya Institute of Technology, +Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan. +Email address: chikami.noboru@nitech.ac.jp +(M. Ikeda) Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, +Kohoku-ku, Yokohama, 223-8522, Japan/ Center for Advanced Intelligence Project +RIKEN, Japan. +Email address: masahiro.ikeda@keio.jp/masahiro.ikeda@riken.jp +(K. Taniguchi) Advanced Institute for Materials Research, Tohoku University, +2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan. +Email address: koichi.taniguchi.b7@tohoku.ac.jp +(S. Tayachi) Universit´e de Tunis El Manar, Facult´e des Sciences de Tunis, D´epartement +de Math´ematiques, Laboratoire ´Equations aux D´eriv´ees Partielles LR03ES04, 2092 +Tunis, Tunisia. +Email address: slim.tayachi@fst.rnu.tn + diff --git a/09AyT4oBgHgl3EQfoPj8/content/tmp_files/load_file.txt b/09AyT4oBgHgl3EQfoPj8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efda51047378dd2952a9e998a1886c1947504e85 --- /dev/null +++ b/09AyT4oBgHgl3EQfoPj8/content/tmp_files/load_file.txt @@ -0,0 +1,1860 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf,len=1859 +page_content='UNCONDITIONAL UNIQUENESS AND NON-UNIQUENESS FOR HARDY-H´ENON PARABOLIC EQUATIONS NOBORU CHIKAMI, MASAHIRO IKEDA, KOICHI TANIGUCHI AND SLIM TAYACHI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' We study the problems of uniqueness for Hardy-H´enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H´enon type) in the nonlinear term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' To deal with the Hardy-H´enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-H´enon parabolic equations in the weighted Lorentz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Introduction and main results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Introduction and our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' We consider the Cauchy problem of the Hardy-H´enon parabolic equation � ∂tu − ∆u = |x|γ|u|α−1u, (t, x) ∈ (0, T) × Rd, u(0) = u0 ∈ Lq,r s (Rd), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) where T > 0, d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞], r ∈ (0, ∞] and s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Here, ∂t := ∂ ∂t is the time derivative, ∆ := �d j=1 ∂2 ∂x2 j is the Laplace operator on Rd, u = u(t, x) is an unknown complex-valued function on (0, T) × Rd, u0 = u0(x) is a prescribed complex-valued function on Rd, and Lq,r s (Rd) is the weighted Lorentz space (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='3), which includes the Lebesgue space Lq(Rd) = Lq,q 0 (Rd) as a special case r = q and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) in the case γ = 0 is the Fujita equation, which has been extensively studied in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) with γ < 0 is known as a Hardy parabolic equation, while that with γ > 0 is known as a H´enon parabolic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The corresponding stationary problem to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1), that is, − ∆U = |x|γ|U|α−1U, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='2) was proposed by H´enon as a model to study the rotating stellar systems (see [19]), and has also been extensively studied in the mathematical context, especially in the fields of nonlinear analysis and variational methods (see [14] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In this paper we study the problem on unconditional uniqueness and non-uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) in weighted Lorentz spaces Lq,r s (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Here, unconditional uniqueness means 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Primary 35A02, 35K58;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Secondary 35B33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Hardy-H´enon parabolic equations, semilinear heat equations, uncon- ditional uniqueness, non-uniqueness, uniqueness criterion, singular stationary solutions, weighted Lorentz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='00506v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='AP] 2 Jan 2023 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' CHIKAMI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' IKEDA, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' TANIGUCHI AND S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' TAYACHI uniqueness of the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) in the sense of the integral form u(t) = et∆u0 + � t 0 e(t−τ)∆(| · |γ|u(τ)|α−1u(τ)) dτ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='3) in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Lq,r s (Rd)) or C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Lq,r s (Rd)), where T > 0 and {et∆}t>0 is the heat semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' We say that non-uniqueness holds for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) if unconditional uniqueness fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In contrast, we say that conditional uniqueness holds if uniqueness of the solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) holds in the entire space with some auxiliary function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In addition, we also study uniqueness criterion which is a necessary and sufficient con- dition on the Duhamel term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' the second term in the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='3)) for uniqueness to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Let us here state previous works on uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' For (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) with γ ≤ 0, the problem on uniqueness has been well studied (see [3, 4, 7–9, 18, 27, 29, 36, 38, 42, 43] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In the study of unconditional uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) in Lebesgue spaces Lq(Rd) or Lorentz spaces Lq,r(Rd), the following two critical exponents are known to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The first one is the so-called scale-critical exponent qc given by qc = qc(d, γ, α) := d(α − 1) 2 + γ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='4) and we say that the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) is scale-critical if q = qc, scale-subcritical if q > qc, and scale-supercritical if q < qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The second one is the critical exponent Qc given by Qc = Qc(d, γ, α) := dα d + γ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='5) which is related to well-definedness of the Duhamel term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='3) in Lq,r(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In fact, the nonlinear term |x|γ|u|α−1u ∈ L1 loc(Rd) for any u ∈ Lq,r(Rd) if and only if “q > Qc” or “q = Qc and r ≤ α”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In the case γ = 0, unconditional uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Lq(Rd)) was proved in the double subcritical case q > max{qc, Qc} by Weissler [42] and in the single critical cases q = Qc > qc and q = qc > Qc by Brezis and Cazenave [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In the double critical case q = qc = Qc, non-uniqueness was proved for some initial data u0 ∈ Lq(Rd) by Terraneo [38], and then, for any initial data u0 ∈ Lq(Rd) by Matos and Terraneo [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In [38], uniqueness criterion was also obtained in the double critical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In the scale-supercritical case q < qc, non-uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) was proved for initial data u0 = 0 by Haraux and Weissler [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Uniqueness and non-uniqueness have also been studied for heat equations with exponential nonlinearities (see [21, 23] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In the Hardy case − min{2, d} < γ < 0, similar results were obtained by [4,36], where the Lorentz spaces Lq,r(Rd) is used to study unconditional uniqueness in the critical case q = Qc in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' In contrast, the H´enon case γ > 0 has not been well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' This is due to the difficulty of treating the increasing potential |x|γ in the nonlinear term at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' To overcome this difficulty, the weighted spaces are effective, and recently, conditional uniqueness was obtained in Lq s(Rd) = Lq,q s (Rd) in [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' however, uncon- ditional uniqueness and non-uniqueness are completely open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The main purpose of this paper is to prove unconditional uniqueness, non-uniqueness and uniqueness criterion for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) with all γ > − min{2, d}, including the H´enon case, in Lq,r s (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' UNIQUENESS FOR THE HARDY-H´ENON PARABOLIC EQUATION 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' The figure shows the domain of (α, q) for d ≥ 3 and γ ≤ 0, where α0 := 1 + γ d , αF := 1 + 2+γ d is the Fujita exponent, α∗ := d+γ d−2 is the Serrin exponent and αHS := d+2+2γ d−2 is the Hardy- Sobolev exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' Table 1 and Table 2 summarize the previous results on uniqueness for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1) with γ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content=' q= qc q=Qc b d-2 1 0 αo 1 αF Q* SHo α Table l: Unconditional Uniqueness q > min[qc, Qc}] >= b ----- >= == YES YES YES NO [42, Thm 1] ( = 0) [7, Thm 4] ( = 0) [7, Thm 4] (= 0) [27, Thm 1] (= 0) [4, Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1] (< 0) [36, Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1] (< 0) [36, Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='1] (< 0) [36, Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/09AyT4oBgHgl3EQfoPj8/content/2301.00506v1.pdf'} +page_content='3] (< 0) Table 2: Conditional Uniqueness q 105 K, there are multiple ionization states for metals which +contribute to the photoionization cross section. +We also include the cross section for the Compton effect, which +becomes particularly important at higher energies. We use the total +Klein-Nishina (KN) cross section (Klein & Nishina 1929; Longair +2011), 𝜎KN, +𝜎KN = 𝜋𝑟2 +𝑒𝑥−1 +�� +1 − 2(𝑥 + 1) +𝑥2 +� +ln(2𝑥 + 1) + 1 +2 + 4 +𝑥 − +1 +2(2𝑥 + 1)2 +� +, +(4) +where 𝑟𝑒 is the classical electron radius and 𝑥 = 𝐸𝛾/(𝑚𝑒𝑐2). While +most applications will be in the limit of Thomson scattering, we +include the full Compton cross sections to enable more flexibility +in the choice of energy bins. In the cross-section plots, we show the +total Compton cross-section weighted by the number of free electrons +contributed by each species. The total cross section, 𝜎𝑥, is thus +𝜎𝑥(𝐸) = +𝑁elem +∑︁ +𝑖 +𝑥𝑖𝜎pi,i(𝐸) + 𝜎KN,i(𝐸), +(5) +where 𝑥𝑖 is the abundance of element 𝑖 with respect to hydrogen and +the sum is carried out including the cross sections for the 𝑁elem ele- +ments. Table 1 shows the elements we include in the photoabsorption +cross section and their fiducial abundances relative to hydrogen. +Figure 2 shows the photoionization cross sections and free-electron +contributions to the Compton cross section as a function of energy for +gas with𝑇 = 105 K. As shown, for some elements, the Compton cross +section becomes more important than photoionization, in particular +for hydrogen and helium above 1 keV, and for carbon and oxygen +above 30 keV. For hydrogen, the Compton effect is dominant due to +the negligible neutral fraction at T = 105 K. Figure 3 shows the total +cross section as a function of energy for 𝑇 = 105 K and each of the +total elemental contributions. Here, the X-ray photoabsorption cross +section is dominated by helium (< 0.3 keV), then carbon (0.3 − 0.6 +keV) and oxygen (0.8−4 keV). At energies above 4 keV, the hydrogen +and helium Compton cross sections dominate with a contribution +from the iron photoionization cross section around 10 keV. However, +many of the metals contribute equally to the total cross section around +1 keV. +Finally, Figure 4 shows 𝜎𝑥 as a function of energy and temperature +from 𝑇 = 104 to 107 K. At low temperatures, we recover the analytic +1 https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+AS/109/125 +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +3 +Table 1. Elements included in our photoabsorption cross section calculation +and their fiducial abundances, 𝐴𝑋, reported as 𝐴𝑋 = log(𝑁𝑥/𝑁𝐻) + 12 +(Asplund et al. 2009). +Element +Abundance (𝐴𝑋) +H +12 +He +10.986 +C +8.443 +O +8.783 +N +7.913 +Ne +8.103 +Na +6.353 +Mg +7.593 +Al +6.523 +Si +7.573 +S +7.193 +Ar +6.553 +Ca +6.383 +Fe +7.503 +Ni +6.283 +cross section previously used, although around 10 keV there is an in- +crease in the cross section due to iron. However, the rather significant +temperature dependence highlights the necessity of including a tem- +perature dependent cross section: at high temperatures, the higher +thermal ionization state leads to a reduction of nearly two orders of +magnitude in the cross section, thereby making the gas significantly +more optically thin to the X-ray radiation, producing less heating and +enabling more X-rays to escape. Below 104 K, the cross section does +not noticeably change since hydrogen is not significantly collisional +ionized. Therefore, for the results of this paper, for colder gas, we +use the 𝑇 = 104 photoabsorption cross section. The module though +allows for the user to define their own temperature dependent cross +sections across any temperature range. +For a given set of energy bins, {(𝐸𝑙,𝑖, 𝐸𝑟,𝑖)}, where 𝑖 = 1, 𝑁bin +for 𝑁bin bins, we define: +𝐸𝑐,𝑖 = 1 +2 +�𝐸𝑙,𝑖 + 𝐸𝑟,𝑖 +� , +(6) +where 𝐸𝑙,𝑖 is the left bound of the 𝑖th bin, 𝐸𝑟,𝑖 is the right bound, +and 𝐸𝑐,𝑖 is the midpoint of the bin. We derive bin-averaged cross +sections, such that +exp +� +− ⟨𝜎𝑋,𝑖⟩ +𝜎𝑐 +� += +1 +𝐸𝑙,𝑖 − 𝐸𝑟,𝑖 +∫ 𝐸𝑟,𝑖 +𝐸𝑙,𝑖 +exp +� +−𝜎𝑥(𝐸𝛾) +𝜎𝑐 +� +𝑑𝐸, +(7) +where ⟨𝜎𝑥,𝑖⟩ is the bin-averaged cross section for bin, 𝑖, and 𝜎𝑐 = +𝜎𝑥(𝐸𝑐,𝑖). Our fiducial tests use 𝑁bin = 8 between 1 – 10 keV using +logarithmically spaced bins. +All of these cross section data, and the initialization and storage +of the bins and bin-averaged cross sections are kept in a new Flash +module, XrayCommon. Flash is a highly module public magneto- +hydrodynamic code (Fryxell et al. 2000) written in Fortran and +highly-scalable with MPI. The scripts necessary to compute the X- +ray cross sections are publicly available on GitHub2. This module +enables the coupling of X-ray physics to multiple other modules. +Plasma models and the necessary X-ray data are also stored within +this module, as a unified location. +2.2 TreeRay +TreeRay is a novel reverse ray tracing scheme, described fully in +2 � https://github.com/AstroBrandt/XRayCrossSections +Wünsch et al. (2021), implemented in Flash. Simply, TreeRay +enables an efficient method to compute the contributions of radiation +from every cell, for every cell. It does so through the combination of +a reverse ray-trace algorithm with a tree (Wünsch et al. 2018), which +also currently is used in the gravity solver. Below we describe briefly +the different aspects of the TreeRay algorithm and XRayTheSpot +extension and refer the reader to Wünsch et al. (2021) for more +details. +2.2.1 Building the Tree +The foundation of the TreeRay algorithm is an octtree which stores +all necessary variables for the various TreeRay modules. At min- +imum, the tree stores the mass and center of mass coordinates for +the respective cell, or leaf, or higher nodes. For XRayTheSpot, two +further quantities are stored onto the tree: the bin-integrated X-ray +luminosity in each energy bin and the gas temperature. While the +bin-integrated X-ray luminosity is purely additive, the temperature is +stored as a mass-weighted average of each set of eight sub-nodes (or +leaves). +2.2.2 Ray Structure +Before the tree walk is executed for a given cell, rays are generated by +casting 𝑁pix = 12𝑁2 +side rays from each cell using directions defined +by the HealPix (Górski et al. 2005) algorithm. HealPix tessellates +the unit sphere into areas representing equal solid angles with a unit +vector pointing to the center of each of these surface areas from the +sphere’s center. TreeRay allows for 𝑁side = 1, 2, 4, 8, ..., with higher +values representing higher angular resolution. The rays are split into +𝑁𝑟 evaluation points, set by the grid resolution, Δ𝑥, the allowed +length of the ray, 𝐿ray, which is set to three-dimensional diagonal of +the computational domain, and a free parameter, 𝜂𝑅. +Along each ray, the radial coordinate point of the ith evaluation +point is +𝑟𝑖 = Δ𝑥𝑖2 +2𝜂2 +𝑅 +, +(8) +leading to segments with increasing lengths. This behavior coincides +well with the geometric acceptance criterion described below for +deciding whether or not to accept a tree node. The total number of +evaluation points is +𝑁𝑅 = 𝜂𝑅 × floor +�√︂ +2𝐿ray +Δ𝑋 +� ++ 1. +(9) +2.2.3 Tree Walk +The mapping of the cells/nodes onto the rays requires two factors: a +multipole acceptance criterion (MAC) and a weighting function to +map from the tree onto the different radial evaluation points. When +the MAC is met, a node is accepted and used. The simplest MAC is +the Barnes-Hut (BH) geometric MAC (Barnes & Hut 1986), where +a node of size ℎ𝑛, at a distance 𝑑, from the cell is opened if +ℎ𝑛/𝑑 < 𝜃lim +(10) +where 𝜃lim is a user-defined opening angle with a sensible choice +being 𝜃lim = +√︃ +4𝜋/𝑁pix3. We also utilize the ‘Src MAC’ (Wünsch +3 The resulting 𝜃lim for 𝑁side = 1, 2, 4, 8 is 1.0, 0.5, 0.25, 0.125, respec- +tively. For the results of this paper, we adopted these recommended values of +MNRAS 000, 1–14 (2022) + +4 +Gaches et al. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ionization Fraction +I +II +I +II +III +IV +V +VI +I +II +III +IV +V +VI +VII +VIII +H +He +C +O +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ionization Fraction +I +II +III +IV +V +VI +VII +I +II +III +IV +V +VI +VII +VIII +IX +X +I +II +III +IV +V +VI +VII +VIII IX X +XI +XII +XIII +I +II +III +IV +V +VI VII +VIII +IX +X +XI +XII +N +Ne +Si +Mg +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ionization Fraction +I +II +III +IV +V +VI +VII +VIII IX X XIXIIXIIIXIV +XV +I +II +III +IV +V +VI +VII +VIII +IX +X XI +XIIXIII +XIV +XV +I +II +III +IV +V +VI +VII +VIII +IX +X +XI +I +II +III +IV +V +VI +VIIVIII +IX +X +XI +XII +S +Fe +Na +Al +104 +105 +106 +107 +Temperature (K) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Ionization Fraction +I +II +III +IV +V VIVII +VIII +IX +X XI XIIXIIIXIVXV +I +II +III +IV +V +VI +VIIVIIIIX +X +XI +XIIXIIIXIVXV +I +II +III +IV +V +VI +VII VIII IX +X +XI +XIIXIII +XIV +XV +Ar +Ca +Ni +Figure 1. Ionization fraction for different elements as a function of temperature. Annotated in the text are the peaks of different ionization levels for each element. +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +5 +10 +15 +10 +12 +10 +9 +10 +6 +10 +3 + (Mbarn/H-nucleus) +Photoionization +Compton +H +He +C +O +10 +10 +10 +8 +10 +6 +10 +4 + (Mbarn/H-nucleus) +N +Ne +Si +Mg +10 +11 +10 +9 +10 +7 +10 +5 + (Mbarn/H-nucleus) +S +Fe +Na +Al +102 +103 +104 +105 +Energy (eV) +10 +10 +10 +8 +10 +6 + (Mbarn/H-nucleus) +Ar +Ca +Ni +Figure 2. Photoionization (solid) and Compton process (dashed) cross sec- +tions for each element as a function of energy, assuming thermal ionization +equilibrium at 𝑇 = 105 K. Each elemental contribution is weighted by the +assumed abundance with respect to hydrogen. +et al. 2021), where a node with sources is opened if +ℎ𝑛/𝑑 < 𝜃src, +(11) +where 𝜃src is a user-defined parameter. +Quantities on the tree are mapped onto the radial evaluation points +of a ray through the use of kernels. We utilize both a piece-wise +third-order polynomial, 𝑊𝑝(𝛿), and a kernel derived to ensure it +meets the requirements of the radiation transfer equation, 𝑊 𝑓 (𝛿), +where 𝛿 = (𝑟𝑖 − 𝑑)/ℎ𝑛 and 𝑑 is the distance from the node center +of mass and the ray evaluation point (see Wünsch et al. 2021) and +ℎ𝑛 is the node’s linear size. The node quantites are weighted by the +overlap of the volume of the ray segment and the node. +𝜃lim for the corresponding 𝑁side. See Wünsch et al. (2018) for 𝜃lim resolution +tests in the context of TreeRay/OpticalDepth +102 +103 +104 +Energy (eV) +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +tot (Mbar/H-nucleus) +H +He +C +O +N +Ne +Si +Mg +S +Fe +Na +Al +Ar +Ca +Ni +Figure 3. Total photo-absorption cross section (black) with each element +contribution highlighted (colors) as a function of energy for gas at temper- +ature, 𝑇 = 105 K. Each elemental contribution is weighted by the assumed +abundance with respect to hydrogen. +102 +103 +104 +105 +Energy (eV) +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Cross section (Mbarn) +103 +104 +Energy (eV) +10 +6 +10 +5 +10 +4 +Cross section (Mbarn) +4.0 +4.5 +5.0 +5.5 +6.0 +6.5 +7.0 +Temperature +Figure 4. Total photoabsorption cross section as a function of energy and +temperature (color). The black dashed line shows the previously used analytic +cross section from Panoglou et al. (2012). Inset: Zoom-in to 1 – 10 keV. +Following the tree walk, the rays from a cell outward store the +mass, center of mass, gas temperature and bin-integrated luminosi- +ties. These provide all the necessary information to solve the equation +of radiation transfer along each ray. +2.2.4 Solving the Radiation Transfer Equation +Along the rays, the one-dimensional radiation transfer equation is +solved: +𝑑𝐼𝜈 +𝑑𝑠 = −𝜖𝜈 + 𝛼𝜈𝐼𝜈 +(12) +where 𝑠 is the distance along the ray, and 𝜖𝜈 and 𝛼𝜈 are the emission +and absorption coefficients. The band-integrated flux, 𝐽(𝐸), irradiat- +MNRAS 000, 1–14 (2022) + +6 +Gaches et al. +ing a cell 𝑖 due to the band-integrated luminosity, 𝐿𝑋, emitting from +node 𝑗 can be simply written +𝐽 𝑗𝑖(𝐸) = 𝐿𝑋, 𝑗 (𝐸) 𝑒−𝜏𝑥 +4𝜋𝑟2 +𝑖 𝑗 +(13) +where 𝑟𝑖 𝑗 is the distance between the centers of cell 𝑖 to node 𝑗 and +𝜏𝑥 = +∫ +𝑛H(𝑠)𝜎𝑥(𝐸,𝑇(𝑠))𝑑𝑠 ≈ +∑︁ +𝑘 +𝜌𝑘 +𝜇𝑚𝐻 +⟨𝜎𝑋 (𝐸,𝑇𝑘)⟩ 𝛿𝑠 +(14) +is the X-ray opacity between cell 𝑖 and node 𝑗 and 𝜌𝑘 is the density +at evaluation point 𝑘 along the ray, 𝑛H is the hydrogen nuclei density, +and 𝛿𝑠 = 𝑟𝑘 − 𝑟𝑘−1. We store the solution as an energy density, +𝜀𝑥 = 𝐽/𝑐, where 𝑐 is the speed of light, onto the grid to be used in +chemistry, described below. The total energy density is the sum over +the HealPix rays: +𝜀𝑖 = +𝑁pix +∑︁ +𝑘 +𝜀𝑘𝑖(𝐸). +(15) +For the solution, we also store the cell mass and temperature onto +the tree and map these to the rays using 𝑊𝑝. In our algorithm, no +assumption is made with respect to what produces the X-ray energy +density, enabling both point sources (with their radiation spread over +their host cells) and diffuse emission produced via cooling of hot gas +in the cell. +2.3 Pre-existing Chemistry +We briefly describe here the previous treatment of X-ray radiation +within the chemical network (see Mackey et al. (2019) for more +details). The chemical network consists of 17 species, of which 9 are +solved numerically and the rest are followed through conservation +equations. We solve the non-equilibrum species H+, H2, C+, CO, +HCO+, CHx, OHx, He+ and M+. CHx is a proxy species for simple +hydrocarbons, e.g. CH, CH2, etc, and simple ions CH+, CH2+, etc. +Similarly, OHx is a proxy for OH, H2O and ions OH+, H2O+, etc. M +is a proxy for metals that can become the primary source of electrons +in shielded regions of molecular clouds, where reaction rates treat +M as Si. The network is primarily based on the ‘NL99’ network of +Glover & Clark (2012), which uses the hydrogen chemistry from +Glover & Mac Low (2007a,b) with the CO chemistry of Nelson & +Langer (1999) including updated reaction rates from Gong et al. +(2017). For this work, all photodissociation rates have been updated +using the KIDA astrochemistry database (Wakelam et al. 2012). +X-ray radiation is coupled to the thermochemistry through the +following primary processes (see also Mackey et al. 2019): +• Dust heating, following the analytic prescription in Yan (1997). +• Primary ionization of a species by X-rays. Note though that is is +relatively unimportant for our considered species, and plays a minor +role in the heating and ionization for hydrogen species and helium. +• Secondary ionization through collisional ionization by fast elec- +trons produced following primary ionizations (e.g. Dalgarno et al. +1999; Meijerink & Spaans 2005). +• Induced FUV radiation generated by H2, which is collisionally +excited by fast electrons and the subsequent ionizations and dissoci- +ations (Prasad & Tarafdar 1983; Gredel et al. 1987; Maloney et al. +1996; Meijerink & Spaans 2005). +• Coulomb heating of the gas via energy exchange between the +produced fast electrons and other charged particles (Dalgarno et al. +1999). +These processes have all been generalized for the arbitrary number +of energy bins and the temperature-dependent cross sections. The +input X-rays are computed by the XRayTheSpot module. Since we +use band-integrated radiative variables, the heating parameter for a +particular cell 𝑖 due to the impinging X-ray radiation is +𝐻𝑥,𝑖 = +𝑁bin +∑︁ +𝑛 +𝑗𝑖(𝐸𝑛) ⟨𝜎𝑥(𝐸𝑛,𝑇𝑖)⟩ . +(16) +3 TESTS AND BENCHMARKING +Here we show various numerical tests of the radiation transfer and a +benchmark of the thermochemsitry against Cloudy. For our bench- +marks, we fiducially use 8 bins, logarithmically spaced between 1 - +10 keV, following Meijerink & Spaans (2005). +3.1 Point Source Test +Our first test is a single central point source with a constant luminosity +distribution, 𝐿𝑥,𝑛 = 1 L⊙ for all 𝑁bin bins, embedded in a volume +with a uniform density of 𝑛(𝐻) = 2 × 103 cm−3 and a spatially +constant temperature 𝑇 = 10 Kelvin in a (30 pc)3 volume. We use +a constant luminosity to better compare the solutions of different +energy bins. We then compute the radial profiles of the energy density +and compare against the analytic solution: +𝐽(𝐸, 𝑟) = 𝐿𝑥(𝐸) 𝑒−𝜎𝑥 (𝐸)𝜌𝑟 +4𝜋𝑟2 +(17) +where the energy density, 𝜖 = 𝐽(𝐸)/𝑐. +Figure 5 shows the performance of XRayTheSpot for a single +bright point source as a function of radius. The results in figure 5 +used 𝜂𝑅 = 4, 𝑁side = 8 and 𝑁block = 8, where 𝑁block is the number of +blocks of cells per spatial dimension, and one block consists of a cube +of 83 cells. The radial range was chosen that for the lowest energy bin, +the emission transitions from optically thin to strongly optically thick, +with the maximum radius corresponding to 𝜏(𝐸 = 1.17eV) ≈ 10. +The left panel shows the comparison between the ray trace solution +and the analytic solution. These solutions agree well with each other +with the lines largely overlapping. The right panel shows the relative +error, defined as +𝛿𝑐 = |𝑐𝜀 − 𝐽(𝐸, 𝑟)| +𝐽(𝐸, 𝑟) +(18) +where 𝜀 is the solution from XRayTheSpot. The relative error is +rather insensitive to the optical depth but more sensitive to how +strongly the radiation field is coupled to the gas (e.g. the magnitude +of the photoabsorption cross section). The 10% error shown for the +most optically thick bin at low energies is due to the mapping of +the density structure onto the rays using the kernel. For X-ray optical +depths greater than 𝜏𝑥 ≈ 10, the error starts to increase towards unity, +but at these energy densities, the X-rays have a negligible impact on +the thermochemistry. Therefore, these relative differences will have +no discernible impact on the thermochemical evolution of the gas. +In order to highlight the differences of the new module with the +previous cross section implementation presented in Mackey et al. +(2019), we perform a second calculation imposing a strong temper- +ature gradient such that the radial temperature profile is +𝑇(𝑟) = 5 × 105 [1 − tanh(𝑟 − 8 pc)] + 100 K. +(19) +This temperature is artificial and chosen such that the X-ray radiation +transitions from optically thin to optically thick due to the change in +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +7 +𝑁block +𝑁side +𝜂𝑅 +Initialization (s/proc) +Evolution (s/proc) +4 +2 +2 +2.3 +1.3 +4 +4 +2 +5.5 +3.6 +4 +4 +4 +6.1 +4.6 +4 +8 +2 +22.1 +14.0 +8 +4 +2 +17.4 +29.3 +8 +8 +4 +94.0 +167.2 +Table 2. Timing for the pont source test for the different runs in Figure 7. +Each row gives the model parameters of 𝑁block, 𝑁side and 𝜂𝑅 and the time +in seconds per processor for the initialization of the tree and a ray trace step. +cross section. Figure 6 shows the result of this comparison and as +expected, the emission for the lower energy bins is up to an order of +magnitude greater than the low-temperature solution and maintains +an 𝑟−2 trend until a much greater radius. +Figure 7 shows the performance of XRayTheSpot for a range of +parameters, exploring both low- and high- spatial and ray resolutions. +We find that grid resolution is the primary source of deviations at +small radius, while the ray resolution increases the accuracy at larger +radii. At large distances from the source, the solution tends to slightly +under predict for low ray and angular resolution due to overestimation +of the column density. At small radii, the solution over-predicts the +resulting flux. For optically thin radiation bins, the solution almost +exactly matches the analytic. Therefore, the deviations come about +due to mapping the mass from the cells and tree nodes onto the rays +using the kernel. +For science uses, the number of blocks, 𝑁block, is determined by +the necessary resolution to resolve crucial gas dynamics (e.g. the +Jeans length for gravity simulations). Increasing both 𝑁side and 𝜂𝑅, +while producing more accurate radiation transfer solutions, leads to +substantially higher computational costs. Table 2 shows the compu- +tational time per processor for the initialization and per ray trace step +for the models in Figure 7. Between the lowest and highest accuracy +tests, (𝑁block, 𝑁side, 𝜂𝑅) = (4, 2, 2) and (8, 8, 4), respectively, the +increase in cost of the initialization and ray trace was a factor of ≈ +40 and ≈ 130, respectively. The time for the raytrace is dominated +(≥ 95%) by the tree walk. We find using 𝑁side = 4 is the best balance +of time and accuracy. +3.2 Shadow Test +Our next test is a shadow test to verify the solution of the solver when +sources are placed near dense regions. Here, we have a 𝐿𝑋 = 10 L⊙ +source with a spectrum, 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2, placed near a dense core with +a hydrogen-nuclei number density of 𝑛H = 103 cm−3. We consider +radiation between 1 - 10 keV, moving from optically thick bands to +optically thin. Figures 8 and 9 show the results of this test, for both +low- and high- ray resolution which use (𝑁block = 8, 𝑁side = 4, +𝜂𝑅 = 2) and (𝑁block = 8, 𝑁side = 8, 𝜂𝑅 = 4), respectively. For both +cases, the test reveals the expected results that the low-energy X-rays +are absorbed by the dense core and this creates a wide-angle shadow, +while higher energy X-rays are barely attenuated, producing smaller +to no shadows. The high-ray resolution test also shows the expected +drop in ray-tracing artifacts. +Figure 10 shows a one-dimensional cut along the z-axis from the +source through the dense blob of gas for the two ray resolutions +compared. The figure shows that higher ray resolution leads to a +smoother attenuation of the flux for the optically thick, lower energy +bins while there is very little change for higher energy bins which +are substantially less attenuated. This is most pronounced for the +𝐸𝑐 = 1.33 keV bin +3.3 Benchmark against Cloudy +The final benchmark tests the X-ray radiation coupling to the ther- +mochemistry. We place a point source with a physical size of 1016 +cm and total X-ray luminosity, 𝐿XR = 1036 erg s−1, and a luminosity +spectrum 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2 between 1 - 10 keV in a (1.3 pc)3 volume +filled with a gas number density of 𝑛H = 103 cm−3, 𝑁side = 4, +and 𝜂𝑅 = 2. and compare with a one-dimensional model using the +Cloudy code. The volume and resolution were chosen such the inner +XDR is resolved by ≈ 20 cells while the optically thick regime is +also traced. In particular, we use 7 maximum adaptive-mesh reso- +lution levels refining on the density and temperature, such that the +maximal resolution is 1.3 × 10−3 pc. The one-dimensional Cloudy +model used a “sphere” geometry, an input power-law spectrum be- +tween 1 - 10 keV for the X-ray radiation source, a cosmic microwave +background and a cosmic-ray ionization rate of 3 × 10−17 s−1. Fur- +ther, we turn off grain physics, induced radiative processes, radiation +pressure, radiation scattering, outward line radiation transfer and +molecule freeze-out, since the Flash simulations do not have these +processes. Finally, we set refractory metal abundances to zero, with +the exception of silicon which Flash uses as the proxy for metals +for the chemistry (as described above). The Cloudy script used is +shown in Appendix A. +Using uniform-spaced grids, even with substantial AMR levels +and, it is in practice difficult to fully capture sharp thermochemical +transition regions, such as that shown below as captured by Cloudy. +Further, the source encompasses several cells at the highest reso- +lution, rather than an infinitely small point source. Capturing such +ionization and dissociation fronts entirely is numerically intensive +and generally requires the use of one-dimensional models tailored to +do so (as with Cloudy). +Figure 11 shows the result of this benchmark, using . Near the +source, the temperature and chemistry solutions well match the +Cloudy solution. The Flash and Cloudy solutions qualitatively +reproduce the chemical structure, although due to the larger cell-size +of the Flash grids, the sharp HI transition seen at 𝑁(𝐻) ≈ 5 × 1020 +cm−2 is not fully captured and is instead smoothed over a few cells. +The temperature solutions agree within a factor of a few. However, +Cloudy solves the line cooling and level excitations in a much more +robust manner than the included Flash thermochemistry, including +a full non-equilibrium solution with many more electronic and ion- +ization states. Such inclusions though are not numerically feasible +for in-situ thermochemistry in three-dimensional MHD simulations. +Further, Cloudy solves the full radiation transfer solution from radio +through X-ray radiation with substantially more bins. Given the con- +straints of these physics, the found solution is deemed to be adequate +and matches the overall trends as determined by Cloudy. +4 PROTOSTELLAR DISK +Evolved protostellar objects, in particular Class II objects in which the +lack of a surrounding gaseous envelope leaves the central protostar +and disk exposed, are known to be X-ray emitters. These X-rays +can become important for disk dynamics and planet formation (e.g. +Ercolano et al. 2008b; Mohanty et al. 2013). For these stars, the +X-ray emission is thought to come from a combination of accretion +and magnetospheric emission (Hartmann et al. 2016). As a first test +science case, we model the X-ray radiation transport from a central +protostar into a protostellar disk. +The surface density follows from the often used truncated power- +law (e.g. Lynden-Bell & Pringle 1974; Andrews et al. 2011; Cleeves +MNRAS 000, 1–14 (2022) + +8 +Gaches et al. +100 +101 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +1.17 keV +1.56 keV +2.07 keV +2.77 keV +TreeRay +3.69 keV +4.92 keV +6.56 keV +8.75 keV +Analytic +100 +101 +E (keV) +10 +24 +10 +23 +10 +22 + (cm +2) +1.17 keV +1.56 keV +2.07 keV +2.77 keV +TreeRay +3.69 keV +4.92 keV +6.56 keV +8.75 keV +Analytic +2 +4 +6 +8 +10 +r (pc) +10 +4 +10 +3 +10 +2 +10 +1 +100 +Relative Error +c +Figure 5. Radial profile test for a single source in a constant density and temperature medium. Left: Radiation density versus radius for each bin for the TreeRay +(solid) and analytic solution (dotted). Left inset: Bin-averaged cross sections (black points) and the analytic cross section in Eq. 1. Right: Relative errors for each +bin as a function of radius. +100 +101 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +1.17 keV +1.56 keV +2.07 keV +2.77 keV +3.69 keV +4.92 keV +6.56 keV +8.75 keV +Const T +Temp Grad +Const T +Temp Grad +103 +104 +105 +106 +Temperature (K) +Figure 6. X-ray energy density versus radius for single point source. The +solid line uses the constant temperature at 𝑇 = 10 K (same as Figure 5) while +the dashed-dotted line uses the temperature profile shown by the red dotted +line. +et al. 2016): +Σ𝑔(𝑅) = Σ𝑐 +� 𝑅 +𝑅𝑐 +�−𝛼 +exp +� +− +� 𝑅 +𝑅𝑐 +�2−𝛼� +(20) +between an inner and outer radius, 𝑅in and 𝑅out, respectively, 𝑅𝑐 +is the critical radius where the surface density distribution becomes +exponential, 𝛼 is the power law index and Σ𝑐 is the characteristic +surface where the disk transitions to an exponential profile. For the +initial conditions, we assume the gas is in hyrostatic equilibrium, +such that the density follows +𝜌𝑔(𝑅, 𝑧) = Σ𝑔(𝑅) +√ +2𝜋ℎ +exp +� +− +� 𝑧2 +2ℎ2 +�� +(21) +where ℎ = 𝑐𝑠/Ω is the disk scale height, 𝑐𝑠 = +√︃ 𝛾𝑘𝑏𝑇𝑔 +𝜇𝑚H , 𝑘𝐵 is +Boltzmann’s constant, 𝛾 = 5/3 is the adiabatic index, 𝑇𝑔 is the gas +temperature, 𝜇 = 2.33 is the mean mass per particle for molecular gas, +𝑚𝐻 is the mass of the hydrogen atom, Ω = 3 +4 +√︃ +𝐺𝑀∗ +𝑅3 +is the Keplerian +rotational frequency and 𝑀∗ is the mass of the central protostellar +object. For this fiducial test, we set 𝑀∗ = 0.7 M⊙, Σ𝑐 = 64 g cm−2, +𝑅𝑐 = 100 AU, 𝛼 = 1. The temperature profile is given by +𝑇(𝑅) = max +� +𝑇0 +� +𝑅 +1𝐴𝑈 +�−0.5 +, 10 K +� +, +(22) +where we fiducially take 𝑇0 = 50 K. The disk is initialized to be +rotating in Keplerian motion around the central protostellar object. +We assume the disk is magnetized with an initial toroidal field such +that the ratio of the magnetic to thermal pressure, 𝜇𝑀 = 10−5. We +simulate the domain in a 240 AU box with a maximal resolution of +1 AU. +The central protostar is put in by hand, with active accretion. For +the X-ray emission, we assume an accretion floor of 10−9 M⊙ yr−1, +similar to rates observed in young stellar objects (e.g. Ingleby et al. +2013). The simulation is run using the Bouchut-5 MHD solver, grav- +ity, and XRayTheSpot. The X-ray emission is derived by assuming +there is an accretion shock, with properties following “hot spot” ac- +cretion (Hartmann et al. 2016) with accretion columns filling 10% of +the protostar surface, thermally emitting X-ray emission. The ther- +mal X-ray emission is computed using a one-temperature Raymond- +Smith plasma model (Raymond & Smith 1977). The implementation +of a coronal model is left for a future work. +Figure 12 shows a slice of the density, gas temperature, X-ray +emission at 1.17 keV (1st bin) and 6.56 keV (8th bin), the heating +rate per H nucleus, 𝐻𝑥, and 𝐻𝑥/𝑛, which is often used as a diagnostic +for the importance of the X-ray heating (Wolfire et al. 2022). We find +that the lowest energy X-rays are all absorbed near the protostar or +escape through the outflow. However, the harder X-rays at 6.56 keV +are able to permeate much of the domain. The 𝐻𝑋 and 𝐻𝑥/𝑛 slices +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +9 +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +Nblock = 4, Nside = 4, +R = 4 +1.17 keV +1.56 keV +2.07 keV +2.77 keV +TreeRay +3.69 keV +4.92 keV +6.56 keV +8.75 keV +Analytic +2 +4 +6 +8 +10 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +1.17 keV +1.56 keV +2.07 keV +2.77 keV +TreeRay +3.69 keV +4.92 keV +6.56 keV +8.75 keV +Analytic +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +Nblock = 4, Nside = 2, +R = 2 +2.5 +5.0 +7.5 +10.0 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +r (pc) +10 +21 +10 +19 +10 +17 +10 +15 +10 +13 + (erg/cm3) +Nblock = 4, Nside = 4, +R = 2 +2.5 +5.0 +7.5 +10.0 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +101 +2 × 100 +3 × 100 +4 × 100 +6 × 100 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +Nblock = 4, Nside = 8, +R = 2 +2.5 +5.0 +7.5 +10.0 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +101 +2 × 100 +3 × 100 4 × 100 +6 × 100 +r (pc) +10 +21 +10 +19 +10 +17 +10 +15 +10 +13 + (erg/cm3) +Nblock = 8, Nside = 4, +R = 2 +2.5 +5.0 +7.5 +10.0 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +101 +2 × 100 +3 × 100 4 × 100 +6 × 100 +r (pc) +10 +20 +10 +18 +10 +16 +10 +14 + (erg/cm3) +Nblock = 8, Nside = 8, +R = 4 +2.5 +5.0 +7.5 +10.0 +r (pc) +10 +3 +10 +2 +10 +1 +100 +c +Figure 7. . Energy density versus radius for the different model parameters, annotated in the top left of each subfigure. Inset: Relative error, 𝛿𝑐, of the numerical +solution against the analytic solution as a function of radius from the source. +MNRAS 000, 1–14 (2022) + +10 +Gaches et al. +15 +10 +5 +0 +5 +10 +15 +y (pc) +Density +1.33 keV +1.78 keV +15 +10 +5 +0 +5 +10 +15 +y (pc) +2.37 keV +3.16 keV +4.22 keV +10 +0 +10 +x (pc) +15 +10 +5 +0 +5 +10 +15 +y (pc) +5.62 keV +10 +0 +10 +x (pc) +7.50 keV +10 +0 +10 +x (pc) +10.00 keV +0 +2 +4 +nH (cm +3) +18.0 +17.5 +17.0 +16.5 +16.0 +15.5 +15.0 +14.5 +14.0 +Fi (erg cm +2 s +1) +Figure 8. Shadow test, consisting of a point source illuminating a constant +density core. Top left corner: Number density distribution for a z-axis slice. +Others: X-ray flux in the given energy band for a z-axis slice using 𝑁block = 8, +𝑁side = 4, 𝜂𝑅 = 2. +15 +10 +5 +0 +5 +10 +15 +y (pc) +Density +1.33 keV +1.78 keV +15 +10 +5 +0 +5 +10 +15 +y (pc) +2.37 keV +3.16 keV +4.22 keV +10 +0 +10 +x (pc) +15 +10 +5 +0 +5 +10 +15 +y (pc) +5.62 keV +10 +0 +10 +x (pc) +7.50 keV +10 +0 +10 +x (pc) +10.00 keV +0 +2 +4 +nH (cm +3) +18.0 +17.5 +17.0 +16.5 +16.0 +15.5 +15.0 +14.5 +14.0 +Fi (erg cm +2 s +1) +Figure 9. Same as Figure 8, but with 𝑁block = 8, 𝑁side = 8 and 𝜂𝑅 = 4. +100 +101 +zsrc (pc) +0 +10 +18 +10 +17 +10 +16 +10 +15 +10 +14 +e (erg/cm3) +1.33 keV +1.78 keV +2.37 keV +3.16 keV +4.22 keV +5.62 keV +7.50 keV +10.00 keV +Fiducial +High Res. +Fiducial +High Res. +100 +101 +102 +103 +104 +Hydrogen Density (cm +3) +Figure 10. X-ray energy density versus distance along the z-axis from the +source for the shadow test. The solid line uses the ray resolution in Figure 8 and +the dashed-dot uses the ray resolution in Figure 9. The dotted red line shows +the hydrogen nuclei density highlighting the location of the high-density blob. +clearly show that the disk midplane is left relatively unheated by the +X-rays, although the X-rays become important in the cavity and outer +disk regions. In particular, most of the cavity exhibits very warm gas, +even with only X-ray emission included, due to the rapid absorption +of soft X-ray emission. The cavity heats to temperatures exceeding +104 Kelvin, potentially becoming bright in hydrogen recombination +lines. The inclusion of EUV radiation will heat the diffuse gas further, +along with further ionizing the surrounding low-density cavity. +5 MOLECULAR CLOUD +We present an example application for XRayTheSpot, to demon- +strate how all the different TreeRay energy bands work together: a +virialized, magnetized turbulent cloud. We consider a 2 pc region of +a molecular cloud resolved with 2563 cells. We produce an initial +turbulent field by stirring the domain with a flat power spectrum +between the largest wave modes 𝑘 = 1...3 for 10 crossing times at +a velocity dispersion of 0.72 km s−1, consistent with the observed +linewidth-size relationship (McKee & Ostriker 2007). During the stir- +ring, we use periodic boundary conditions and chemistry to achieve +more accurate initial conditions for the abundances before collapse. +The choice of stirring for 10 crossing times is to ensure the chemistry +has reached a more quiescent state, with the kinetic energy spectrum +generally being reached after two crossing times (Federrath et al. +2010). We assume the cloud is nearly virialized, such that the virial +parameter +𝛼 ≡ 5𝜎2𝑅 +𝐺𝜌𝐿3 = 2 +(23) +where 𝑅 = 𝐿 is the box length, resulting in 𝜌 = 5 × 10−21 (g cm−3) +and a total box mass of 𝑀 = 590 M⊙. Before stirring, we initialize +a magnetic field in the 𝑧-axis with a magnitude such that the plasma +beta, +𝛽 ≡ +𝜌𝑐2𝑠 +𝐵2/8𝜋 = 103. +(24) +After the turbulence is initialized, gravity and source particles (stars) +are turned on and the boundary conditions are changed to “diode” +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +11 +1019 +1020 +1021 +Hydrogen Column Density +101 +102 +103 +104 +Temperature (K) +Cloudy +Flash +1019 +1020 +1021 +Hydrogen Column Density +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +Abundances +H/Htot +2H2/Htot +Figure 11. Flash vs Cloudy benchmark. Left: Temperature versus hydrogen column density from the central point source for Flash (black) and Cloudy +(blue). Right: Atomic (solid) and molecular (dashed) hydrogen abundances versus total hydrogen column density from the source, where Htot = H+ + H + 2H2. +t = 100 yr +10 +20 +10 +19 +10 +18 +10 +17 +10 +16 +10 +15 +10 +14 +102 +103 +104 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +10 +23 +10 +22 +10 +21 +10 +20 +10 +19 +10 +18 +10 +17 +10 +29 +10 +28 +10 +27 +10 +26 +10 +25 +10 +24 +10 +23 +10 +10 +10 +9 +10 +8 +10 +7 +10 +6 +Figure 12. Protostellar disk example case usage. Top row: Slice plots at 𝑧 = 0 for the density (left), gas temperature (middle) and 1.17 keV radiation energy +density. Bottom row: X-ray heating rate, H𝑥 (left), H𝑥/n diagnostic term (middle) and 6.56 keV radiation energy density. +such that gas can flow out of the domain. During the simulation, the +cloud is irradiated by an FUV radiation field of 𝜒 = 1.7 in units +of the Habing field (Habing 1968). The simulation is run using the +chemistry described above, and all TreeRay modules: +• OpticalDepth for the external radiation field (Wünsch et al. +2018). OpticalDepth solves for the column density from a cell to +the external boundary and attenuates a prescribes external radiation +flux (𝜒 = 1.7). In this study, it is only used for the FUV radiation, +while (Mackey et al. 2019) implemented the ability to include an +impinging X-ray flux. +• OnTheSpot for the EUV emission (Wünsch et al. 2021). This +module solves for UV-ionizing radiation from arbitrary sources and +MNRAS 000, 1–14 (2022) + +12 +Gaches et al. +iterates to convergence. The UV photon flux is coupled to the ther- +mochemistry to model photochemistry. +• RadPressure to account for the thermal radiation and radiation +pressure (Klepitko et al. 2022). This module enables the inclusion +of thermal radiation from point and diffuse sources and the resulting +radiation pressure. The thermal radiation is included in the chemistry +through radiative dust heating. +• XRayTheSpot, described above. +Sink particles representing protostars are injected when the den- +sity exceeds 𝜌thresh ≥ 4.59×10−18 g cm−3. Further criteria are used: +there are checks to ensure a local gravitational potential and a con- +verging flow. The protostar evolution follows the Offner et al. (2009) +model and implemented in Flash (Klepitko et al. 2022). Protostellar +emission consists of the intrinsic and accretion luminosities, where +the total accretion luminosity is +𝐿acc = 𝑓acc +𝐺𝑀∗ �𝑀∗ +𝑅∗ +, +(25) +where 𝑀∗ is the mass of the protostar, +�𝑀∗ is the accretion rate, +𝑅∗ is the protostar’s radius and we take 𝑓acc = 0.33. The X-ray +spectrum was computed by assuming hot-spot accretion, described +above, which provides the temperature and the density of the accre- +tion shocks near the protostellar surface (Calvet & Gullbring 1998; +Hartmann et al. 2016) and a single temperature plasma model (Ray- +mond & Smith 1977). Due to the low resolution, we set a minimum +of �𝑀∗ = 10−9 M⊙ yr−1. This is needed since when the protostar +particles first form, the burst of accretion blows out HII regions, and +the low resolution inhibits resolving the proper structure around the +cores. The infrared to EUV spectrum, used for RadPressure and +OnTheSpot is computed assuming the emission is composed of two +blackbodies: one for the intrinsic spectrum of the protostar at the +photosphere, such that +𝑇∗ = +� +𝐿∗ +4𝜋𝜎sb𝑅2∗ +�1/4 +, +(26) +which is provided by the protostellar evolution model, and another +assuming the accretion luminosityisreprocessedprimarily as a black- +body with temperature 𝑇acc, such that +𝑇acc = +� +𝐿acc +4𝜋𝜎sb𝑅2∗ +�1/4 +, +(27) +where 𝜎sb is the Stefan-Boltzmann constant. Therefore, the total +infrared luminosity from the protostar is described as +𝐿∗,IR = 𝑓∗,IR(𝑇∗)𝐿∗ + 𝑓acc,IR(𝑇acc)𝐿acc +(28) +and the EUV luminosity as +𝐿∗,EUV = 𝑓∗,EUV(𝑇∗)𝐿∗ + 𝑓acc,EUV(𝑇acc)𝐿acc +(29) +where 𝑓IR(𝑇) and 𝑓UV(𝑇) are the fraction of the blackbody emis- +sion in each of these bands (𝐸 < 13.6 eV and 13.6 eV ≤ 𝐸 ≤ 100 +eV, respectively). The X-ray emission was computed assuming the +“hot-spot” model, described above. While there may be some double +counting of emission by treating the total spectrum in the two differ- +ent methods, we find this impact is marginal as the X-ray emission +generally accounts for only a small fraction (≤ 10%) of the total +protostellar luminosity. +Figure 13 shows the column density, and density-weighted projec- +tions of the gas and temperature, radiation temperature, EUV photon +density and X-ray energy densities after ≈ 1 Myr of evolution with +gravity. The star formation, as traced by heated knots of gas, is oc- +curring along a main filament structure. The high temperatures here +are primarily caused by the EUV photons, which are rapidly ab- +sorbed in the nearby gas. The X-ray emission is found to be highly +absorbed along the main filament structure, and instead traces out the +more diffuse turbulent structure of the molecular cloud. As expected, +higher energy X-ray bands showcase more extended emission with +the brightest emission in the 3.7 keV band. In all X-ray bands, the tur- +bulent structure of the molecular cloud is seen in the density-weighted +integrated emission. This case study highlights the new capabilities +of including protostellar radiative feedback from infrared to X-ray in +star formation simulations. +6 DISCUSSION/FUTURE WORK +We have presented the new X-ray radiation transfer module, +XRayTheSpot using the reverse ray-tracing scheme TreeRay im- +plemented in Flash (Wünsch et al. 2021). XRayTheSpot enables +an arbitrary number of point or diffusive sources of X-ray emission, +and an arbitrary number and position of energy bins. The module +uses temperature dependent cross sections assuming gas in thermal +ionization equilibrium. However, the module is flexible enough such +that the user can provide their own cross section data to be used. The +module produces the expected behavior for X-ray point sources and +shadow tests and is able to reasonably reproduce the thermochemistry +compared to Cloudy, despite the significantly simpler treatment of +X-ray chemistry and grain-processes in Flash. +We demonstrated the utility of this module with two example sci- +ence cases focusing on protostellar X-ray emission. First, we mod- +elled the emission of an 0.7 M⊙ protostar with an accretion rate of +10−9 M⊙ yr−1 through a protostellar disk. We find that soft X-rays +are rapidly absorbed at the disk surfance, with most of the emission +escaping through the outflow cavity. However, harder X-rays are able +to permeate the disk due to their significantly lower optical depth. +The X-ray heating was also strong within the outflow cavity, with no +X-ray heating towards the midplane of the disk, as expected. Second, +we perform a low-resolution star formation simulation of a turbulent +molecular cloud. In this simulation, protostars are self-consistently +formed and the X-ray emission modelled on the fly. This simulation +includes the entire range of different TreeRay radiation modules: +diffuse FUV (OpticalDepth Wünsch et al. (2018)), EUV (OnTheS- +pot Wünsch et al. (2021)), thermal radiation and radiation pressure +(RadPressure Klepitko et al. (2022)) and X-ray emission from 1 +keV to 10 keV. Since the X-ray emission in the simulation comes en- +tirely from accretion onto the protostars, the X-ray emission is highly +variable. Due to the lower resolution and the inclusion of ionizing +radiation, the accretion occurs in bursts followed by the expansion of +HII regions, which cut off accretion. With higher resolution, accre- +tion may still be able to occur through disks, instabilities and more +porous density structures. In future work, we will perform higher +resolution simulations to model star formation including chemistry +and radiation feedback across the electromagnetic spectrum. +In this work, we focus primarily on point sources. However, +XRayTheSpot makes no differentiation between point sources ver- +sus extended more diffusion emission. Future studies will include +diffuse X-ray emission from cooling hot gas and shocked gas. Our +module currently includes the computation of X-ray emission from +accretion onto protostars, and future work will include X-ray models +for more types of point sources such as X-ray binaries. The module +presented in this work will allow the first-generation of simulations +of star formation and galaxies with the inclusion of a wide range of +X-ray sources. +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +13 +t = 1.08 Myr +10 +2 +10 +1 +100 +101 +102 +10 +15 +20 +25 +30 +8 +10 +12 +14 +16 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +10 +22 +10 +21 +10 +20 +10 +19 +10 +18 +10 +17 +10 +19 +10 +18 +10 +17 +10 +19 +10 +18 +10 +17 +10 +16 +10 +18 +10 +17 +10 +16 +Figure 13. Panel plots highlighting the features of a 2 pc piece of a molecular cloud after 𝑡 = 1.08 Myr of evolution. For all fields except the column density, the +panel is showing the density-weighted projection. All projections are along the z-axis. The figure shows a simulated molecular cloud after 1 Myr of gravitational +evolution including protostar sink particles and radiation feedback from infrared to X-rays. While the EUV radiation is rapidly absorbed (indicated by the black +background color), the infrared and X-ray emission is able to penetrate much further into the cloud. +ACKNOWLEDGEMENTS +BALG and SWG acknowledges support by the ERC starting grant +No. 679852 ‘RADFEEDBACK’. SWG and BALG thank the German +Science Foundation (DFG) for funding through SFB956 project C5. +We also thank the Regional Computing Center Cologne (RRZK) +for hosting our HPC cluster, Odin, on which the simulations have +been performed. RW acknowledges the support by project 20- +19854S of the Czech Science Foundation and by the institutional +project RVO:67985815. JM acknowledges support from a Royal +Society-Science Foundation Ireland University Research Fellowship +(20/RS-URF-R/3712) and an Irish Research Council Starting Lau- +reate Award (IRCLA\2017\83). The authors thank Andre Klepitko +for many helpful discussions. Andre Klepitko also implemented the +protostellar evolution model into the code. 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We turn off most +induced and grain processes and set the abundances for most metals +to zero to better match the methods used in our Flash simulations. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–14 (2022) + +XRayTheSpot: X-raying Molecular Gas +15 +1 t i t l e XDR source +2 ## r a d i a t i o n +s o u rces +3 CMB +4 t a b l e SED " plaw . sed " +5 l u m i n o s i t y +35 range +73.5 +to +735 Ryd +6 ##Geometry +7 r a d i u s +16.1938 +8 hden +3.0 +9 sphere +10 ## Stopping +and +i t e r a t e +11 stop H2 column +d e n s i t y +24.0 +12 stop +t e m p e r a t u r e +l i n e a r +3.0 +13 i t e r a t e +to +convergence +14 ##ISM and +Grain +physics +15 cosmic +ray +r a t e +−16.523 +16 abundances ISM +17 g r a i n s ISM no +qheat +0.56 +18 no +g r a i n x−ray +t r e a t m e n t +19 no induced +p r o c e s s e s +20 no +r a d i a t i o n +p r e s s u r e +21 no +s c a t t e r i n g +o p a c i t y +22 no +g r a i n +p h y s i c s +23 no +g r a i n +molecules +24 no +l i n e +t r a n s f e r +25 ##Abundances +26 element +carbon +abundance +−3.853872 +27 element +helium +abundance −1 +28 element +oxygen +abundance +−3.494850 +29 element +s i l i c o n +abundance −7 +30 element +n i t r o g e n +o f f +31 element +s u l p h u r +o f f +32 element +neon +o f f +33 element +aluminium +o f f +34 element +phosphor +o f f +35 element +c h l o r i n e +o f f +36 element +argon +o f f +37 element +calcium +o f f +38 element +chromium +o f f +39 element +n i c k e l +o f f +40 element +l i t h i u m +o f f +41 element +b e r y l l i u m +o f f +42 element +f l u o r i n e +o f f +43 element +potassium +o f f +44 element +scandium +o f f +45 element +t i t a n i u m +o f f +46 element +vanadium +o f f +47 element +manganese +o f f +48 element +c o b a l t +o f f +49 element +copper +o f f +50 element +zinc +o f f +51 ## output +52 save +overview +l a s t +" xdr . ovr " +53 save +molecules +l a s t +" xdr . mol " +54 save +abundances +l a s t +" xdr . abund " +55 save +continuum +l a s t +" xdr . cont " +56 save PDR l a s t +" xdr . pdr " +Listing 1: Input file for Cloudy benchmark +MNRAS 000, 1–14 (2022) + diff --git a/1NFQT4oBgHgl3EQfETVZ/content/tmp_files/load_file.txt b/1NFQT4oBgHgl3EQfETVZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b9fa09495f8830fcf7796f4b9dea377b9e4f23c --- /dev/null +++ b/1NFQT4oBgHgl3EQfETVZ/content/tmp_files/load_file.txt @@ -0,0 +1,1270 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf,len=1269 +page_content='MNRAS 000, 1–14 (2022) Preprint 1 February 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Tree-based solvers for adaptive mesh refinement code FLASH - IV: An X-ray radiation scheme to couple discrete and diffuse X-ray emission sources to the thermochemistry of the interstellar medium Brandt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Gaches,1,2★ Stefanie Walch,1,3 Richard Wünsch4 and Jonathan Mackey5 1I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, 50937, Köln, Germany 2Department of Space, Earth and Environment, Chalmers University of Technology, Gothenburg SE-412 96, Sweden 3Center for Data and Simulation Science (CDS), University of Cologne, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='de, Germany 4Astronomical Institute, Czech Academy of Sciences, Bo˘ciní II 1401, 141 00 Prague, Czech Republic 5Centre for AstroParticle Physics and Astrophysics, DIAS Dunsink Observatory, Dunsink Lane, Dublin 15, Ireland Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' in original form ZZZ ABSTRACT X-ray radiation, in particular radiation between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 keV and 10 keV, is evident from both point-like sources, such as compact objects and T-Tauri young stellar objects, and extended emission from hot, cooling gas, such as in supernova remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray radiation is absorbed by nearby gas, providing a source of both heating and ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While protoplanetary chemistry models now often include X-ray emission from the central young stellar object, simulations of star-forming regions have yet to include X-ray emission coupled to the chemo-dynamical evolution of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We present an extension of the TreeRay reverse raytrace algorithm implemented in the Flash magneto-hydrodynamic code which enables the inclusion of X-ray radiation from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 keV < 𝐸𝛾 < 100 keV, dubbed XrayTheSpot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' XrayTheSpot allows for the use of an arbitrary number of bins, minimum and maximum energies, and both temperature-independent and temperature-dependent user-defined cross sections, along with the ability to include both point and extended diffuse emission and is coupled to the thermochemical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We demonstrate the method with several multi-bin benchmarks testing the radiation transfer solution and coupling to the thermochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Finally, we show two example star formation science cases for this module: X-ray emission from protostellar accretion irradiating an accretion disk and simulations of molecular clouds with active chemistry, radiation pressure, protostellar radiation feedback from infrared to X-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Key words: astrochemistry -– radiative transfer -– methods:numerical — ISM:clouds -– X-rays: general — X-rays: ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1 INTRODUCTION Molecular gas is subjected to radiation across the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Hard radiation, such as X-ray and gamma-ray radiation, can penetrate deep into molecular gas and drive the thermochemistry of dense gas (Spitzer & Tomasko 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Maloney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Yan 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Wolfire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' X-rays provide an important source of ion- ization in dense gas, driving the ion-neutral chemistry and providing heating through photo-electrons (Lepp & Shull 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Maloney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Dalgarno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Using the typical molecular gas photoab- sorption cross sections (Maloney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1996), the 𝜏 = 1 surface for 1 keV photons is approximately 4 × 1021 cm−2 (compared to ≈ 10−18 cm−2 for UV radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, the photoabsorption cross sections scale roughly as 𝐸−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 (Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019), so harder radiation pen- etrates much further into the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Therefore, in regions near bright X-ray sources, the cloud structure can become dominated through- out by the X-ray radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Regions in which the thermochemistry is regulated primarily through X-ray radiation are often denoted as X-ray Dominated Regions (XDRs) (coined by Maloney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1996), in analogue to photo-dissociation regions (PDRs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' ★ E-mail: gaches@ph1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='uni-koeln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='de (BALG) X-ray radiation drives ionization primarily through secondary, in- duced processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While the primary ionization cross sections are low, the resulting ejected fast electrons can produce a cascade of secondary ionizations and pumped far ultraviolet (FUV) radiation through the excitation and subsequent de-exictation of H and H2 (Prasad & Tarafdar 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Dalgarno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink & Spaans 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These fast electrons can also provide heating through pho- toelectric heating of dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In this way, X-ray radiation acts in a very similar manner as cosmic rays, and untangling the two contributions can be difficult (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Molecular clouds are immersed in a bath of X-ray radiation, with contributions from both external and internal sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Externally, molecular gas can be irradiated through supernovae and their rem- nants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Yamane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Brose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2022), X-ray binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Remillard & McClintock 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Reig 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Mineo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Lutovinov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Giacobbo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2018), nearby activate galactic nuclei (AGN) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Sunyaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Sun- yaev & Churazov 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Churazov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Mingozzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Cruz-González et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Internally, young stellar objects, including embedded accreting protostars and more evolved T-Tauri stars (Calvet & Gullbring 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Feigelson & Mont- merle 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Feigelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2007), and high-mass stars just reaching © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='13237v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='IM] 30 Jan 2023 2 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' the main sequence can become X-ray bright (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Cassinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1994), whether through accretion or magnetic powered radiation or coronal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Finally, gas heated through feedback processes, such as winds and supernovae, can become warm enough to emit X-ray radiation while they cool (Raymond & Smith 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Observa- tional X-ray surveys of molecular gas and star-forming regions show substantial amounts of diffuse emission and a sizable number of point sources (Sunyaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Feigelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Townsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2014, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Despite their potential importance, their inclusion into simulations of molecular clouds has been sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' There has been substantial focus on thermochemical models of protoplanetary disks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Glassgold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Igea & Glassgold 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2008a, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Owen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Cleeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Picogna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Waggoner & Cleeves 2019) and models of molecular gas near external sources or compact objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Krolik & Kallman 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Lepp & McCray 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Draine & Woods 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' García-Burillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Hocuk & Spaans 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Odaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These methods typically utilize Monte Carlo methods (Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2008a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Odaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Molaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Walls et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Cleeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2017), or ray-trace schemes and focus primarily either on the inclusion of point sources or external radiation fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Wise & Abel 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Khabibullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2020) In this paper, we will present an X-ray extension of the reverse ray tracing scheme TreeRay (Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2021), which allows for the inclusion of an arbitrary number of point sources and diffuse radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The module is a TreeRay extension of the diffuse X-ray module presented in Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2019), which enabled diffuse X- ray irradiation at the domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Our implementation enables up to 100 energy bins at arbitrary locations between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 and 100 keV and temperature-dependent photoabsorption cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In Section 2 we give an overview of the X-ray TreeRay algorithm, called XRayTheSpot, and the coupling of it to the X-ray-driven chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In Section 3 we show the performance of the module with different radiation transfer tests and a benchmark against the Cloudy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In Sections 4 and 5 we demonstrate the use of this module for protostellar emission irradiating a surrounding disk and in a star formation simulation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Finally, in Section 6 we discuss the future extensions and scientific applications of XRayTheSpot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2 METHODS Our new XRayTheSpot module is able to treat radiation from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 keV to 100 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We describe below in detail the adopted pho- toabsorption cross sections and the module’s implementation within TreeRay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 XRay Cross Sections X-ray radiation is attenuated as it propagates through gas via a com- bination of photoionization, at lower energies, and the Compton pro- cess, at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The previous module, described in Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2019), used the low-energy approximation for the X-ray cross section, 𝜎𝑥, from Panoglou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2012): 𝜎𝑥 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='27 × 10−22𝐸−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='485 𝛾 cm2 (1) per H-nucleus, where 𝐸𝛾 is the photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, this cross section is valid only for cold, neutral gas and for solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We include now, as input during run time, temperature-dependent cross sections, which can be re-computed for problems with different metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For photoionization, we use the analytic fits from Verner & Yakovlev (1995), 𝜎pi, where 𝜎pi = 𝜎0𝐹(𝐸𝛾/𝐸0), (2) where 𝐹(𝑦) = � (𝑦 − 1)2 + 𝑦2 𝑤 � 𝑦−𝑄 � 1 + √︁ 𝑦/𝑦𝑎 �−𝑃 , (3) 𝑦 = 𝐸𝛾/𝐸0, 𝑄 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 + 𝑙 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5P, 𝑙 = 0, 1, 2 is the subshell orbital quantum number, and 𝜎0, 𝐸0, 𝑦𝑤, 𝑦𝑎 and P are fit parameters from the associated public ViZieR catalog1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, to utilize these cross sections, the ionization level populations must be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We assume collisional ionization equilibrium and use the ChiantiPy package (Dere 2013), using version 9 of the Chianti atomic database (Dere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1997, 2019) to compute the ionization fraction as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 1 shows the equilibrium ionization fractions as a function of temperature for the 15 different elements (see Table 1) we include in the cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These computations show that, particularly for 𝑇 > 105 K, there are multiple ionization states for metals which contribute to the photoionization cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We also include the cross section for the Compton effect, which becomes particularly important at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We use the total Klein-Nishina (KN) cross section (Klein & Nishina 1929;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Longair 2011), 𝜎KN, 𝜎KN = 𝜋𝑟2 𝑒𝑥−1 �� 1 − 2(𝑥 + 1) 𝑥2 � ln(2𝑥 + 1) + 1 2 + 4 𝑥 − 1 2(2𝑥 + 1)2 � , (4) where 𝑟𝑒 is the classical electron radius and 𝑥 = 𝐸𝛾/(𝑚𝑒𝑐2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While most applications will be in the limit of Thomson scattering, we include the full Compton cross sections to enable more flexibility in the choice of energy bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In the cross-section plots, we show the total Compton cross-section weighted by the number of free electrons contributed by each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The total cross section, 𝜎𝑥, is thus 𝜎𝑥(𝐸) = 𝑁elem ∑︁ 𝑖 𝑥𝑖𝜎pi,i(𝐸) + 𝜎KN,i(𝐸), (5) where 𝑥𝑖 is the abundance of element 𝑖 with respect to hydrogen and the sum is carried out including the cross sections for the 𝑁elem ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Table 1 shows the elements we include in the photoabsorption cross section and their fiducial abundances relative to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 2 shows the photoionization cross sections and free-electron contributions to the Compton cross section as a function of energy for gas with𝑇 = 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' As shown, for some elements, the Compton cross section becomes more important than photoionization, in particular for hydrogen and helium above 1 keV, and for carbon and oxygen above 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For hydrogen, the Compton effect is dominant due to the negligible neutral fraction at T = 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 3 shows the total cross section as a function of energy for 𝑇 = 105 K and each of the total elemental contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Here, the X-ray photoabsorption cross section is dominated by helium (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 keV), then carbon (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 keV) and oxygen (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='8−4 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' At energies above 4 keV, the hydrogen and helium Compton cross sections dominate with a contribution from the iron photoionization cross section around 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, many of the metals contribute equally to the total cross section around 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Finally, Figure 4 shows 𝜎𝑥 as a function of energy and temperature from 𝑇 = 104 to 107 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' At low temperatures, we recover the analytic 1 https://cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='fr/viz-bin/cat/J/A+AS/109/125 MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Elements included in our photoabsorption cross section calculation and their fiducial abundances, 𝐴𝑋, reported as 𝐴𝑋 = log(𝑁𝑥/𝑁𝐻) + 12 (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Element Abundance (𝐴𝑋) H 12 He 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='986 C 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='443 O 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='783 N 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='913 Ne 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='103 Na 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='353 Mg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='593 Al 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='523 Si 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='573 S 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='193 Ar 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='553 Ca 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='383 Fe 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='503 Ni 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='283 cross section previously used, although around 10 keV there is an in- crease in the cross section due to iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, the rather significant temperature dependence highlights the necessity of including a tem- perature dependent cross section: at high temperatures, the higher thermal ionization state leads to a reduction of nearly two orders of magnitude in the cross section, thereby making the gas significantly more optically thin to the X-ray radiation, producing less heating and enabling more X-rays to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Below 104 K, the cross section does not noticeably change since hydrogen is not significantly collisional ionized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Therefore, for the results of this paper, for colder gas, we use the 𝑇 = 104 photoabsorption cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The module though allows for the user to define their own temperature dependent cross sections across any temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For a given set of energy bins, {(𝐸𝑙,𝑖, 𝐸𝑟,𝑖)}, where 𝑖 = 1, 𝑁bin for 𝑁bin bins, we define: 𝐸𝑐,𝑖 = 1 2 �𝐸𝑙,𝑖 + 𝐸𝑟,𝑖 � , (6) where 𝐸𝑙,𝑖 is the left bound of the 𝑖th bin, 𝐸𝑟,𝑖 is the right bound, and 𝐸𝑐,𝑖 is the midpoint of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We derive bin-averaged cross sections, such that exp � − ⟨𝜎𝑋,𝑖⟩ 𝜎𝑐 � = 1 𝐸𝑙,𝑖 − 𝐸𝑟,𝑖 ∫ 𝐸𝑟,𝑖 𝐸𝑙,𝑖 exp � −𝜎𝑥(𝐸𝛾) 𝜎𝑐 � 𝑑𝐸, (7) where ⟨𝜎𝑥,𝑖⟩ is the bin-averaged cross section for bin, 𝑖, and 𝜎𝑐 = 𝜎𝑥(𝐸𝑐,𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Our fiducial tests use 𝑁bin = 8 between 1 – 10 keV using logarithmically spaced bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' All of these cross section data, and the initialization and storage of the bins and bin-averaged cross sections are kept in a new Flash module, XrayCommon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Flash is a highly module public magneto- hydrodynamic code (Fryxell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2000) written in Fortran and highly-scalable with MPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The scripts necessary to compute the X- ray cross sections are publicly available on GitHub2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This module enables the coupling of X-ray physics to multiple other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Plasma models and the necessary X-ray data are also stored within this module, as a unified location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 TreeRay TreeRay is a novel reverse ray tracing scheme, described fully in 2 � https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='com/AstroBrandt/XRayCrossSections Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2021), implemented in Flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Simply, TreeRay enables an efficient method to compute the contributions of radiation from every cell, for every cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' It does so through the combination of a reverse ray-trace algorithm with a tree (Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2018), which also currently is used in the gravity solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Below we describe briefly the different aspects of the TreeRay algorithm and XRayTheSpot extension and refer the reader to Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 Building the Tree The foundation of the TreeRay algorithm is an octtree which stores all necessary variables for the various TreeRay modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' At min- imum, the tree stores the mass and center of mass coordinates for the respective cell, or leaf, or higher nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For XRayTheSpot, two further quantities are stored onto the tree: the bin-integrated X-ray luminosity in each energy bin and the gas temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While the bin-integrated X-ray luminosity is purely additive, the temperature is stored as a mass-weighted average of each set of eight sub-nodes (or leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 Ray Structure Before the tree walk is executed for a given cell, rays are generated by casting 𝑁pix = 12𝑁2 side rays from each cell using directions defined by the HealPix (Górski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2005) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' HealPix tessellates the unit sphere into areas representing equal solid angles with a unit vector pointing to the center of each of these surface areas from the sphere’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' TreeRay allows for 𝑁side = 1, 2, 4, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=', with higher values representing higher angular resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The rays are split into 𝑁𝑟 evaluation points, set by the grid resolution, Δ𝑥, the allowed length of the ray, 𝐿ray, which is set to three-dimensional diagonal of the computational domain, and a free parameter, 𝜂𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Along each ray, the radial coordinate point of the ith evaluation point is 𝑟𝑖 = Δ𝑥𝑖2 2𝜂2 𝑅 , (8) leading to segments with increasing lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This behavior coincides well with the geometric acceptance criterion described below for deciding whether or not to accept a tree node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The total number of evaluation points is 𝑁𝑅 = 𝜂𝑅 × floor �√︂ 2𝐿ray Δ𝑋 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 Tree Walk The mapping of the cells/nodes onto the rays requires two factors: a multipole acceptance criterion (MAC) and a weighting function to map from the tree onto the different radial evaluation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' When the MAC is met, a node is accepted and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The simplest MAC is the Barnes-Hut (BH) geometric MAC (Barnes & Hut 1986), where a node of size ℎ𝑛, at a distance 𝑑, from the cell is opened if ℎ𝑛/𝑑 < 𝜃lim (10) where 𝜃lim is a user-defined opening angle with a sensible choice being 𝜃lim = √︃ 4𝜋/𝑁pix3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We also utilize the ‘Src MAC’ (Wünsch 3 The resulting 𝜃lim for 𝑁side = 1, 2, 4, 8 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='125, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For the results of this paper, we adopted these recommended values of MNRAS 000, 1–14 (2022) 4 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Ionization Fraction I II I II III IV V VI I II III IV V VI VII VIII H He C O 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Ionization Fraction I II III IV V VI VII VIII IX X XIXIIXIIIXIV XV I II III IV V VI VII VIII IX X XI XIIXIII XIV XV I II III IV V VI VII VIII IX X XI I II III IV V VI VIIVIII IX X XI XII S Fe Na Al 104 105 106 107 Temperature (K) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Ionization Fraction I II III IV V VIVII VIII IX X XI XIIXIIIXIVXV I II III IV V VI VIIVIIIIX X XI XIIXIIIXIVXV I II III IV V VI VII VIII IX X XI XIIXIII XIV XV Ar Ca Ni Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Ionization fraction for different elements as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Annotated in the text are the peaks of different ionization levels for each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 5 10 15 10 12 10 9 10 6 10 3 (Mbarn/H-nucleus) Photoionization Compton H He C O 10 10 10 8 10 6 10 4 (Mbarn/H-nucleus) N Ne Si Mg 10 11 10 9 10 7 10 5 (Mbarn/H-nucleus) S Fe Na Al 102 103 104 105 Energy (eV) 10 10 10 8 10 6 (Mbarn/H-nucleus) Ar Ca Ni Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Photoionization (solid) and Compton process (dashed) cross sec- tions for each element as a function of energy, assuming thermal ionization equilibrium at 𝑇 = 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Each elemental contribution is weighted by the assumed abundance with respect to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2021), where a node with sources is opened if ℎ𝑛/𝑑 < 𝜃src, (11) where 𝜃src is a user-defined parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Quantities on the tree are mapped onto the radial evaluation points of a ray through the use of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We utilize both a piece-wise third-order polynomial, 𝑊𝑝(𝛿), and a kernel derived to ensure it meets the requirements of the radiation transfer equation, 𝑊 𝑓 (𝛿), where 𝛿 = (𝑟𝑖 − 𝑑)/ℎ𝑛 and 𝑑 is the distance from the node center of mass and the ray evaluation point (see Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2021) and ℎ𝑛 is the node’s linear size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The node quantites are weighted by the overlap of the volume of the ray segment and the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 𝜃lim for the corresponding 𝑁side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' See Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2018) for 𝜃lim resolution tests in the context of TreeRay/OpticalDepth 102 103 104 Energy (eV) 10 10 10 8 10 6 10 4 10 2 tot (Mbar/H-nucleus) H He C O N Ne Si Mg S Fe Na Al Ar Ca Ni Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Total photo-absorption cross section (black) with each element contribution highlighted (colors) as a function of energy for gas at temper- ature, 𝑇 = 105 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Each elemental contribution is weighted by the assumed abundance with respect to hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 102 103 104 105 Energy (eV) 10 6 10 5 10 4 10 3 10 2 10 1 Cross section (Mbarn) 103 104 Energy (eV) 10 6 10 5 10 4 Cross section (Mbarn) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Temperature Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Total photoabsorption cross section as a function of energy and temperature (color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The black dashed line shows the previously used analytic cross section from Panoglou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Inset: Zoom-in to 1 – 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Following the tree walk, the rays from a cell outward store the mass, center of mass, gas temperature and bin-integrated luminosi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These provide all the necessary information to solve the equation of radiation transfer along each ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='4 Solving the Radiation Transfer Equation Along the rays, the one-dimensional radiation transfer equation is solved: 𝑑𝐼𝜈 𝑑𝑠 = −𝜖𝜈 + 𝛼𝜈𝐼𝜈 (12) where 𝑠 is the distance along the ray, and 𝜖𝜈 and 𝛼𝜈 are the emission and absorption coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The band-integrated flux, 𝐽(𝐸), irradiat- MNRAS 000, 1–14 (2022) 6 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' ing a cell 𝑖 due to the band-integrated luminosity, 𝐿𝑋, emitting from node 𝑗 can be simply written 𝐽 𝑗𝑖(𝐸) = 𝐿𝑋, 𝑗 (𝐸) 𝑒−𝜏𝑥 4𝜋𝑟2 𝑖 𝑗 (13) where 𝑟𝑖 𝑗 is the distance between the centers of cell 𝑖 to node 𝑗 and 𝜏𝑥 = ∫ 𝑛H(𝑠)𝜎𝑥(𝐸,𝑇(𝑠))𝑑𝑠 ≈ ∑︁ 𝑘 𝜌𝑘 𝜇𝑚𝐻 ⟨𝜎𝑋 (𝐸,𝑇𝑘)⟩ 𝛿𝑠 (14) is the X-ray opacity between cell 𝑖 and node 𝑗 and 𝜌𝑘 is the density at evaluation point 𝑘 along the ray, 𝑛H is the hydrogen nuclei density, and 𝛿𝑠 = 𝑟𝑘 − 𝑟𝑘−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We store the solution as an energy density, 𝜀𝑥 = 𝐽/𝑐, where 𝑐 is the speed of light, onto the grid to be used in chemistry, described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The total energy density is the sum over the HealPix rays: 𝜀𝑖 = 𝑁pix ∑︁ 𝑘 𝜀𝑘𝑖(𝐸).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (15) For the solution, we also store the cell mass and temperature onto the tree and map these to the rays using 𝑊𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In our algorithm, no assumption is made with respect to what produces the X-ray energy density, enabling both point sources (with their radiation spread over their host cells) and diffuse emission produced via cooling of hot gas in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 Pre-existing Chemistry We briefly describe here the previous treatment of X-ray radiation within the chemical network (see Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2019) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The chemical network consists of 17 species, of which 9 are solved numerically and the rest are followed through conservation equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We solve the non-equilibrum species H+, H2, C+, CO, HCO+, CHx, OHx, He+ and M+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' CHx is a proxy species for simple hydrocarbons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' CH, CH2, etc, and simple ions CH+, CH2+, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Similarly, OHx is a proxy for OH, H2O and ions OH+, H2O+, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' M is a proxy for metals that can become the primary source of electrons in shielded regions of molecular clouds, where reaction rates treat M as Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The network is primarily based on the ‘NL99’ network of Glover & Clark (2012), which uses the hydrogen chemistry from Glover & Mac Low (2007a,b) with the CO chemistry of Nelson & Langer (1999) including updated reaction rates from Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For this work, all photodissociation rates have been updated using the KIDA astrochemistry database (Wakelam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' X-ray radiation is coupled to the thermochemistry through the following primary processes (see also Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019): Dust heating, following the analytic prescription in Yan (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Primary ionization of a species by X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Note though that is is relatively unimportant for our considered species, and plays a minor role in the heating and ionization for hydrogen species and helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Secondary ionization through collisional ionization by fast elec- trons produced following primary ionizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Dalgarno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink & Spaans 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Induced FUV radiation generated by H2, which is collisionally excited by fast electrons and the subsequent ionizations and dissoci- ations (Prasad & Tarafdar 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Gredel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Maloney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Meijerink & Spaans 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Coulomb heating of the gas via energy exchange between the produced fast electrons and other charged particles (Dalgarno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These processes have all been generalized for the arbitrary number of energy bins and the temperature-dependent cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The input X-rays are computed by the XRayTheSpot module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Since we use band-integrated radiative variables, the heating parameter for a particular cell 𝑖 due to the impinging X-ray radiation is 𝐻𝑥,𝑖 = 𝑁bin ∑︁ 𝑛 𝑗𝑖(𝐸𝑛) ⟨𝜎𝑥(𝐸𝑛,𝑇𝑖)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (16) 3 TESTS AND BENCHMARKING Here we show various numerical tests of the radiation transfer and a benchmark of the thermochemsitry against Cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For our bench- marks, we fiducially use 8 bins, logarithmically spaced between 1 - 10 keV, following Meijerink & Spaans (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 Point Source Test Our first test is a single central point source with a constant luminosity distribution, 𝐿𝑥,𝑛 = 1 L⊙ for all 𝑁bin bins, embedded in a volume with a uniform density of 𝑛(𝐻) = 2 × 103 cm−3 and a spatially constant temperature 𝑇 = 10 Kelvin in a (30 pc)3 volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We use a constant luminosity to better compare the solutions of different energy bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We then compute the radial profiles of the energy density and compare against the analytic solution: 𝐽(𝐸, 𝑟) = 𝐿𝑥(𝐸) 𝑒−𝜎𝑥 (𝐸)𝜌𝑟 4𝜋𝑟2 (17) where the energy density, 𝜖 = 𝐽(𝐸)/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 5 shows the performance of XRayTheSpot for a single bright point source as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The results in figure 5 used 𝜂𝑅 = 4, 𝑁side = 8 and 𝑁block = 8, where 𝑁block is the number of blocks of cells per spatial dimension, and one block consists of a cube of 83 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The radial range was chosen that for the lowest energy bin, the emission transitions from optically thin to strongly optically thick, with the maximum radius corresponding to 𝜏(𝐸 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17eV) ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The left panel shows the comparison between the ray trace solution and the analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These solutions agree well with each other with the lines largely overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The right panel shows the relative error, defined as 𝛿𝑐 = |𝑐𝜀 − 𝐽(𝐸, 𝑟)| 𝐽(𝐸, 𝑟) (18) where 𝜀 is the solution from XRayTheSpot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The relative error is rather insensitive to the optical depth but more sensitive to how strongly the radiation field is coupled to the gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' the magnitude of the photoabsorption cross section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The 10% error shown for the most optically thick bin at low energies is due to the mapping of the density structure onto the rays using the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For X-ray optical depths greater than 𝜏𝑥 ≈ 10, the error starts to increase towards unity, but at these energy densities, the X-rays have a negligible impact on the thermochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Therefore, these relative differences will have no discernible impact on the thermochemical evolution of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In order to highlight the differences of the new module with the previous cross section implementation presented in Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2019), we perform a second calculation imposing a strong temper- ature gradient such that the radial temperature profile is 𝑇(𝑟) = 5 × 105 [1 − tanh(𝑟 − 8 pc)] + 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (19) This temperature is artificial and chosen such that the X-ray radiation transitions from optically thin to optically thick due to the change in MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 7 𝑁block 𝑁side 𝜂𝑅 Initialization (s/proc) Evolution (s/proc) 4 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 4 4 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 4 4 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 4 8 2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 8 4 2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='4 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 8 8 4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Timing for the pont source test for the different runs in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Each row gives the model parameters of 𝑁block, 𝑁side and 𝜂𝑅 and the time in seconds per processor for the initialization of the tree and a ray trace step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 6 shows the result of this comparison and as expected, the emission for the lower energy bins is up to an order of magnitude greater than the low-temperature solution and maintains an 𝑟−2 trend until a much greater radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 7 shows the performance of XRayTheSpot for a range of parameters, exploring both low- and high- spatial and ray resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We find that grid resolution is the primary source of deviations at small radius, while the ray resolution increases the accuracy at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' At large distances from the source, the solution tends to slightly under predict for low ray and angular resolution due to overestimation of the column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' At small radii, the solution over-predicts the resulting flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For optically thin radiation bins, the solution almost exactly matches the analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Therefore, the deviations come about due to mapping the mass from the cells and tree nodes onto the rays using the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For science uses, the number of blocks, 𝑁block, is determined by the necessary resolution to resolve crucial gas dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' the Jeans length for gravity simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Increasing both 𝑁side and 𝜂𝑅, while producing more accurate radiation transfer solutions, leads to substantially higher computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Table 2 shows the compu- tational time per processor for the initialization and per ray trace step for the models in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Between the lowest and highest accuracy tests, (𝑁block, 𝑁side, 𝜂𝑅) = (4, 2, 2) and (8, 8, 4), respectively, the increase in cost of the initialization and ray trace was a factor of ≈ 40 and ≈ 130, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The time for the raytrace is dominated (≥ 95%) by the tree walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We find using 𝑁side = 4 is the best balance of time and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='2 Shadow Test Our next test is a shadow test to verify the solution of the solver when sources are placed near dense regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Here, we have a 𝐿𝑋 = 10 L⊙ source with a spectrum, 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2, placed near a dense core with a hydrogen-nuclei number density of 𝑛H = 103 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We consider radiation between 1 - 10 keV, moving from optically thick bands to optically thin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figures 8 and 9 show the results of this test, for both low- and high- ray resolution which use (𝑁block = 8, 𝑁side = 4, 𝜂𝑅 = 2) and (𝑁block = 8, 𝑁side = 8, 𝜂𝑅 = 4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For both cases, the test reveals the expected results that the low-energy X-rays are absorbed by the dense core and this creates a wide-angle shadow, while higher energy X-rays are barely attenuated, producing smaller to no shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The high-ray resolution test also shows the expected drop in ray-tracing artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 10 shows a one-dimensional cut along the z-axis from the source through the dense blob of gas for the two ray resolutions compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The figure shows that higher ray resolution leads to a smoother attenuation of the flux for the optically thick, lower energy bins while there is very little change for higher energy bins which are substantially less attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This is most pronounced for the 𝐸𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33 keV bin 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 Benchmark against Cloudy The final benchmark tests the X-ray radiation coupling to the ther- mochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We place a point source with a physical size of 1016 cm and total X-ray luminosity, 𝐿XR = 1036 erg s−1, and a luminosity spectrum 𝑑𝐿/𝑑𝐸 ∝ 𝐸−2 between 1 - 10 keV in a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 pc)3 volume filled with a gas number density of 𝑛H = 103 cm−3, 𝑁side = 4, and 𝜂𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' and compare with a one-dimensional model using the Cloudy code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The volume and resolution were chosen such the inner XDR is resolved by ≈ 20 cells while the optically thick regime is also traced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In particular, we use 7 maximum adaptive-mesh reso- lution levels refining on the density and temperature, such that the maximal resolution is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 × 10−3 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The one-dimensional Cloudy model used a “sphere” geometry, an input power-law spectrum be- tween 1 - 10 keV for the X-ray radiation source, a cosmic microwave background and a cosmic-ray ionization rate of 3 × 10−17 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Fur- ther, we turn off grain physics, induced radiative processes, radiation pressure, radiation scattering, outward line radiation transfer and molecule freeze-out, since the Flash simulations do not have these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Finally, we set refractory metal abundances to zero, with the exception of silicon which Flash uses as the proxy for metals for the chemistry (as described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The Cloudy script used is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Using uniform-spaced grids, even with substantial AMR levels and, it is in practice difficult to fully capture sharp thermochemical transition regions, such as that shown below as captured by Cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Further, the source encompasses several cells at the highest reso- lution, rather than an infinitely small point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Capturing such ionization and dissociation fronts entirely is numerically intensive and generally requires the use of one-dimensional models tailored to do so (as with Cloudy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 11 shows the result of this benchmark, using .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Near the source, the temperature and chemistry solutions well match the Cloudy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The Flash and Cloudy solutions qualitatively reproduce the chemical structure, although due to the larger cell-size of the Flash grids, the sharp HI transition seen at 𝑁(𝐻) ≈ 5 × 1020 cm−2 is not fully captured and is instead smoothed over a few cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The temperature solutions agree within a factor of a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, Cloudy solves the line cooling and level excitations in a much more robust manner than the included Flash thermochemistry, including a full non-equilibrium solution with many more electronic and ion- ization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Such inclusions though are not numerically feasible for in-situ thermochemistry in three-dimensional MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Further, Cloudy solves the full radiation transfer solution from radio through X-ray radiation with substantially more bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Given the con- straints of these physics, the found solution is deemed to be adequate and matches the overall trends as determined by Cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 4 PROTOSTELLAR DISK Evolved protostellar objects, in particular Class II objects in which the lack of a surrounding gaseous envelope leaves the central protostar and disk exposed, are known to be X-ray emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' These X-rays can become important for disk dynamics and planet formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Ercolano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Mohanty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For these stars, the X-ray emission is thought to come from a combination of accretion and magnetospheric emission (Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' As a first test science case, we model the X-ray radiation transport from a central protostar into a protostellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The surface density follows from the often used truncated power- law (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Lynden-Bell & Pringle 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Cleeves MNRAS 000, 1–14 (2022) 8 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 100 101 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='07 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='77 keV TreeRay 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='69 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='92 keV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='75 keV Analytic 100 101 E (keV) 10 24 10 23 10 22 (cm 2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='07 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='77 keV TreeRay 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='69 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='92 keV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='75 keV Analytic 2 4 6 8 10 r (pc) 10 4 10 3 10 2 10 1 100 Relative Error c Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Radial profile test for a single source in a constant density and temperature medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Left: Radiation density versus radius for each bin for the TreeRay (solid) and analytic solution (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Left inset: Bin-averaged cross sections (black points) and the analytic cross section in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Right: Relative errors for each bin as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 100 101 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='07 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='77 keV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='69 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='92 keV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='75 keV Const T Temp Grad Const T Temp Grad 103 104 105 106 Temperature (K) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' X-ray energy density versus radius for single point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The solid line uses the constant temperature at 𝑇 = 10 K (same as Figure 5) while the dashed-dotted line uses the temperature profile shown by the red dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016): Σ𝑔(𝑅) = Σ𝑐 � 𝑅 𝑅𝑐 �−𝛼 exp � − � 𝑅 𝑅𝑐 �2−𝛼� (20) between an inner and outer radius, 𝑅in and 𝑅out, respectively, 𝑅𝑐 is the critical radius where the surface density distribution becomes exponential, 𝛼 is the power law index and Σ𝑐 is the characteristic surface where the disk transitions to an exponential profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For the initial conditions, we assume the gas is in hyrostatic equilibrium, such that the density follows 𝜌𝑔(𝑅, 𝑧) = Σ𝑔(𝑅) √ 2𝜋ℎ exp � − � 𝑧2 2ℎ2 �� (21) where ℎ = 𝑐𝑠/Ω is the disk scale height, 𝑐𝑠 = √︃ 𝛾𝑘𝑏𝑇𝑔 𝜇𝑚H , 𝑘𝐵 is Boltzmann’s constant, 𝛾 = 5/3 is the adiabatic index, 𝑇𝑔 is the gas temperature, 𝜇 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33 is the mean mass per particle for molecular gas, 𝑚𝐻 is the mass of the hydrogen atom, Ω = 3 4 √︃ 𝐺𝑀∗ 𝑅3 is the Keplerian rotational frequency and 𝑀∗ is the mass of the central protostellar object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For this fiducial test, we set 𝑀∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='7 M⊙, Σ𝑐 = 64 g cm−2, 𝑅𝑐 = 100 AU, 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The temperature profile is given by 𝑇(𝑅) = max � 𝑇0 � 𝑅 1𝐴𝑈 �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 , 10 K � , (22) where we fiducially take 𝑇0 = 50 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The disk is initialized to be rotating in Keplerian motion around the central protostellar object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We assume the disk is magnetized with an initial toroidal field such that the ratio of the magnetic to thermal pressure, 𝜇𝑀 = 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We simulate the domain in a 240 AU box with a maximal resolution of 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The central protostar is put in by hand, with active accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For the X-ray emission, we assume an accretion floor of 10−9 M⊙ yr−1, similar to rates observed in young stellar objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Ingleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The simulation is run using the Bouchut-5 MHD solver, grav- ity, and XRayTheSpot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray emission is derived by assuming there is an accretion shock, with properties following “hot spot” ac- cretion (Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016) with accretion columns filling 10% of the protostar surface, thermally emitting X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The ther- mal X-ray emission is computed using a one-temperature Raymond- Smith plasma model (Raymond & Smith 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The implementation of a coronal model is left for a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 12 shows a slice of the density, gas temperature, X-ray emission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV (1st bin) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV (8th bin), the heating rate per H nucleus, 𝐻𝑥, and 𝐻𝑥/𝑛, which is often used as a diagnostic for the importance of the X-ray heating (Wolfire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We find that the lowest energy X-rays are all absorbed near the protostar or escape through the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, the harder X-rays at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV are able to permeate much of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The 𝐻𝑋 and 𝐻𝑥/𝑛 slices MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 9 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) Nblock = 4, Nside = 4, R = 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='07 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='77 keV TreeRay 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='69 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='92 keV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='75 keV Analytic 2 4 6 8 10 r (pc) 10 3 10 2 10 1 100 c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='07 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='77 keV TreeRay 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='69 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='92 keV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='75 keV Analytic 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) Nblock = 4, Nside = 2, R = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 r (pc) 10 3 10 2 10 1 100 c 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 21 10 19 10 17 10 15 10 13 (erg/cm3) Nblock = 4, Nside = 4, R = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 r (pc) 10 3 10 2 10 1 100 c 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) Nblock = 4, Nside = 8, R = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 r (pc) 10 3 10 2 10 1 100 c 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 21 10 19 10 17 10 15 10 13 (erg/cm3) Nblock = 8, Nside = 4, R = 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 r (pc) 10 3 10 2 10 1 100 c 101 2 × 100 3 × 100 4 × 100 6 × 100 r (pc) 10 20 10 18 10 16 10 14 (erg/cm3) Nblock = 8, Nside = 8, R = 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 r (pc) 10 3 10 2 10 1 100 c Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Energy density versus radius for the different model parameters, annotated in the top left of each subfigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Inset: Relative error, 𝛿𝑐, of the numerical solution against the analytic solution as a function of radius from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) 10 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 15 10 5 0 5 10 15 y (pc) Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='78 keV 15 10 5 0 5 10 15 y (pc) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='37 keV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='16 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='22 keV 10 0 10 x (pc) 15 10 5 0 5 10 15 y (pc) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='62 keV 10 0 10 x (pc) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='50 keV 10 0 10 x (pc) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='00 keV 0 2 4 nH (cm 3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Fi (erg cm 2 s 1) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Shadow test, consisting of a point source illuminating a constant density core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Top left corner: Number density distribution for a z-axis slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Others: X-ray flux in the given energy band for a z-axis slice using 𝑁block = 8, 𝑁side = 4, 𝜂𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 15 10 5 0 5 10 15 y (pc) Density 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='78 keV 15 10 5 0 5 10 15 y (pc) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='37 keV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='16 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='22 keV 10 0 10 x (pc) 15 10 5 0 5 10 15 y (pc) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='62 keV 10 0 10 x (pc) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='50 keV 10 0 10 x (pc) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='00 keV 0 2 4 nH (cm 3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 Fi (erg cm 2 s 1) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Same as Figure 8, but with 𝑁block = 8, 𝑁side = 8 and 𝜂𝑅 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 100 101 zsrc (pc) 0 10 18 10 17 10 16 10 15 10 14 e (erg/cm3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33 keV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='78 keV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='37 keV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='16 keV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='22 keV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='62 keV 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='50 keV 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='00 keV Fiducial High Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Fiducial High Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 100 101 102 103 104 Hydrogen Density (cm 3) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' X-ray energy density versus distance along the z-axis from the source for the shadow test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The solid line uses the ray resolution in Figure 8 and the dashed-dot uses the ray resolution in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The dotted red line shows the hydrogen nuclei density highlighting the location of the high-density blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' clearly show that the disk midplane is left relatively unheated by the X-rays, although the X-rays become important in the cavity and outer disk regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In particular, most of the cavity exhibits very warm gas, even with only X-ray emission included, due to the rapid absorption of soft X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The cavity heats to temperatures exceeding 104 Kelvin, potentially becoming bright in hydrogen recombination lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The inclusion of EUV radiation will heat the diffuse gas further, along with further ionizing the surrounding low-density cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 5 MOLECULAR CLOUD We present an example application for XRayTheSpot, to demon- strate how all the different TreeRay energy bands work together: a virialized, magnetized turbulent cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We consider a 2 pc region of a molecular cloud resolved with 2563 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We produce an initial turbulent field by stirring the domain with a flat power spectrum between the largest wave modes 𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='3 for 10 crossing times at a velocity dispersion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='72 km s−1, consistent with the observed linewidth-size relationship (McKee & Ostriker 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' During the stir- ring, we use periodic boundary conditions and chemistry to achieve more accurate initial conditions for the abundances before collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The choice of stirring for 10 crossing times is to ensure the chemistry has reached a more quiescent state, with the kinetic energy spectrum generally being reached after two crossing times (Federrath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We assume the cloud is nearly virialized, such that the virial parameter 𝛼 ≡ 5𝜎2𝑅 𝐺𝜌𝐿3 = 2 (23) where 𝑅 = 𝐿 is the box length, resulting in 𝜌 = 5 × 10−21 (g cm−3) and a total box mass of 𝑀 = 590 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Before stirring, we initialize a magnetic field in the 𝑧-axis with a magnitude such that the plasma beta, 𝛽 ≡ 𝜌𝑐2𝑠 𝐵2/8𝜋 = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (24) After the turbulence is initialized, gravity and source particles (stars) are turned on and the boundary conditions are changed to “diode” MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 11 1019 1020 1021 Hydrogen Column Density 101 102 103 104 Temperature (K) Cloudy Flash 1019 1020 1021 Hydrogen Column Density 10 7 10 6 10 5 10 4 10 3 10 2 10 1 100 Abundances H/Htot 2H2/Htot Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Flash vs Cloudy benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Left: Temperature versus hydrogen column density from the central point source for Flash (black) and Cloudy (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Right: Atomic (solid) and molecular (dashed) hydrogen abundances versus total hydrogen column density from the source, where Htot = H+ + H + 2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' t = 100 yr 10 20 10 19 10 18 10 17 10 16 10 15 10 14 102 103 104 10 10 10 9 10 8 10 7 10 6 10 23 10 22 10 21 10 20 10 19 10 18 10 17 10 29 10 28 10 27 10 26 10 25 10 24 10 23 10 10 10 9 10 8 10 7 10 6 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Protostellar disk example case usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Top row: Slice plots at 𝑧 = 0 for the density (left), gas temperature (middle) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='17 keV radiation energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Bottom row: X-ray heating rate, H𝑥 (left), H𝑥/n diagnostic term (middle) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 keV radiation energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' such that gas can flow out of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' During the simulation, the cloud is irradiated by an FUV radiation field of 𝜒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='7 in units of the Habing field (Habing 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The simulation is run using the chemistry described above, and all TreeRay modules: OpticalDepth for the external radiation field (Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' OpticalDepth solves for the column density from a cell to the external boundary and attenuates a prescribes external radiation flux (𝜒 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In this study, it is only used for the FUV radiation, while (Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2019) implemented the ability to include an impinging X-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' OnTheSpot for the EUV emission (Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This module solves for UV-ionizing radiation from arbitrary sources and MNRAS 000, 1–14 (2022) 12 Gaches et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' iterates to convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The UV photon flux is coupled to the ther- mochemistry to model photochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' RadPressure to account for the thermal radiation and radiation pressure (Klepitko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This module enables the inclusion of thermal radiation from point and diffuse sources and the resulting radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The thermal radiation is included in the chemistry through radiative dust heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' XRayTheSpot, described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Sink particles representing protostars are injected when the den- sity exceeds 𝜌thresh ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='59×10−18 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Further criteria are used: there are checks to ensure a local gravitational potential and a con- verging flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The protostar evolution follows the Offner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2009) model and implemented in Flash (Klepitko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Protostellar emission consists of the intrinsic and accretion luminosities, where the total accretion luminosity is 𝐿acc = 𝑓acc 𝐺𝑀∗ �𝑀∗ 𝑅∗ , (25) where 𝑀∗ is the mass of the protostar, �𝑀∗ is the accretion rate, 𝑅∗ is the protostar’s radius and we take 𝑓acc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray spectrum was computed by assuming hot-spot accretion, described above, which provides the temperature and the density of the accre- tion shocks near the protostellar surface (Calvet & Gullbring 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2016) and a single temperature plasma model (Ray- mond & Smith 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Due to the low resolution, we set a minimum of �𝑀∗ = 10−9 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This is needed since when the protostar particles first form, the burst of accretion blows out HII regions, and the low resolution inhibits resolving the proper structure around the cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The infrared to EUV spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' used for RadPressure and OnTheSpot is computed assuming the emission is composed of two blackbodies: one for the intrinsic spectrum of the protostar at the photosphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' such that 𝑇∗ = � 𝐿∗ 4𝜋𝜎sb𝑅2∗ �1/4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (26) which is provided by the protostellar evolution model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' and another assuming the accretion luminosityisreprocessedprimarily as a black- body with temperature 𝑇acc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' such that 𝑇acc = � 𝐿acc 4𝜋𝜎sb𝑅2∗ �1/4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (27) where 𝜎sb is the Stefan-Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Therefore, the total infrared luminosity from the protostar is described as 𝐿∗,IR = 𝑓∗,IR(𝑇∗)𝐿∗ + 𝑓acc,IR(𝑇acc)𝐿acc (28) and the EUV luminosity as 𝐿∗,EUV = 𝑓∗,EUV(𝑇∗)𝐿∗ + 𝑓acc,EUV(𝑇acc)𝐿acc (29) where 𝑓IR(𝑇) and 𝑓UV(𝑇) are the fraction of the blackbody emis- sion in each of these bands (𝐸 < 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 eV and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='6 eV ≤ 𝐸 ≤ 100 eV, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray emission was computed assuming the “hot-spot” model, described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While there may be some double counting of emission by treating the total spectrum in the two differ- ent methods, we find this impact is marginal as the X-ray emission generally accounts for only a small fraction (≤ 10%) of the total protostellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Figure 13 shows the column density, and density-weighted projec- tions of the gas and temperature, radiation temperature, EUV photon density and X-ray energy densities after ≈ 1 Myr of evolution with gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The star formation, as traced by heated knots of gas, is oc- curring along a main filament structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The high temperatures here are primarily caused by the EUV photons, which are rapidly ab- sorbed in the nearby gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray emission is found to be highly absorbed along the main filament structure, and instead traces out the more diffuse turbulent structure of the molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' As expected, higher energy X-ray bands showcase more extended emission with the brightest emission in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='7 keV band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In all X-ray bands, the tur- bulent structure of the molecular cloud is seen in the density-weighted integrated emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This case study highlights the new capabilities of including protostellar radiative feedback from infrared to X-ray in star formation simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 6 DISCUSSION/FUTURE WORK We have presented the new X-ray radiation transfer module, XRayTheSpot using the reverse ray-tracing scheme TreeRay im- plemented in Flash (Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' XRayTheSpot enables an arbitrary number of point or diffusive sources of X-ray emission, and an arbitrary number and position of energy bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The module uses temperature dependent cross sections assuming gas in thermal ionization equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, the module is flexible enough such that the user can provide their own cross section data to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The module produces the expected behavior for X-ray point sources and shadow tests and is able to reasonably reproduce the thermochemistry compared to Cloudy, despite the significantly simpler treatment of X-ray chemistry and grain-processes in Flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We demonstrated the utility of this module with two example sci- ence cases focusing on protostellar X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' First, we mod- elled the emission of an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='7 M⊙ protostar with an accretion rate of 10−9 M⊙ yr−1 through a protostellar disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We find that soft X-rays are rapidly absorbed at the disk surfance, with most of the emission escaping through the outflow cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, harder X-rays are able to permeate the disk due to their significantly lower optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The X-ray heating was also strong within the outflow cavity, with no X-ray heating towards the midplane of the disk, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Second, we perform a low-resolution star formation simulation of a turbulent molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In this simulation, protostars are self-consistently formed and the X-ray emission modelled on the fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' This simulation includes the entire range of different TreeRay radiation modules: diffuse FUV (OpticalDepth Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2018)), EUV (OnTheS- pot Wünsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2021)), thermal radiation and radiation pressure (RadPressure Klepitko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' (2022)) and X-ray emission from 1 keV to 10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Since the X-ray emission in the simulation comes en- tirely from accretion onto the protostars, the X-ray emission is highly variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Due to the lower resolution and the inclusion of ionizing radiation, the accretion occurs in bursts followed by the expansion of HII regions, which cut off accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' With higher resolution, accre- tion may still be able to occur through disks, instabilities and more porous density structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In future work, we will perform higher resolution simulations to model star formation including chemistry and radiation feedback across the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' In this work, we focus primarily on point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' However, XRayTheSpot makes no differentiation between point sources ver- sus extended more diffusion emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Future studies will include diffuse X-ray emission from cooling hot gas and shocked gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Our module currently includes the computation of X-ray emission from accretion onto protostars, and future work will include X-ray models for more types of point sources such as X-ray binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The module presented in this work will allow the first-generation of simulations of star formation and galaxies with the inclusion of a wide range of X-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 13 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='08 Myr 10 2 10 1 100 101 102 10 15 20 25 30 8 10 12 14 16 10 10 10 8 10 6 10 4 10 2 10 22 10 21 10 20 10 19 10 18 10 17 10 19 10 18 10 17 10 19 10 18 10 17 10 16 10 18 10 17 10 16 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Panel plots highlighting the features of a 2 pc piece of a molecular cloud after 𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='08 Myr of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' For all fields except the column density, the panel is showing the density-weighted projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' All projections are along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The figure shows a simulated molecular cloud after 1 Myr of gravitational evolution including protostar sink particles and radiation feedback from infrared to X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' While the EUV radiation is rapidly absorbed (indicated by the black background color), the infrared and X-ray emission is able to penetrate much further into the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' ACKNOWLEDGEMENTS BALG and SWG acknowledges support by the ERC starting grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' 679852 ‘RADFEEDBACK’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' SWG and BALG thank the German Science Foundation (DFG) for funding through SFB956 project C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' We also thank the Regional Computing Center Cologne (RRZK) for hosting our HPC cluster, Odin, on which the simulations have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' RW acknowledges the support by project 20- 19854S of the Czech Science Foundation and by the institutional project RVO:67985815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' JM acknowledges support from a Royal Society-Science Foundation Ireland University Research Fellowship (20/RS-URF-R/3712) and an Irish Research Council Starting Lau- reate Award (IRCLA\\2017\\83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The authors thank Andre Klepitko for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' Andre Klepitko also implemented the protostellar evolution model into the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Centre at the University of Chicago (Fryxell et al.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' MNRAS 000, 1–14 (2022) XRayTheSpot: X-raying Molecular Gas 15 1 t i t l e XDR source 2 ## r a d i a t i o n s o u rces 3 CMB 4 t a b l e SED " plaw .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' sed " 5 l u m i n o s i t y 35 range 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='5 to 735 Ryd 6 ##Geometry 7 r a d i u s 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='1938 8 hden 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 9 sphere 10 ## Stopping and i t e r a t e 11 stop H2 column d e n s i t y 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 12 stop t e m p e r a t u r e l i n e a r 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='0 13 i t e r a t e to convergence 14 ##ISM and Grain physics 15 cosmic ray r a t e −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='523 16 abundances ISM 17 g r a i n s ISM no qheat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='56 18 no g r a i n x−ray t r e a t m e n t 19 no induced p r o c e s s e s 20 no r a d i a t i o n p r e s s u r e 21 no s c a t t e r i n g o p a c i t y 22 no g r a i n p h y s i c s 23 no g r a i n molecules 24 no l i n e t r a n s f e r 25 ##Abundances 26 element carbon 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='51 ## output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='52 save ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='overview ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='l a s t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content='" xdr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' ovr " 53 save molecules l a s t " xdr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' mol " 54 save abundances l a s t " xdr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' abund " 55 save continuum l a s t " xdr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' cont " 56 save PDR l a s t " xdr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} +page_content=' pdr " Listing 1: Input file for Cloudy benchmark MNRAS 000, 1–14 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFQT4oBgHgl3EQfETVZ/content/2301.13237v1.pdf'} diff --git a/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf b/29E2T4oBgHgl3EQfNwZM/content/2301.03740v1.pdf new file mode 100644 index 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Spivak1, and Ryan Wisnesky2 +1 Massachusetts Institute of Technology +2 Conexus AI +Abstract. We survey the field of model management and describe a +new model management approach based on algebraic specification. +1 +Introduction +In this paper we survey the field of model management and describe a new +model management approach based on techniques from the field of algebraic +specification, with the hope of establishing an interlingua between the two fields. +By “model management” we mean “meta-data intensive” database management +in the sense of Bernstein & Melnik [4], which we define in Section 2. By “a new +algebraic model management approach” we mean our particular way [19] [20] of +specifying database schemas and instances using algebraic (equational) theories. +We first noticed a connection between model management and algebraic spec- +ification while investigating applications of category theory [3] to data integra- +tion [5]. These investigations are described in [19] and [20], and we present no +substantial new results in this paper. We assume readers have basic proficiency +with category theory [3], algebraic specification [17], and SQL. +Outline. In Section 2 we describe the traditional approach to model manage- +ment and in Section 3 we describe our algebraic approach. Also in Section 3 we +describe the open-source CQL (Categorical Query Language) tool, available for +download at http://categoricaldata.net, which implements our approach in +software. We conclude in Section 4 by comparing our approach with the tradi- +tional approach. +2 +Model Management +To quote from Melnik [16]: +Many challenging problems facing information systems engineering in- +volve the manipulation of complex metadata artifacts, or models, such +as database schemas, interface specifications, or object diagrams, and +mappings between models. The applications that solve metadata ma- +nipulation problems are complex and hard to build. The goal of generic +model management is to reduce the amount of programming needed +to develop such applications by providing a database infrastructure in +which a set of high-level algebraic operators, such as Match, Merge, and +Compose, are applied to models and mappings as a whole rather than +to their individual building blocks. +arXiv:2301.04846v1 [cs.LO] 12 Jan 2023 + +In the paragraph above the word “model” is defined to mean a metadata ar- +tifact such as a schema, which conflicts with the definition of the word “model” +as a structure satisfying a theory. In this paper, we use the phrase “model man- +agement” to mean the field identified above, and use the word “model” to mean +a structure satisfying a theory. +Today model management is a large sub-field of information management +with a research literature containing hundreds of published articles [4]. There is +a consensus in that literature [4] that model management is concerned with at +least the problems described in the next sections. +2.1 +Schema mapping +Given two database schemas S and T, the schema mapping problem [7] is to +construct a “mapping” F : S → T that captures some user-specified relationship +between S and T. Different model management systems use different notions +of schema, including SQL, XML, and RDF [4]. The most common mapping +formalism studied in the literature is that of “embedded dependencies” (EDs) [5]: +formulae in a fragment of first-order logic with useful computational properties. +We will use SQL schemas and EDs in our examples in this section. Consider +the following SQL schema S, consisting of two tables connected by a foreign key: +CREATE TABLE N2(ID INT PRIMARY KEY, age INT) +CREATE TABLE N1(ID INT PRIMARY KEY, name STRING, salary INT, +f INT FOREIGN KEY REFERENCES N2(ID)) +and the following SQL schema T, consisting of one table: +CREATE TABLE N(ID INT PRIMARY KEY, age INT, +name STRING, salary INT). +These two SQL schemas are displayed graphically in Figure 1. +An example schema mapping F : S → T expressing that the target table N +is the join of source tables N1 and N2 along the column f is: +∀id1, id2, a, n, s. N1(id1, n, s, id2) ∧ N2(id2, a) → N(id1, a, n, s). +Two instances satisfying the above ED are shown in Figure 1. In general, many +EDs can map between two SQL schemas. +2.2 +Query generation +Given a schema mapping F : S → T, the query generation problem [4] is to +construct a query which converts databases on S to databases on T in a way +that satisfies F. The query languages typically studied include SQL, XQuery, +and various comprehension- and λ-calculi [4]. +A SQL query to implement the example mapping from Section 2.1 is: + +String +◦ +N1• +name +� +salary +� +f +� N2• +age +� +◦ +Int +F +−−−→ +String +◦ +N• +name +� +age +� +salary � ◦ +Int +N1 +ID name salary f +1 Alice $100 1 +2 +Bob $250 2 +3 +Sue +$300 3 +N2 +ID age +1 20 +2 20 +3 30 +�∆F � +←−−−−−− +�ΠF �,�ΣF � +−−−−−−−−−−→ +N +ID name salary age +1 Alice $100 20 +2 +Bob $250 20 +3 +Sue +$300 30 +Fig. 1. Example Data Migrations, with Foreign Keys (see Sections 2.1, 3.2) +INSERT INTO N +SELECT N1.ID, N1.age, N2.name, N1.sal +FROM N1, N2 +WHERE N1.f = N2.ID +Technically, the INSERT portion of the above SQL code is not a “query”, but +rather an “update”, and in practice the code generated from a query generation +task will often store the results of the query. An example of running the above +SQL is shown as the left-to-right direction of Figure 1. In general, many or no +SQL queries may implement a set of EDs [5]. EDs can also be directly executed +by an algorithm called “the chase” [5]. +2.3 +Mapping Inversion +Given a schema mapping F : S → T, the mapping inversion problem [10] is to +construct a schema mapping F −1 : T → S that undoes F with respect to query +generation (i.e. the queries generated from F and F −1 should be inverses). +The natural candidate ED to invert the schema mapping of Section 2.1 ex- +presses that N projects onto N1 and N2: +∀id1, a, n, s. N(id1, a, n, s) → ∃id2. N1(id, n, s, id2) ∧ N2(id2, a) +and a possible SQL implementation of this ED is: +INSERT INTO N1 +SELECT ID, name, sal, ID +FROM N +INSERT INTO N2 +SELECT ID, age +FROM N + +However, the above ED is not an inverse to the ED of Section 2.1, as is seen by +taking ∅ = N1 ̸= N2. Indeed, it is rare for an ED, or set of EDs, to be invertible, +and weaker notions of inverse, such as “quasi-inverse” [10], are common in the +literature [10]. An example of running the above SQL is shown as the right-to-left +direction of Figure 1. +2.4 +Mapping Composition +Given schema mappings F : S → T and G : T → U, the mapping composition +problem [8] is to construct a schema mapping G ◦ F : S → U that is equivalent +with respect to the query generation problem (i.e. running the query generated +from G ◦ F should have the same effect as running the query generated from G +on the results of the query generated from F). +The composition of the ED from Section 2.1 with the ED from Section 2.3 is +∀id1, id2, n, s, a. N1(id1, n, s, id2) ∧ N2(id2, a) → ∃x. N1′(id, n, s, x) ∧ N2′(x, a) +where N1’, N2’ are target “copies” of source tables N1, N2. This composed ED is +not the identity, thereby showing that the ED from Section 2.3 does not invert the +ED from Section 2.1. In the case of EDs, composed mappings may not exist [8], +but some restrictions and extensions of EDs are closed under composition [8]. +2.5 +Schema matching +Given two database schemas S and T, the schema matching problem [5] is to au- +tomatically find “correspondences” between S and T and to automatically infer +schema mappings S → T from these correspondences. In general, inference of en- +tire mappings cannot be fully automated and the focus of the matching problem +is to reduce the human effort required to construct a schema mapping by e.g., +suggesting partial mappings that can be completed by users. There are many +techniques for schema matching ranging from comparison of column names by +string similarity to machine learning algorithms; for an overview, see [5]. In the +example from Section 2.1, two correspondences that are easy to automatically +find are (N1, N) and (N2, N) and tools such as Clio [14] can create the ED from +Section 2.1 from these two correspondences. +2.6 +Further References +In this paper we will focus on the problems described in the previous sec- +tions, but many other problems are studied in the model management litera- +ture [4], and many of these problems are related to algebraic specification. For +example, schema/instance merge problems [4], which arise often in data inte- +gration scenarios [2], can be formalized as pushouts in suitable categories of +schemas/instances [20], and such pushouts are related to model-theoretic con- +cepts such as model amalgamation [15]. + +Many software products solve model management problems [4], including +ETL (Extract, Transform, Load) tools [5], which extract data from separate +databases, apply user-specified transformations, and then load the result into +a target system such as a data warehouse; query mediators [5], which answer +queries about a “virtual” integrated database by combining queries about sep- +arate source databases; and visual schema mapping tools [14] which allow users +to create schema mappings by visually connecting related schema elements with +lines, as shown in Figure 2. +There have been at least two attempts to provide a “meta semantics” for +model management operations. In [16] Melnik gives a “state based” meta se- +mantics to some of the above operations by defining a schema mapping S → T +to be an arbitrary binary relation between instances on S and instances on T; +the ED-based semantics described above is an instantiation of this meta seman- +tics. In [2] and [13] the authors give an “institution theoretic” meta semantics +to some of the above operations by defining a schema mapping S → T to be a +morphism in a suitable category of schemas; CQL’s semantics is an instantiation +of this meta semantics. +Fig. 2. A schema mapping in Clio [14] + +file:students.xsml (managed) +口 +Source +Target +S File:StudentsSource.xsd +S File:StudentsTarget,xsd +qa +e? targetDB +e gradEnrolls +e Evaluations +* gradEnroll [o,*] +[* eval [0.*] +e: sid (xsistring)- +e: name (xsistring)-- +e: grade (xsiint) +(buuisisn) pp a +e? File (μsistring) +.. +@ Enrollment +e: File (xsistring)- +e* Student [0.*] +e* underGrad [o,*] +e: sid (xsistring) +e: name (rsistring) +. +(ouuisisi) pis a +e Courses +e: name (xsistring) +e: address (xsistring) +[* course [o,*] +e enrolls +e eid (xsint) ..- +[ enroll [o,*] +e: addr (xsistring) +e cid (xsistring) +e: sid (rsistring)3 +Algebraic Model Management +Our approach to model management is based on the algebraic approach to +databases, data migration, and data integration we describe in [19] and [20]. +Those works, and hence this work, extend a particular category-theoretic data +model that originated in the late 1990s [11] and was later extended in [21] and [23] +and implemented in CQL (http://categoricaldata.net). +In the next section we describe our formalism for database schemas and +instances and introduce CQL. The subsequent sections implement the model +management operations from Section 2 using our formalism. In this section we +abbreviate “algebraic theory” as “theory”. +3.1 +Algebraic Databases +In our formalism [20], database schemas and instances are defined as theories of +a certain kind, which we describe in the next sections. For ease of exposition, we +will sometimes conflate schemas and instances as defined in our formalism with +their CQL equivalents. +Type sides We first fix a theory, Ty, called the type side of our formalism. The +sorts of Ty are called types and the functions of Ty are the functions that can +appear in schemas and instances. +CQL allows arbitrary theories to be used as type sides. But we have found +that in practice, CQL users almost always want to use the theory of an existing +programming language, say java, for their type side. The ability to “bind” CQL +to an existing language is particularly important in model management because +input data may only be accessible through, e.g., a java API. For this reason, +CQL allows a type side to be defined by specifying, for each sort s, a java class +Cs and a java function String → Cs that tells CQL how to interpret the strings +it encounters in CQL programs as objects of Cs. +An example CQL type side about integers and strings is shown in Figure 3. +This type side defines a theory with two sorts and infinitely many constants – +all the java strings and integers – and no equations. The java code for Int says +that whenever a string x is encountered in an CQL program and a term of sort +Int is required, that java’s parseInt function should be applied to x to yield +the desired Int. The keyword literal, used in many places in CQL, indicates +a literal (user-defined constant) definition. +Schemas A schema on type side Ty is a theory extending Ty with new sorts +(called entities), new unary functions from entities to types (called attributes), +new unary functions from entities to entities (called foreign keys), and new equa- +tions (called data integrity constraints) of the form ∀v : s. t = t′, where s is an +entity and t, t′ are terms of the same type, each containing a single free variable v. +The restrictions in the preceding sentence (e.g., no functions from types to enti- +ties) are necessary to use our formalism for model management purposes [19] [20]. +Figure 4 shows the CQL schemas corresponding to Figure 1. These schemas +contain no equations and are both on the type side Ty defined in Figure 3. + +typeside Ty = literal { +java_types +String = "java.lang.String" +Int = "java.lang.Integer" +java_constants +String = "return input[0]" +Int = "return java.lang.Integer.parseInt(input[0])" +} +Fig. 3. CQL type side Ty +schema S = literal : Ty { +entities +N1 +N2 +foreign_keys +f : N1 -> N2 +attributes +name : N1 -> String +salary : N1 -> Int +age : N2 -> Int +} +schema T = literal : Ty { +entities +N +attributes +name : N -> String +salary : N -> Int +age : N -> Int +} +Fig. 4. CQL schemas S and T on type side Ty +instance I = literal : S { +generators +1 2 3 : N1 +equations +name(1) = Alice +salary(1) = 100 +age(f(1)) = 20 +name(2) = Bob +salary(2) = 250 +age(f(2)) = 20 +name(3) = Sue +salary(3) = 300 +age(f(3)) = 30 +} +Fig. 5. CQL instance I on schema S +Fig. 6. Initial algebra for CQL instance I + +Delta - 9:36:11 PM 1 (exec: 0s)(gui: 0s) +Select: +Tables +Type Algebra +DP +Text +typeside Ty +N1 (3) +N2 (3) +schema S +ID +name +salary +f +ID +age +schema T +[1] +Alice +100 +[1.f] +[1.f] +20 +mapping F : S -> T +[2] +Bob +250 +[2.f] +[2.f] +20 +instance I : S +[3] +Sue +300 +[3.f] +[3.f] +30Instances An instance I on schema S is a theory extending S with new 0-ary +function (constant) symbols called generators and non-quantified equations. An +example CQL instance on schema S (Figure 4) is shown in Figure 5. +The intended meaning of an instance I, written �I�, is the term model (i.e., +initial algebra) for I which contains, for each sort s, a carrier set consisting of the +closed terms of sort s modulo provability in I. A morphism of instances I → J +is a homomorphism (natural transformation) of algebras �I� → �J�. +Figure 6 shows the meaning of the instance I from Figure 5 in the CQL tool. +The CQL tool visually displays term models as sets of tables, one per entity e, +each with an ID column corresponding to the carrier set for e. The tables in +Figure 6 are isomorphic to the left tables in Figure 1. +In the following sections we implement the model management operations +from Section 2 using the preceding definitions of schema and instance. +3.2 +Schema mapping +Given schemas S, T, the schema mapping problem (Section 2.1) is to construct +a “mapping” F : S → T that captures some relationship between S and T. +Let S and T be CQL schemas on the same type side Ty. An CQL schema +mapping F : S → T is defined as a “derived signature morphism” [18] from +S to T that is the identity on Ty. That is, F : S → T assigns to each entity +e ∈ S an entity F(e) ∈ T, and to each attribute / foreign key f : s → s′ a +term F(f), of type F(s′) and with one free variable of type F(s), in a way that +respects equality: if S ⊢ t = t′, then T ⊢ F(t) = F(t′). We have found that +many mappings arising in practice cannot be expressed using plain signature +morphisms and require the more general notion of “derived” signature morphism. +Whereas a schema mapping in Section 2.1 was an ED (formula in a fragment +of first-order logic), which induces a single binary satisfaction relation between +instances, CQL schema mappings are derived signature morphisms and induce +three relations between instances, which we will describe in the next section. +An example CQL schema mapping F : S → T is shown in Figure 7, where +CQL schemas S and T are defined in Figure 4. This mapping is also shown +graphically in Figure 1. +3.3 +Query generation +Given a mapping F : S → T, the query generation problem (Section 2.2) is to +use F to construct a query which converts databases on S to databases on T. +In our formalism, the database instances and morphisms on a schema S +constitute a category, denoted S–Inst, and a schema mapping F : S → T induces +a functor ΣF : S–Inst → T–Inst defined by substitution. The functor ΣF has a +right adjoint, ∆F : T–Inst → S–Inst, which corresponds to the “model reduct +functor” when our formalism is described in institution-theoretic terms [2]. The +functor ∆F has a right adjoint, ΠF : S–Inst → T–Inst. See [19] for proof +that ∆F always has left and right adjoints. As adjoints, ∆F , ΠF preserve limits +and ∆F , ΣF preserve colimits, implying many useful properties; for example, +ΣF (I + J) ∼= ΣF (I) + ΣF (J) and ΠF (I × J) ∼= ΠF (I) × ΠF (J). + +mapping F = literal : S -> T { +entities +N1 -> N +N2 -> N +foreign_keys +f -> lambda x:N. x +attributes +name -> lambda x:N. name(x) +salary -> lambda x:N. salary(x) +age -> lambda x:N. age(x) +} +Fig. 7. CQL schema mapping F : S → T +Note that unlike Section 2.1, where there was a single query associated with a +schema mapping (ED), in our algebraic approach there are three queries, one for +each of ∆F , ΣF , ΠF . The conditions under which ∆F ,ΣF , ΠF can be expressed +in SQL and vice-versa are characterized in [23]. +Although it is possible to give explicit formulae to define ∆F , ΣF , ΠF [19] +we instead give examples in Figures 1 and 8. Note that in these examples we are +not showing instances (theories) as defined in Section 3.1; we are showing term +models. For this reason, we surround ∆F , ΣF , ΠF with denotation brackets �� in +these examples. In addition, as adjoints ∆, Σ, Π are only defined up to unique +isomorphism, so we arbitrarily make up names for IDs and in these examples. +Figures 1 and 8 show an CQL schema mapping F which takes two distinct source +entities, N1 and N2, to the target entity N. The �∆F � functor projects in the +opposite direction of F: it projects columns from the single table for N to two +separate tables for N1 and N2, similar to FROM N AS N1 and FROM N AS N2 in +SQL. When there is a foreign key from N1 to N2, the �∆F � functor populates it +so that N can be recovered by joining N1 and N2. The �ΠF � functor takes the +cartesian product of N1 and N2 when there is no foreign key between N1 and +N2, and joins N1 and N2 along the foreign key when there is. The �ΣF � functor +disjointly unions N1 and N2; because N1 and N2 are not union compatible (have +different columns), �ΣF � creates null values. When there is a foreign key between +N1 and N2, �ΣF � merges the tuples that are related by the foreign key, resulting +in a join. As these examples illustrate, ∆F can be thought of as projection, +ΠF can be thought of as a product followed by a filter (which can result in a +join), and ΣF can be thought of as a disjoint union (which does not require +union-compatibility) followed by a merge (which can also result in a join). +3.4 +Mapping Composition +Given schema mappings F : S → T and G : T → U, the mapping composition +problem (Section 2.4) is to construct a schema mapping G ◦ F : S → U that is +equivalent with respect to query generation. +In one sense, the mapping composition problem is trivial [19] for our for- +malism: ∆F ◦G ∼= ∆G ◦ ∆F , ΠF ◦G ∼= ΠF ◦ ΠG, and ΣF ◦G ∼= ΣF ◦ ΣG. But + +String +◦ +N1• +name +� +salary +� +N2• +age +� +◦ +Int +F +−−−→ +String +◦ +N• +name +� +age +� +salary � ◦ +Int +N1 +ID name salary +1 Alice $100 +2 +Bob $250 +3 +Sue +$300 +N2 +ID age +4 20 +5 20 +6 30 +�∆F � +←−−−−−− +N +ID name salary age +1 Alice $100 20 +2 +Bob $250 20 +3 +Sue +$300 30 +N1 +ID name salary +1 Alice $100 +2 +Bob $250 +3 +Sue +$300 +N2 +ID age +4 20 +5 20 +6 30 +�ΣF � +−−−−−−→ +N +ID +name +salary +age +1 +Alice +$100 +age(1) +2 +Bob +$250 +age(2) +3 +Sue +$300 +age(3) +4 name(4) salary(4) +20 +5 name(5) salary(5) +20 +6 name(6) salary(6) +30 +N1 +ID name salary +1 Alice $100 +2 +Bob $250 +3 +Sue +$300 +N2 +ID age +4 20 +5 20 +6 30 +�ΠF � +−−−−−−→ +N +ID name salary age +1 Alice $100 20 +2 +Bob $250 20 +3 +Sue +$300 20 +4 Alice $100 20 +5 +Bob $250 20 +6 +Sue +$300 20 +7 Alice $100 30 +8 +Bob $250 30 +9 +Sue +$300 30 +Fig. 8. Example Data Migrations (see Section 3.2) + +this solution is not wholly satisfactory because in practice a mixture of ∆, Σ, Π +functors may be needed to accomplish any particular task (similarly, in SQL a +mixture of joins and unions may be needed to accomplish any particular task). +The following results are proved in [19] and [23]: +– Every composition ΣF ◦∆G is isomorphic to ∆F ′ ◦ΣG′ for some F ′, G′. This +statement is also true if ΣF is replaced with ΠF . +– Pairs of the form (F, G), denoting ΣF ◦ ∆G, are closed under composition. +This statement is also true if ΣF is replaced with ΠF . Such pairs can be spec- +ified in an intuitive “select-from-where” syntax, described in [19] and [20]. +– Triples of the form (F, G, H), denoting ΣF ◦ ΠG ◦ ∆H, are closed under +composition, provided that F is a discrete op-fibration [3], which is exactly +the “union compatibility” condition [5] that ΣF performs unions over tables +whose columns match; Figure 1 is not a discrete op-fibration. +3.5 +Mapping Inversion +Given a schema mapping F : S → T, the mapping inversion problem (Sec- +tion 2.3) is to construct a mapping F −1 : T → S that somehow “undoes” F. +Our formalism has strong inversion properties but does not have inverses per +se. When there exists F −1 : T → S such that F ◦ F −1 = id and F −1 ◦ F = id, +then ∆F ◦ ∆F −1 ∼= id, ΣF ◦ ΣF −1 ∼= id, and ΠF ◦ ΠF −1 ∼= id. In general F +need not have an inverse, but when S and T have finite initial algebras / term +models (which is a priori undecidable, and implies decidability of S and T) it is +possible to construct F −1 whenever it exists by considering all possible functors +T → S. When F has a right adjoint G : T → S, a weaker condition than having +an inverse, there are canonical morphisms ΣF → ∆G and ∆F → ΠG. +In practice “round-tripping” [5] of data is desirable even when inverses do not +exist. For example, projection, because it forgets information, typically cannot +be inverted, but we may want to remember where the projected data originated. +In our formalism the adjunctions between Σ,∆,Π provide round-tripping. For +example, for every F : S → T and S-instance I there is a canonical morphism +I → ∆F (ΣF (I)), the unit of the ΣF ⊣ ∆F adjunction, which describes where +each ID in I is sent to by ΣF (and similarly for ΠF ). Dually, for every T- +instance J there is a canonical morphism ΣF (∆F (J)) → J, the co-unit of the +ΣF ⊣ ∆F adjunction, which describes where the IDs in ∆F (J) originate (and +similarly for ΠF ). The unit and co-unit can be used to obtain, for every morphism +h : ΣF (I) → J, a mate h′ : I → ∆F (J) and vice-versa (and similarly for +ΠF ). Relating adjointness to existing relaxed notions of inverse such as quasi- +inverse [9] is an important area for future work. +3.6 +Schema matching +Given database schemas S and T, the schema matching problem (Section 2.5) +is to automatically suggest schema mappings S → T to the user. + +In this section, we define two schema matching techniques used by CQL. +Our techniques compare entities, and foreign keys and attributes (“symbols”) +by name, as strings, and so our techniques depend on having (probably user- +provided) names whose similarity as strings reflects their semantic similarity. Let +σ : String, String → [0, 1] be any string similarity function [5] where a value of 1 +indicates a “good” match and a value of 0 indicates a “bad” match. +– The first technique attempts to infer a schema mapping F : S → T. For each +entity s ∈ S, we define F(s) := t where t ∈ T is an entity that maximizes +σ(s, t). For each symbol f : s → s′ ∈ S, we then consider the set X of +symbols F(s) → F(s′). If X is non-empty, we choose a symbol g ∈ X that +maximizes σ(f, g) and set F(f) := g. If X is empty but there is a shortest +path p from F(s) to F(s′), we set F(f) := p. If no shortest path exists, the +match fails. The F so constructed is only a candidate schema mapping: CQL +must verify that F preserves provable equality in S. +– The second technique attempts to infer a schema A and schema mappings +F : A → S and G : A → T. Such a span of mappings can be interpreted +as a query of the form ΣF ◦ ∆G or ΠF ◦ ∆G. Let c be some user-provided +string similarity cutoff. The entities of A are those pairs of S-entities and +T-entities (s, t) such that σ(s, t) > c. The symbols (s, t) → (s′, t′) of A are +those pairs of S-symbols and T-symbols (f : s → s′, g : t → t′) such that +σ(f, g) > c. The mappings F and G are projections. +4 +Conclusion +When comparing our algebraic approach to model management with other ap- +proaches originating in relational database theory [1] it is important to note that +our databases are “deductive databases” [1]. That is, we define databases “inten- +sionally”, as sets of equations, rather than as sets of tables. As such, care must +be taken when mediating between our definitions and relational definitions. For +example, our instances can be “inconsistent” in the sense that an instance can +prove 1 = 2 for two distinct constant symbols 1 and 2. Such situations are often, +but not always [12], errors, and the CQL tool checks for such situations using +standard techniques based on “conservative theory extensions” [12]. In addition, +our schemas do not define a set of constants (a “domain”) that all the instances +on that schema share, as is customary in relational database theory [7]. Hence +our approach is closer in spirit to traditional logic [6] than database theory [1]. +There are many connections between our algebraic approach to model man- +agement and the ED-based approach described in Section 2. EDs are more ex- +pressive than our purely equational data integrity constraints and can be added +to our formalism in a simple way, described in [22] (although in [22], EDs are +called “lifting problems”). In ED-based approaches the “chase” [5] operation has +a similar semantics to our Σ operation, and a formal comparison between the +chase and Σ is forthcoming. + +Acknowledgements. +The authors thank Lucian Popa, Eswaran Subrah- +manian, and Peter Gates and were supported by NIST SBIR grant 70NANB +16H178, AFOSR grant FA9550–14–1–0031 and NASA grant NNL14AA05C. +This paper appears in WADT 2016: Recent Trends in Algebraic +Development Techniques, pp 56–69. +References +1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley- +Longman (1995) +2. Alagic, S., Bernstein, P.: A model theory for generic schema management. DBPL +(2001) +3. Barr, M., Wells, C.: Category Theory for Computing Science. Prentice Hall Inter- +national (1995) +4. Bernstein, P.A., Melnik, S.: Model management 2.0: Manipulating richer mappings. +ICMD (2007) +5. Doan, H., Halevy, A., Ives, Z.: Principles of Data Integration. Morgan Kaufmann +(2012) +6. 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Haas, L.M., Hern´andez, M.A., Ho, H., Popa, L., Roth, M.: Clio grows up: From +research prototype to industrial tool. ICMD (2005) +15. Hodges, W.: A Shorter Model Theory. Cambridge University Press (1997) +16. Melnik, S.: Generic Model Management: Concepts And Algorithms (Lecture Notes +in Computer Science). Springer-Verlag (2004) +17. Mitchell, J.C.: Foundations of Programming Languages. MIT Press (1996) +18. Mossakowski, T., Krumnack, U., Maibaum, T.: What is a derived signature mor- +phism? RTADT (2014) +19. Schultz, P., Spivak, D.I., Vasilakopoulou, C., Wisnesky, R.: Algebraic databases. +Theory and Applications of Categories (2017) +20. Schultz, +P., +Wisnesky, +R.: +Algebraic +data +integration +(unpublished). +http://arxiv.org/abs/1503.03571 (2016) +21. Spivak, D.I.: Functorial data migration. Information and Computation (2012) +22. Spivak, D.I.: Database queries and constraints via lifting problems. Mathematical +Structures in Computer Science (2014) +23. Spivak, D.I., Wisnesky, R.: Relational foundations for functorial data migration. +DBPL (2015) + diff --git a/4NE4T4oBgHgl3EQfAwuD/content/tmp_files/load_file.txt b/4NE4T4oBgHgl3EQfAwuD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3271e0a4cefa2901ec3664b540f46dbc777c2305 --- /dev/null +++ b/4NE4T4oBgHgl3EQfAwuD/content/tmp_files/load_file.txt @@ -0,0 +1,506 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf,len=505 +page_content='Algebraic Model Management: A Survey Patrick Schultz1, David I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Spivak1, and Ryan Wisnesky2 1 Massachusetts Institute of Technology 2 Conexus AI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We survey the field of model management and describe a new model management approach based on algebraic specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 1 Introduction In this paper we survey the field of model management and describe a new model management approach based on techniques from the field of algebraic specification, with the hope of establishing an interlingua between the two fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' By “model management” we mean “meta-data intensive” database management in the sense of Bernstein & Melnik [4], which we define in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' By “a new algebraic model management approach” we mean our particular way [19] [20] of specifying database schemas and instances using algebraic (equational) theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We first noticed a connection between model management and algebraic spec- ification while investigating applications of category theory [3] to data integra- tion [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' These investigations are described in [19] and [20], and we present no substantial new results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We assume readers have basic proficiency with category theory [3], algebraic specification [17], and SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In Section 2 we describe the traditional approach to model manage- ment and in Section 3 we describe our algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Also in Section 3 we describe the open-source CQL (Categorical Query Language) tool, available for download at http://categoricaldata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='net, which implements our approach in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We conclude in Section 4 by comparing our approach with the tradi- tional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2 Model Management To quote from Melnik [16]: Many challenging problems facing information systems engineering in- volve the manipulation of complex metadata artifacts, or models, such as database schemas, interface specifications, or object diagrams, and mappings between models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The applications that solve metadata ma- nipulation problems are complex and hard to build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The goal of generic model management is to reduce the amount of programming needed to develop such applications by providing a database infrastructure in which a set of high-level algebraic operators, such as Match, Merge, and Compose, are applied to models and mappings as a whole rather than to their individual building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='04846v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='LO] 12 Jan 2023 In the paragraph above the word “model” is defined to mean a metadata ar- tifact such as a schema, which conflicts with the definition of the word “model” as a structure satisfying a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In this paper, we use the phrase “model man- agement” to mean the field identified above, and use the word “model” to mean a structure satisfying a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Today model management is a large sub-field of information management with a research literature containing hundreds of published articles [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' There is a consensus in that literature [4] that model management is concerned with at least the problems described in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Schema mapping Given two database schemas S and T, the schema mapping problem [7] is to construct a “mapping” F : S → T that captures some user-specified relationship between S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Different model management systems use different notions of schema, including SQL, XML, and RDF [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The most common mapping formalism studied in the literature is that of “embedded dependencies” (EDs) [5]: formulae in a fragment of first-order logic with useful computational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We will use SQL schemas and EDs in our examples in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Consider the following SQL schema S, consisting of two tables connected by a foreign key: CREATE TABLE N2(ID INT PRIMARY KEY, age INT) CREATE TABLE N1(ID INT PRIMARY KEY, name STRING, salary INT, f INT FOREIGN KEY REFERENCES N2(ID)) and the following SQL schema T, consisting of one table: CREATE TABLE N(ID INT PRIMARY KEY, age INT, name STRING, salary INT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' These two SQL schemas are displayed graphically in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example schema mapping F : S → T expressing that the target table N is the join of source tables N1 and N2 along the column f is: ∀id1, id2, a, n, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' N1(id1, n, s, id2) ∧ N2(id2, a) → N(id1, a, n, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Two instances satisfying the above ED are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In general, many EDs can map between two SQL schemas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 Query generation Given a schema mapping F : S → T, the query generation problem [4] is to construct a query which converts databases on S to databases on T in a way that satisfies F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The query languages typically studied include SQL, XQuery, and various comprehension- and λ-calculi [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' A SQL query to implement the example mapping from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 is: String N1• name � salary � f � N2• age � Int F −−−→ String N• name � age � salary � ◦ Int N1 ID name salary f 1 Alice $100 1 2 Bob $250 2 3 Sue $300 3 N2 ID age 1 20 2 20 3 30 �∆F � ←−−−−−− �ΠF �,�ΣF � −−−−−−−−−−→ N ID name salary age 1 Alice $100 20 2 Bob $250 20 3 Sue $300 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Example Data Migrations, with Foreign Keys (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2) INSERT INTO N SELECT N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID, N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='age, N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='name, N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='sal FROM N1, N2 WHERE N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f = N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID Technically, the INSERT portion of the above SQL code is not a “query”, but rather an “update”, and in practice the code generated from a query generation task will often store the results of the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example of running the above SQL is shown as the left-to-right direction of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In general, many or no SQL queries may implement a set of EDs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' EDs can also be directly executed by an algorithm called “the chase” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 Mapping Inversion Given a schema mapping F : S → T, the mapping inversion problem [10] is to construct a schema mapping F −1 : T → S that undoes F with respect to query generation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' the queries generated from F and F −1 should be inverses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The natural candidate ED to invert the schema mapping of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 ex- presses that N projects onto N1 and N2: ∀id1, a, n, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' N(id1, a, n, s) → ∃id2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' N1(id, n, s, id2) ∧ N2(id2, a) and a possible SQL implementation of this ED is: INSERT INTO N1 SELECT ID, name, sal, ID FROM N INSERT INTO N2 SELECT ID, age FROM N However, the above ED is not an inverse to the ED of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1, as is seen by taking ∅ = N1 ̸= N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Indeed, it is rare for an ED, or set of EDs, to be invertible, and weaker notions of inverse, such as “quasi-inverse” [10], are common in the literature [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example of running the above SQL is shown as the right-to-left direction of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 Mapping Composition Given schema mappings F : S → T and G : T → U, the mapping composition problem [8] is to construct a schema mapping G ◦ F : S → U that is equivalent with respect to the query generation problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' running the query generated from G ◦ F should have the same effect as running the query generated from G on the results of the query generated from F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The composition of the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 with the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 is ∀id1, id2, n, s, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' N1(id1, n, s, id2) ∧ N2(id2, a) → ∃x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' N1′(id, n, s, x) ∧ N2′(x, a) where N1’, N2’ are target “copies” of source tables N1, N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This composed ED is not the identity, thereby showing that the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 does not invert the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In the case of EDs, composed mappings may not exist [8], but some restrictions and extensions of EDs are closed under composition [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 Schema matching Given two database schemas S and T, the schema matching problem [5] is to au- tomatically find “correspondences” between S and T and to automatically infer schema mappings S → T from these correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In general, inference of en- tire mappings cannot be fully automated and the focus of the matching problem is to reduce the human effort required to construct a schema mapping by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=', suggesting partial mappings that can be completed by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' There are many techniques for schema matching ranging from comparison of column names by string similarity to machine learning algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' for an overview, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In the example from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1, two correspondences that are easy to automatically find are (N1, N) and (N2, N) and tools such as Clio [14] can create the ED from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 from these two correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 Further References In this paper we will focus on the problems described in the previous sec- tions, but many other problems are studied in the model management litera- ture [4], and many of these problems are related to algebraic specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For example, schema/instance merge problems [4], which arise often in data inte- gration scenarios [2], can be formalized as pushouts in suitable categories of schemas/instances [20], and such pushouts are related to model-theoretic con- cepts such as model amalgamation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Many software products solve model management problems [4], including ETL (Extract, Transform, Load) tools [5], which extract data from separate databases, apply user-specified transformations, and then load the result into a target system such as a data warehouse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' query mediators [5], which answer queries about a “virtual” integrated database by combining queries about sep- arate source databases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' and visual schema mapping tools [14] which allow users to create schema mappings by visually connecting related schema elements with lines, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' There have been at least two attempts to provide a “meta semantics” for model management operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In [16] Melnik gives a “state based” meta se- mantics to some of the above operations by defining a schema mapping S → T to be an arbitrary binary relation between instances on S and instances on T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' the ED-based semantics described above is an instantiation of this meta seman- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In [2] and [13] the authors give an “institution theoretic” meta semantics to some of the above operations by defining a schema mapping S → T to be a morphism in a suitable category of schemas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL’s semantics is an instantiation of this meta semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' A schema mapping in Clio [14] file:students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='xsml (managed) 口 Source Target S File:StudentsSource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='xsd S File:StudentsTarget,xsd qa e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' targetDB e gradEnrolls e Evaluations gradEnroll [o,*] [* eval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' *] e: sid (xsistring)- e: name (xsistring)-- e: grade (xsiint) (buuisisn) pp a e?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' File (μsistring) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='. @ Enrollment e: File (xsistring)- e* Student [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' *] e* underGrad [o,*] e: sid (xsistring) e: name (rsistring) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' (ouuisisi) pis a e Courses e: name (xsistring) e: address (xsistring) [* course [o,*] e enrolls e eid (xsint) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='.- [ enroll [o,*] e: addr (xsistring) e cid (xsistring) e: sid (rsistring)3 Algebraic Model Management Our approach to model management is based on the algebraic approach to databases, data migration, and data integration we describe in [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Those works, and hence this work, extend a particular category-theoretic data model that originated in the late 1990s [11] and was later extended in [21] and [23] and implemented in CQL (http://categoricaldata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In the next section we describe our formalism for database schemas and instances and introduce CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The subsequent sections implement the model management operations from Section 2 using our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In this section we abbreviate “algebraic theory” as “theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Algebraic Databases In our formalism [20], database schemas and instances are defined as theories of a certain kind, which we describe in the next sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For ease of exposition, we will sometimes conflate schemas and instances as defined in our formalism with their CQL equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Type sides We first fix a theory, Ty, called the type side of our formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The sorts of Ty are called types and the functions of Ty are the functions that can appear in schemas and instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL allows arbitrary theories to be used as type sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' But we have found that in practice, CQL users almost always want to use the theory of an existing programming language, say java, for their type side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The ability to “bind” CQL to an existing language is particularly important in model management because input data may only be accessible through, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=', a java API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For this reason, CQL allows a type side to be defined by specifying, for each sort s, a java class Cs and a java function String → Cs that tells CQL how to interpret the strings it encounters in CQL programs as objects of Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example CQL type side about integers and strings is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This type side defines a theory with two sorts and infinitely many constants – all the java strings and integers – and no equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The java code for Int says that whenever a string x is encountered in an CQL program and a term of sort Int is required, that java’s parseInt function should be applied to x to yield the desired Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The keyword literal, used in many places in CQL, indicates a literal (user-defined constant) definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Schemas A schema on type side Ty is a theory extending Ty with new sorts (called entities), new unary functions from entities to types (called attributes), new unary functions from entities to entities (called foreign keys), and new equa- tions (called data integrity constraints) of the form ∀v : s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' t = t′, where s is an entity and t, t′ are terms of the same type, each containing a single free variable v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The restrictions in the preceding sentence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=', no functions from types to enti- ties) are necessary to use our formalism for model management purposes [19] [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Figure 4 shows the CQL schemas corresponding to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' These schemas contain no equations and are both on the type side Ty defined in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' typeside Ty = literal { java_types String = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='String" Int = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Integer" java_constants String = "return input[0]" Int = "return java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='parseInt(input[0])" } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL type side Ty schema S = literal : Ty { entities N1 N2 foreign_keys f : N1 -> N2 attributes name : N1 -> String salary : N1 -> Int age : N2 -> Int } schema T = literal : Ty { entities N attributes name : N -> String salary : N -> Int age : N -> Int } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL schemas S and T on type side Ty instance I = literal : S { generators 1 2 3 : N1 equations name(1) = Alice salary(1) = 100 age(f(1)) = 20 name(2) = Bob salary(2) = 250 age(f(2)) = 20 name(3) = Sue salary(3) = 300 age(f(3)) = 30 } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL instance I on schema S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Initial algebra for CQL instance I Delta - 9:36:11 PM 1 (exec: 0s)(gui: 0s) Select: Tables Type Algebra DP Text typeside Ty N1 (3) N2 (3) schema S ID name salary f ID age schema T [1] Alice 100 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] 20 mapping F : S -> T [2] Bob 250 [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] 20 instance I : S [3] Sue 300 [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='f] 30Instances An instance I on schema S is a theory extending S with new 0-ary function (constant) symbols called generators and non-quantified equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example CQL instance on schema S (Figure 4) is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The intended meaning of an instance I, written �I�, is the term model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=', initial algebra) for I which contains, for each sort s, a carrier set consisting of the closed terms of sort s modulo provability in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' A morphism of instances I → J is a homomorphism (natural transformation) of algebras �I� → �J�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Figure 6 shows the meaning of the instance I from Figure 5 in the CQL tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The CQL tool visually displays term models as sets of tables, one per entity e, each with an ID column corresponding to the carrier set for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The tables in Figure 6 are isomorphic to the left tables in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In the following sections we implement the model management operations from Section 2 using the preceding definitions of schema and instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 Schema mapping Given schemas S, T, the schema mapping problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1) is to construct a “mapping” F : S → T that captures some relationship between S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Let S and T be CQL schemas on the same type side Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An CQL schema mapping F : S → T is defined as a “derived signature morphism” [18] from S to T that is the identity on Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' That is, F : S → T assigns to each entity e ∈ S an entity F(e) ∈ T, and to each attribute / foreign key f : s → s′ a term F(f), of type F(s′) and with one free variable of type F(s), in a way that respects equality: if S ⊢ t = t′, then T ⊢ F(t) = F(t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' We have found that many mappings arising in practice cannot be expressed using plain signature morphisms and require the more general notion of “derived” signature morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Whereas a schema mapping in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 was an ED (formula in a fragment of first-order logic), which induces a single binary satisfaction relation between instances, CQL schema mappings are derived signature morphisms and induce three relations between instances, which we will describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' An example CQL schema mapping F : S → T is shown in Figure 7, where CQL schemas S and T are defined in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This mapping is also shown graphically in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 Query generation Given a mapping F : S → T, the query generation problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2) is to use F to construct a query which converts databases on S to databases on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In our formalism, the database instances and morphisms on a schema S constitute a category, denoted S–Inst, and a schema mapping F : S → T induces a functor ΣF : S–Inst → T–Inst defined by substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The functor ΣF has a right adjoint, ∆F : T–Inst → S–Inst, which corresponds to the “model reduct functor” when our formalism is described in institution-theoretic terms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The functor ∆F has a right adjoint, ΠF : S–Inst → T–Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' See [19] for proof that ∆F always has left and right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' As adjoints, ∆F , ΠF preserve limits and ∆F , ΣF preserve colimits, implying many useful properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' for example, ΣF (I + J) ∼= ΣF (I) + ΣF (J) and ΠF (I × J) ∼= ΠF (I) × ΠF (J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' mapping F = literal : S -> T { entities N1 -> N N2 -> N foreign_keys f -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' x attributes name -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' name(x) salary -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' salary(x) age -> lambda x:N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' age(x) } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' CQL schema mapping F : S → T Note that unlike Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1, where there was a single query associated with a schema mapping (ED), in our algebraic approach there are three queries, one for each of ∆F , ΣF , ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The conditions under which ∆F ,ΣF , ΠF can be expressed in SQL and vice-versa are characterized in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Although it is possible to give explicit formulae to define ∆F , ΣF , ΠF [19] we instead give examples in Figures 1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Note that in these examples we are not showing instances (theories) as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' we are showing term models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For this reason, we surround ∆F , ΣF , ΠF with denotation brackets �� in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In addition, as adjoints ∆, Σ, Π are only defined up to unique isomorphism, so we arbitrarily make up names for IDs and in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Figures 1 and 8 show an CQL schema mapping F which takes two distinct source entities, N1 and N2, to the target entity N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The �∆F � functor projects in the opposite direction of F: it projects columns from the single table for N to two separate tables for N1 and N2, similar to FROM N AS N1 and FROM N AS N2 in SQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' When there is a foreign key from N1 to N2, the �∆F � functor populates it so that N can be recovered by joining N1 and N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The �ΠF � functor takes the cartesian product of N1 and N2 when there is no foreign key between N1 and N2, and joins N1 and N2 along the foreign key when there is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The �ΣF � functor disjointly unions N1 and N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' because N1 and N2 are not union compatible (have different columns), �ΣF � creates null values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' When there is a foreign key between N1 and N2, �ΣF � merges the tuples that are related by the foreign key, resulting in a join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' As these examples illustrate, ∆F can be thought of as projection, ΠF can be thought of as a product followed by a filter (which can result in a join), and ΣF can be thought of as a disjoint union (which does not require union-compatibility) followed by a merge (which can also result in a join).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 Mapping Composition Given schema mappings F : S → T and G : T → U, the mapping composition problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4) is to construct a schema mapping G ◦ F : S → U that is equivalent with respect to query generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In one sense, the mapping composition problem is trivial [19] for our for- malism: ∆F ◦G ∼= ∆G ◦ ∆F , ΠF ◦G ∼= ΠF ◦ ΠG, and ΣF ◦G ∼= ΣF ◦ ΣG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' But ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='String N1• ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='salary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Alice $100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID name salary age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Alice $100 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID name salary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Alice $100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='�ΣF � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='−−−−−−→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='salary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Alice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='age(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='age(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='age(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 name(4) salary(4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 name(5) salary(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 name(6) salary(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID name salary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Alice $100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='�ΠF � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='−−−−−−→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='ID name salary age ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='1 Alice $100 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='4 Alice $100 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='7 Alice $100 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Bob $250 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Sue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='$300 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Example Data Migrations (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='2) this solution is not wholly satisfactory because in practice a mixture of ∆, Σ, Π functors may be needed to accomplish any particular task (similarly, in SQL a mixture of joins and unions may be needed to accomplish any particular task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The following results are proved in [19] and [23]: – Every composition ΣF ◦∆G is isomorphic to ∆F ′ ◦ΣG′ for some F ′, G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This statement is also true if ΣF is replaced with ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' – Pairs of the form (F, G), denoting ΣF ◦ ∆G, are closed under composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This statement is also true if ΣF is replaced with ΠF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Such pairs can be spec- ified in an intuitive “select-from-where” syntax, described in [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' – Triples of the form (F, G, H), denoting ΣF ◦ ΠG ◦ ∆H, are closed under composition, provided that F is a discrete op-fibration [3], which is exactly the “union compatibility” condition [5] that ΣF performs unions over tables whose columns match;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Figure 1 is not a discrete op-fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5 Mapping Inversion Given a schema mapping F : S → T, the mapping inversion problem (Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='3) is to construct a mapping F −1 : T → S that somehow “undoes” F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Our formalism has strong inversion properties but does not have inverses per se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' When there exists F −1 : T → S such that F ◦ F −1 = id and F −1 ◦ F = id, then ∆F ◦ ∆F −1 ∼= id, ΣF ◦ ΣF −1 ∼= id, and ΠF ◦ ΠF −1 ∼= id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In general F need not have an inverse, but when S and T have finite initial algebras / term models (which is a priori undecidable, and implies decidability of S and T) it is possible to construct F −1 whenever it exists by considering all possible functors T → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' When F has a right adjoint G : T → S, a weaker condition than having an inverse, there are canonical morphisms ΣF → ∆G and ∆F → ΠG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In practice “round-tripping” [5] of data is desirable even when inverses do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For example, projection, because it forgets information, typically cannot be inverted, but we may want to remember where the projected data originated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In our formalism the adjunctions between Σ,∆,Π provide round-tripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For example, for every F : S → T and S-instance I there is a canonical morphism I → ∆F (ΣF (I)), the unit of the ΣF ⊣ ∆F adjunction, which describes where each ID in I is sent to by ΣF (and similarly for ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Dually, for every T- instance J there is a canonical morphism ΣF (∆F (J)) → J, the co-unit of the ΣF ⊣ ∆F adjunction, which describes where the IDs in ∆F (J) originate (and similarly for ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The unit and co-unit can be used to obtain, for every morphism h : ΣF (I) → J, a mate h′ : I → ∆F (J) and vice-versa (and similarly for ΠF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Relating adjointness to existing relaxed notions of inverse such as quasi- inverse [9] is an important area for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='6 Schema matching Given database schemas S and T, the schema matching problem (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='5) is to automatically suggest schema mappings S → T to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In this section, we define two schema matching techniques used by CQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Our techniques compare entities, and foreign keys and attributes (“symbols”) by name, as strings, and so our techniques depend on having (probably user- provided) names whose similarity as strings reflects their semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Let σ : String, String → [0, 1] be any string similarity function [5] where a value of 1 indicates a “good” match and a value of 0 indicates a “bad” match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' – The first technique attempts to infer a schema mapping F : S → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For each entity s ∈ S, we define F(s) := t where t ∈ T is an entity that maximizes σ(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For each symbol f : s → s′ ∈ S, we then consider the set X of symbols F(s) → F(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' If X is non-empty, we choose a symbol g ∈ X that maximizes σ(f, g) and set F(f) := g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' If X is empty but there is a shortest path p from F(s) to F(s′), we set F(f) := p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' If no shortest path exists, the match fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The F so constructed is only a candidate schema mapping: CQL must verify that F preserves provable equality in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' – The second technique attempts to infer a schema A and schema mappings F : A → S and G : A → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Such a span of mappings can be interpreted as a query of the form ΣF ◦ ∆G or ΠF ◦ ∆G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Let c be some user-provided string similarity cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The entities of A are those pairs of S-entities and T-entities (s, t) such that σ(s, t) > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The symbols (s, t) → (s′, t′) of A are those pairs of S-symbols and T-symbols (f : s → s′, g : t → t′) such that σ(f, g) > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The mappings F and G are projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' 4 Conclusion When comparing our algebraic approach to model management with other ap- proaches originating in relational database theory [1] it is important to note that our databases are “deductive databases” [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' That is, we define databases “inten- sionally”, as sets of equations, rather than as sets of tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' As such, care must be taken when mediating between our definitions and relational definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' For example, our instances can be “inconsistent” in the sense that an instance can prove 1 = 2 for two distinct constant symbols 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Such situations are often, but not always [12], errors, and the CQL tool checks for such situations using standard techniques based on “conservative theory extensions” [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In addition, our schemas do not define a set of constants (a “domain”) that all the instances on that schema share, as is customary in relational database theory [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Hence our approach is closer in spirit to traditional logic [6] than database theory [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' There are many connections between our algebraic approach to model man- agement and the ED-based approach described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' EDs are more ex- pressive than our purely equational data integrity constraints and can be added to our formalism in a simple way, described in [22] (although in [22], EDs are called “lifting problems”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' In ED-based approaches the “chase” [5] operation has a similar semantics to our Σ operation, and a formal comparison between the chase and Σ is forthcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' The authors thank Lucian Popa, Eswaran Subrah- manian, and Peter Gates and were supported by NIST SBIR grant 70NANB 16H178, AFOSR grant FA9550–14–1–0031 and NASA grant NNL14AA05C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' This paper appears in WADT 2016: Recent Trends in Algebraic Development Techniques, pp 56–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Abiteboul, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' : Functorial data migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Information and Computation (2012) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Spivak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' : Database queries and constraints via lifting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Mathematical Structures in Computer Science (2014) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' Spivak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=', Wisnesky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=': Relational foundations for functorial data migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} +page_content=' DBPL (2015)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE4T4oBgHgl3EQfAwuD/content/2301.04846v1.pdf'} diff --git a/5dFJT4oBgHgl3EQfkywG/content/tmp_files/2301.11580v1.pdf.txt b/5dFJT4oBgHgl3EQfkywG/content/tmp_files/2301.11580v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d0dbf9d54b55d2a28b13376c48fbd2a2506660fa --- /dev/null +++ b/5dFJT4oBgHgl3EQfkywG/content/tmp_files/2301.11580v1.pdf.txt @@ -0,0 +1,818 @@ +A Characterization of Complexity in Public Goods Games +MATAN GILBOA +We complete the characterization of the computational complexity of equilibrium in public goods games on +graphs by proving that the problem is NP-complete for every finite non-monotone best-response pattern. This +answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by +[Papadimitriou and Peng, 2021], for all finite best-response patterns. +Manuscript submitted for review to the 24nd ACM Conference on Economics & Computation (EC'23). +arXiv:2301.11580v1 [cs.GT] 27 Jan 2023 + +M. Gilboa +1 +1 +INTRODUCTION +Public goods games describe scenarios where multiple agents face a decision of whether or not +to produce some "good", such that producing this good benefits not only themselves, but also +other (though not necessarily all) agents. Typically, we consider the good to be costly to produce, +and therefore an agent might choose not to produce it, depending on the actions of the agents +that affect her. This type of social scenarios can be found in various real-life examples, such as +vaccination efforts (an individual pays some personal cost for being vaccinated but she and other +people in her proximity gain from it) and research efforts (a research requires many resources, +but the researcher benefits from the result along with other researchers in similar areas). As is +common in the literature, to model this we use an undirected graph, where each node represents +an agent and an edge between two nodes captures the fact that these nodes directly affect one +another by their strategy. As in [Kempe et al., 2021, Maiti and Dey, 2022, Papadimitriou and Peng, +2021, Yang and Wang, 2020, Yu et al., 2021], in our model the utility of an agent is completely +determined by the number of productive neighbors she has, as well as by her own action. We focus +on a specific version of the game which has the following characteristics. Firstly, our strategy space +is binary, i.e. an agent can only choose whether or not to produce the good, rather than choose +a quantity (we call an agent who produces the good a productive agent); secondly, our game is +fully-homogeneous, meaning that all agents share the same utility function and cost of producing +the good; and thirdly, our game is strict, which means that an agent has a single best response to +any number of productive neighbors she might have (i.e. we do not allow indifference between the +actions). +The game is formally defined by some fixed cost 𝑐 of producing the good, and by some "social" +function 𝑋 (𝑠𝑖,𝑛𝑖), which takes into account the boolean strategy of agent 𝑖 and the number of +productive neighbors she has (marked as 𝑠𝑖 and 𝑛𝑖 respectively), and outputs a number representing +how much the agent gains. The utility 𝑢𝑖 of agent 𝑖 is then given by the social function 𝑋 (𝑠𝑖,𝑛𝑖), +reduced by the cost 𝑐 if the agent produces the good, i.e. 𝑢𝑖 (𝑠𝑖,𝑛𝑖) = 𝑋 (𝑠𝑖,𝑛𝑖) −𝑐 ·𝑠𝑖. However, since +any number of productive neighbors yields a unique best response (i.e. the game is strict), we can +capture the essence of the utility function and the cost using what we call (as in [Gilboa and Nisan, +2022]), a Best-Response Pattern𝑇 : IN → {0, 1}. We think of the Best-Response Pattern as a boolean +vector in which the 𝑘𝑡ℎ entry represents the best response to exactly 𝑘 productive neighbors. We +are interested in the problem of determining the existence of a non-trivial pure Nash equilibrium +in these games, which is defined as follows. +Equilibrium decision problem in a public goods game: For a fixed Best-Response Pattern +𝑇 : IN → {0, 1}, and with an undirected graph 𝐺 = (𝑉, 𝐸) given as input, determine whether there +exists a pure non-trivial Nash equilibrium of the public goods game defined by 𝑇 on 𝐺, i.e. an +assignment 𝑠 : 𝑉 → {0, 1} that is not all 0, such that for every 1 ≤ 𝑖 ≤ |𝑉 | we have that +𝑠𝑖 = 𝑇 [ +∑︁ +𝑗 +𝑠.𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸 +𝑠𝑗]. +The first Best-Response Pattern for which this problem was studies was the so-called Best-Shot +pattern (where an agent’s best response is to produce the good only if she has no productive +neighbors), which was shown in [Bramoullé and Kranton, 2007] to have a pure Nash equilibrium +in any graph. In [Bramoullé and Kranton, 2007], they also show algorithmic results for "convex" +patterns, which are monotonically increasing (best response is 1 if you have at least 𝑘 productive +neighbors). The question of characterizing the complexity of this problem for all possible patterns + +M. Gilboa +2 +was first raised by [Papadimitriou and Peng, 2021], where they manage to fully answer an equivalent +problem on directed graphs, showing NP-completeness for most patterns, and algorithmic results +for the remaining few. The open question on undirected graphs was then partially answered in +[Gilboa and Nisan, 2022], where they show NP-completeness for several classes of patterns, and a +polynomial-time algorithm for one other pattern. There have been several studies concerning other +versions of this problem as well. In [Yang and Wang, 2020], the general version of this problem +(where the pattern is part of the input rather than being fixed) was shown to be NP-complete +when removing the strictness assumption, (i.e. allowing indifference between actions, such that +both 0 and 1 are best responses in certain cases) 1. In [Yu et al., 2021], NP-completeness is shown +for the general version of the problem in the heterogeneous public goods game, in which the +utility function varies between agents. In [Kempe et al., 2021], they show NP-completeness of the +equilibrium problem when restricting the equilibrium to have at least 𝑘 productive agents, or at +least some specific subset of agents. In [Maiti and Dey, 2022], the parameterized complexity of the +equilibrium problem is studied, for a number of parameters of the graph on which the game is +defined. +Papadimitirou and Peng raised the problem of characterizing all Best-Response Patterns, and +Gilboa and Nisan suggested two specific open problems regarding two specific patterns. One of +these patterns has been recently solved by Max Klimm (personal communication) who showed +that all monotonically decreasing patterns can be viewed as potential games, and thus always have +a pure Nash equilibrium2. +Our main contribution is completing the characterization of the equilibrium decision problem +for all finite patterns, by showing that for all non-monotone patterns the problem is NP-complete. +Theorem: For any Best-Response Pattern that is non-monotone and finite (i.e., has a finite number +of entries with value 1), the equilibrium decision problem in a public goods game is NP-complete +(under Turing reductions). +The first step along this way was to prove NP-completeness for the specific open problem by +[Gilboa and Nisan, 2022]. It has come to our attention that an alternative proof to this specific open +problem was obtained independently and concurrently by Max Klimm and Maximilian Stahlberg +(private communication). +We note that we only focus on finite patterns, which we believe to be more applicable to real-life +problems that can be modeled by this game. We believe that the characterization for all infinite +patterns is of interest, and remains open, though some results can be found in Corollary 3.9. +The rest of this paper is organized as follows. In Section 2 we introduce the formal model and +some relevant definitions. We then set out to show hardness of all remaining patterns, dividing +them into classes. In Section 3 we present a solution for an open question from [Gilboa and Nisan, +2022], showing hardness of a pattern we call the 0-Or-2-Neighbors Best Response Pattern, and +expanding the result to a larger sub-class of patterns that begin with 1,0,1. In Section 4 we show +hardness of all patterns beginning with 1,0,0 (where we also have a slightly more subtle division +into sub-classes), and in Section 5 we show hardness of all patterns beginning with 1,0,1 that were +not covered in Section 3, thus completing the characterization for all finite patterns. The outline of +this paper is also depicted3 in Figure 1. +1The paper [Yu et al., 2020] had an earlier version [Yu et al., 2021] which presented a proof for this case as well, but an error +in the proof was pointed out by [Yang and Wang, 2020], who then also provided an alternative proof. +2Alternatively, Sigal Oren (personal communication) observed that known results about 𝑘-Dominating and 𝑘-independent +sets [Chellali et al., 2012] (Theorem 19) can be used to prove this. +3Some patterns which start with 1,0 were solved in [Gilboa and Nisan, 2022], though for simplicity we omit them from +Figure 1. + +M. Gilboa +3 +Fig. 1. Outline of this paper. +2 +MODEL AND DEFINITIONS +A Public Goods Game (PGG) is defined on an undirected graph 𝐺 = (𝑉, 𝐸), 𝑉 = {𝑣1, ..., 𝑣𝑛}, where +each node represents an agent. The strategy space, which is identical for all agents, is 𝑆 = {0, 1}, +where 1 represent producing the good and 0 represents not producing it. The utility of node +𝑣𝑖 (which is assumed to be the same for all agents) is completely determined by the number of +productive neighbors 𝑣𝑖 has, as well as by 𝑣𝑖’s own strategy. Moreover, our model is restricted to +utility functions where an agent always has a single best response to the strategies of its neighbors, +i.e. there is no indifference between actions in the game. Therefore, rather than defining a PGG +with an explicit utility function and cost for producing the good, we can simply consider the best +response of an agent for any number of productive agents in its neighborhood. Essentially, this can +be modeled as a function 𝑇 : IN → {0, 1}, which, as in [Gilboa and Nisan, 2022], we represent in +the form of a Best Response Pattern: +Definition 2.1. A Best-Response Pattern (BRP) of a PGG, denoted by 𝑇, is an infinite boolean +vector in which the 𝑘𝑡ℎ entry indicates the best response for each agent 𝑣𝑖 given that exactly 𝑘 +neighbors of 𝑣𝑖 (excluding 𝑣𝑖) produce the good: +∀𝑘 ≥ 0 𝑇 [𝑘] = best response to 𝑘 productive neighbors. +Definition 2.2. Given a Public Goods Game defined on a graph 𝐺 = (𝑉, 𝐸) with respect to a +BRP 𝑇, a strategy profile 𝑠 = (𝑠1, ...,𝑠𝑛) ∈ 𝑆𝑛 (where 𝑠𝑖 ∈ {0, 1} represents the strategy of node +𝑣𝑖 ∈ 𝑉 ) is a pure Nash equilibrium (PNE) if all agents play the best response to the strategies of +their neighbors: +∀1 ≤ 𝑖 ≤ 𝑛 𝑠𝑖 = 𝑇 [ +∑︁ +𝑗 +𝑠.𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸 +𝑠𝑗]. + +finite,non-monotonepatterns +starts with 0 +starts with 1,0 +starts with 1,1 +Solved +Solved +[GilboaandNisan,2022] +[GilboaandNisan,2022] +starts with1,0,1 +starts with1,0,0 +of the form: +all other forms +has"isolated" odd 1 +doesn't have +1,0,1,0,1,0,..,1,0,0,0,... +"isolated"odd 1 +or +1,0,1,0,1,0, .,1,1,?,?.. +Section 3 +Section 5 +Section 4.1 +Section 4.2M. Gilboa +4 +In addition, if there exists 1 ≤ 𝑖 ≤ 𝑛 s.t 𝑠𝑖 = 1, then 𝑠 is called a non-trivial pure Nash equilibrium +(NTPNE). +We note that throughout the paper we also use the notation 𝑣𝑖 = 0 and 𝑣𝑖 = 1 to indicate the +strategy of some node 𝑣𝑖, rather than use 𝑠𝑖 = 0 and 𝑠𝑖 = 1, respectively. +Definition 2.3. For a fixed BRP 𝑇, the non-trivial4 pure Nash equilibrium decision problem +corresponding to 𝑇, denoted by NTPNE(𝑇), is defined as follows: The input is an undirected graph +𝐺. The output is ’True’ if there exists an NTPNE in the PGG defined on 𝐺 with respect to 𝑇, and +’False’ otherwise. +Definition 2.4. A BRP 𝑇 is called monotonically increasing (resp. decreasing) if ∀𝑘 ∈ IN, 𝑇 [𝑘] ≤ +𝑇 [𝑘 + 1] (resp. 𝑇 [𝑘] ≥ 𝑇 [𝑘 + 1]). +Definition 2.5. A BRP 𝑇 is called finite if it has a finite number of entries with value 1: +∃𝑁 ∈ IN 𝑠.𝑡 ∀𝑛 > 𝑁 𝑇 [𝑛] = 0 +As seen in Figure 1, the only patterns for which the equilibrium decision problem remains open +are patterns that begin with 1,0. We divide those into the two following classes of patterns. +Definition 2.6. A BRP 𝑇 is called semi-sharp if: +(1) 𝑇 [0] = 1 +(2) 𝑇 [1] = 𝑇 [2] = 0 +i.e. 𝑇 begins with 1, 0, 0. +Definition 2.7. A BRP 𝑇 is called spiked if: +(1) 𝑇 [0] = 𝑇 [2] = 1 +(2) 𝑇 [1] = 0 +i.e. 𝑇 begins with 1, 0, 1. +3 +HARDNESS OF THE 0-OR-2-NEIGHBORS PATTERN +In this section we show that the equilibrium problem is NP-complete for the 0-Or-2-Neighbors +pattern, and provide some intuition about the problem. This result answers an open question by +Gilboa and Nisan [Gilboa and Nisan, 2022]. We then expand this to show hardness of a slightly +more general class of patterns. In the 0-Or-2-Neighbors BRP the best response is 1 only to zero or +two productive neighbors, as we now define. +Definition 3.1. The 0-Or-2-Neighbors Best Response Pattern is defined as follows: +∀𝑘 ∈ IN 𝑇 [𝑘] = +� +1 +if 𝑘 = 0 𝑜𝑟 𝑘 = 2 +0 +otherwise +i.e. +𝑇 = [1, 0, 1, 0, 0, 0, ...]. +Theorem 3.2. Let 𝑇 be the 0-Or-2 Neighbors BRP. Then NTPNE(𝑇) is NP-complete. +4In this paper, we only study BRPs where the best response for zero productive neighbors is 1, for which there never exists +a trivial all-zero PNE (as these are the only BRPs left to solve). However, we sometimes reduce from patterns where this is +not the case, and therefore include the non-triviality restriction in our problem definition, in order to correspond with the +literature. + +M. Gilboa +5 +Before proving the theorem, we wish to provide basic intuition about the 0-Or-2-Neighbors BRP, +by examining several simple graphs. Take for example a simple cycle graph. Since𝑇 [2] = 1 (i.e. best +response for two productive neighbors is 1), we have that any simple cycle admits a non-trivial pure +Nash equilibrium, assigning 1 to all nodes (see Figure 2. However, looking at a simple path with 𝑛 +nodes, we see that the all-ones assignment is never a pure Nash equilibrium. The reason for this is +that 𝑇 [1] = 0 (i.e. best response for one productive neighbors is 0), and so the two nodes at both +edges of the path, having only one productive neighbor, do not play best response. Nevertheless, +any simple path does admit a pure Nash equilibrium. To see why, let us observe the three smallest +paths, of length 2,3 and 4. Notice that in a path of length two a PNE is given by the assignment 0,1; +in a path of length three a PNE is given by the assignment 0,1,0; and in a path of length four a PNE +is given by the assignment 1,0,0,1. We can use these assignment to achieve a PNE in any simple +path: given a simple path of length 𝑛, if 𝑛 ≡ 0 (mod 3) we use the path of length three as our basis, +adding 0,1,0 to it as many times as needed; if 𝑛 ≡ 1 (mod 3) we use the path of length four as our +basis, adding 0,0,1 to it as many times as needed; and if 𝑛 ≡ 2 (mod 3) we use the path of length +two as our basis, adding 0,0,1 to it as many times as needed (see example in Figure 3). +Fig. 2. PNE in cycles. +Fig. 3. PNE in paths of lengths 2 and 5. +In contrast to the graphs we have discussed so far, there are graphs in which a pure Nash +equilibrium doesn’t exist for the 0-Or-2-Neighbors pattern. An example of this can be seen in a +graph composed of four triangles, connected as a chain where each two neighboring triangles have +a single overlapping vertex, as demonstrated in Figure 4. One may verify that no PNE exists in this +graph. This specific graph will also be of use to us during our proof5. +Fig. 4. No PNE exists in this graph. +Having provided some intuition regarding the problem, we move on to prove Theorem 3.2. The +reduction is from ONE-IN-THREE 3SAT, which is a well known NP-complete problem [Schaefer, +1978]. In ONE-IN-THREE 3SAT, the input is a CNF formula with 3 literals in each clause, and the +5The Negation Gadget defined throughout the proof of Theorem 3.2 is constructed similarly to the graph described here. + +10 +0 +0 +0M. Gilboa +6 +Fig. 5. Clause Gadget. +goal is to determine whether there exists a boolean assignment to the variables such that in each +clause exactly one of the literals is assigned True. We begin by introducing our Clause Gadget, +which is a main component of the proof. Given a CNF formula, for each of its clauses we construct +a 21-nodes Clause Gadget, in which three of the nodes, denoted 𝑙1,𝑙2,𝑙3 (also referred to as the +literal nodes) represent the three literals of the matching clause. The purpose of this gadget is to +enforce the property that in any NTPNE, exactly one literal node in the gadget will be assigned 1, +which easily translates to the key property of a satisfying assignment in the ONE-IN-THREE 3SAT +problem. The three literal nodes are connected to one another, forming a triangle. Additionally, for +each 𝑖 ∈ {1, 2, 3}, 𝑙𝑖 is connected to two other nodes 𝑥𝑖,𝑦𝑖, which are also connected to one another, +forming another triangle. Lastly, 𝑥𝑖 and 𝑦𝑖 each form yet another triangle, along with nodes 𝑎𝑖,𝑏𝑖 +and 𝑐𝑖,𝑑𝑖 respectively. We refer to 𝑥𝑖,𝑦𝑖,𝑎𝑖,𝑏𝑖,𝑐𝑖,𝑑𝑖 as the sub-gadget of 𝑙𝑖. We note that out of the +nodes of the Clause Gadget, only the literal nodes may be connected to other nodes outside of their +gadget, a property on which we rely throughout the proof. The structure of the Clause Gadget is +demonstrated in Figure 5, where each sub-gadget is colored differently. +The next four lemmas lead us to the conclusion that the gadget indeed has the desired property +mentioned above. +Lemma 3.3. In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if a literal node 𝑙𝑖 of 𝑐𝑔 +is assigned 1 then so are its two neighbors from its respective sub-gadget, 𝑥𝑖,𝑦𝑖. +Proof. Divide into cases. +Case 1: If 𝑥𝑖 = 𝑦𝑖 = 0, then if 𝑎𝑖 ≠ 𝑏𝑖 (meaning only one of them is assigned 1) then 𝑥𝑖 would +have two productive neighbors and would not be playing its best response. However, if 𝑎𝑖 = 𝑏𝑖 then +𝑎𝑖 and 𝑏𝑖 would not be playing their best response, and we reach a contradiction. +Case 2: If 𝑥𝑖 = 1,𝑦𝑖 = 0 (the case where 𝑥𝑖 = 0,𝑦𝑖 = 1 is, of course, symmetrical) then 𝑥𝑖 must +have exactly one more productive neighbor (either 𝑎𝑖 or 𝑏𝑖) in order to be playing best response. +But then that node would not be playing best response, in contradiction. + +C1 +d1 +a1 +X1 +Y1 +V3 +3 +3 +V2 +a3M. Gilboa +7 +Case 3: We are left with the option where 𝑥𝑖 = 𝑦𝑖 = 1, where it is easy to verify that all nodes of +the sub-gadget of 𝑙𝑖 are playing their best response if we set 𝑎𝑖 = 𝑏𝑖 = 𝑐𝑖 = 𝑑𝑖 = 0. +□ +Lemma 3.4. In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if one of the literal +nodes 𝑙𝑖 of 𝑐𝑔 is assigned 1 then the other two literal nodes of 𝑐𝑔 must be assigned 0. +Proof. Since 𝑙𝑖 = 1, from Lemma 3.3 we have that 𝑥𝑖 = 𝑦𝑖 = 1. Therefore, 𝑙𝑖 has two productive +neighbors and cannot have any more, and so we have that the other two literal nodes must play +0. +□ +Lemma 3.5. In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if exactly one of the literal nodes of +𝑐𝑔 is assigned 1 then there exists a unique assignment to the other nodes of 𝑐𝑔 such that they all play +their best response.6 +Proof. W.l.o.g assume that 𝑙1 = 1,𝑙2 = 𝑙3 = 0. Then, focusing first on the sub-gadget of 𝑙1, +according to Lemma 3.3 we have that 𝑥1 = 𝑦1 = 1. Since 𝑥1,𝑦1 already have two productive +neighbors, they mustn’t have any others, and so it must be that 𝑎1 = 𝑏1 = 𝑐1 = 𝑑1 = 0. We move +on to the sub-gadget of 𝑙2. If 𝑥2 ≠ 𝑦2 then 𝑙2 would have 2 productive neighbors and would not be +playing its best response. If 𝑥2 = 𝑦2 = 1 then there is no assignment to 𝑎2,𝑏2 s.t 𝑎2,𝑏2,𝑥2 all play +their best response. Therefore 𝑥2 = 𝑦2 = 0. We are left only with the option of setting 𝑎2 ≠ 𝑏2 and +𝑐2 ≠ 𝑑2 (for instance 𝑎2 = 𝑐2 = 1,𝑏2 = 𝑑2 = 0). The sub-gadget of 𝑙3 is symmetrical to that of 𝑙2. One +may verify that in this assignment all nodes of 𝑐𝑔 indeed play their best response. +□ +Lemma 3.6. In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if all three of the literal nodes of +𝑐𝑔 are assigned 0, and the literal nodes do not have any productive neighbors outside of 𝑐𝑔, then the +assignment is not a PNE. +Proof. Assume by way of contradiction that there exists a PNE where 𝑙1 = 𝑙2 = 𝑙3 = 0, and all +three of them have no productive neighbors outside 𝑐𝑔. It must be that the other two neighbors of +𝑙1, 𝑥1,𝑦1, are assigned with different values (otherwise 𝑙1 is not playing its best response). W.l.o.g +assume 𝑥1 = 1,𝑦1 = 0. Now, If the remaining neighbors of 𝑦1 (𝑐1 and 𝑑1) are both assigned with 0 or +both assigned with 1, then they themselves would not be playing their best response. On the other +hand, if we assign them with different values then 𝑦1 would not be playing its best response, and +so we have reached a contradiction. +□ +So far, we have seen that in any PNE which includes a Clause Gadget, it must be that exactly +one of the literal nodes of that gadget is assigned with 1, as long as the literal nodes don’t have +productive neighbors outside of their Clause Gadget. As we introduce the external nodes that will +be connected to the literal nodes, we will show that they all must be assigned with 0 in any PNE, +and thus a literal node cannot have any productive neighbor outside of its Clause Gadget, which +will finalize the property we were looking to achieve with the Clause Gadget. +Our next goal is to make sure the translation between solutions from one domain to the other is +always valid. Specifically, we wish to ensure that in any PNE in our constructed graph, if any two +literal nodes represent the same variable in the CNF formula then they will be assigned the same +value, and if they represent a variable and its negation then they will be assigned opposite values. +We begin with the latter, introducing our Negation Gadget. The goal of the Negation Gadget is to +force opposite assignments to two chosen nodes, in any Nash equilibrium. The Negation Gadget +is composed of 9 nodes: five ’bottom’ nodes 𝑏1,𝑏2,𝑏3,𝑏4,𝑏5, and four ’top’ nodes 𝑡1,𝑡2,𝑡3,𝑡4, and +for each 𝑘 ≤ 4 we create the edges (𝑏𝑖,𝑏𝑖+1), (𝑡𝑖,𝑏𝑖) and (𝑡𝑖,𝑏𝑖+1). It can intuitively be described as +6We ignore the possibility of changing between the assignments of 𝑎𝑖 and 𝑏𝑖, or 𝑐𝑖 and 𝑑𝑖 for 𝑖 ∈ {1, 2, 3}, as it does not +affect anything in our proof. + +M. Gilboa +8 +four triangles that are connected as a chain. Say we have two nodes 𝑢, 𝑣 which we want to force +to have opposite assignments, we simply connect them both to node 𝑡2 of a Negation Gadget, as +demonstrated in Figure 6. +Lemma 3.7. In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through +a Negation Gadget 𝑛𝑔, 𝑢 and 𝑣 must have different assignments. In addition, the node 𝑡2 of 𝑛𝑔, to which +𝑢 and 𝑣 are connected, must be assigned 0. +Proof. We first show that 𝑢 and 𝑣 must have different assignments, dividing into two cases. +Case 1: Assume by way of contradiction that 𝑢 = 𝑣 = 0. We divide into two sub-cases, where in +the first one 𝑡2 = 0; in this case, exactly one of the two remaining neighbors of 𝑡2 must be assigned +1 in order for 𝑡2 itself to be playing best response. If 𝑏2 = 0 then 𝑏3 = 1 and so, looking at 𝑡1,𝑏1 (the +remaining neighbors of 𝑏2), we see that any assignment to them results either in 𝑏2 not playing +best response, or in 𝑡1,𝑏1 not playing best response, in contradiction. If, however, 𝑏3 = 0, then +𝑏2 = 1, and so, looking at 𝑡3,𝑏4 (the remaining neighbors of 𝑏3), the same logic leads us to a similar +contradiction. In the second sub-case, where 𝑡2 = 1, we have that its two remaining neighbors must +be assigned the same value in order for 𝑡2 itself to be playing best response. If 𝑏2 = 𝑏3 = 0 then +again there is no assignment to 𝑏1,𝑡1 s.t all of 𝑏1,𝑡1,𝑏2 play best response, and if 𝑏2 = 𝑏3 = 1 then +one may verify that there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response, +and so we reach a contradiction. +Case 2: Assume 𝑢 = 𝑣 = 1. Then we again divide into sub-cases according to 𝑡2’s assignment. +If 𝑡2 = 0, it must have at least one more productive neighbor in order to play best response. The +assignments where 𝑏2 = 𝑏3 = 1 or 𝑏2 = 0,𝑏1 = 1 are easily disqualified, seeing as there is no +assignment to 𝑡1,𝑏1 s.t 𝑡1,𝑏1,𝑏2 all play best response. If 𝑏2 = 1,𝑏3 = 0 then it must hold that 𝑡3 = 𝑏4 +in order for 𝑏3 to play best response, but this would mean that 𝑡3 is not playing best response, in +contradiction. If 𝑡2 = 1, then its two remaining neighbors 𝑏2,𝑏3 must be set to 0 in order for it to +play best response, and then there is no assignment to 𝑏1,𝑡1 s.t all of 𝑡1,𝑏1,𝑏2 play best response, in +contradiction. +And so it cannot be that 𝑢 = 𝑣. We move on to show that 𝑡2 must play 0. Assume by way of +contradiction that 𝑡2 = 1. Then, seeing as exactly one of 𝑢, 𝑣 is productive, 𝑡2 must have exactly one +more productive neighbor in order to play best response. If 𝑏2 = 1,𝑏3 = 0 we reach a contradiction +as there is no assignment to 𝑡1,𝑏1 s.t 𝑡1,𝑏1,𝑏2 all play best response. If 𝑏2 = 0,𝑏3 = 1 we reach a +contradiction as there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response. +Lastly, one may verify that in the assignment where 𝑡1 = 𝑏4 = 1,𝑏1 = 𝑏2 = 𝑏3 = 𝑏5 = 𝑡2 = 𝑡3 = 𝑡4 = 0 +all nodes of the gadget play best response. +□ +Now, for each variable that appears in the CNF formula, we choose one instance of it and one +instance of its negation7 and connect the literal nodes representing these instances via a Negation +Gadget, thus ensuring they are assigned opposite values in any PNE, according to Lemma 3.7. We +note that this is not the only place where we use this gadget, as we will see shortly. +We move on to introduce our Copy Gadget, which we will use to force literal nodes which +represent the same variable to have the same assignment in any PNE. The Copy Gadget is composed +of two negation gadgets 𝑛𝑔1,𝑛𝑔2, and two additional nodes 𝑥,𝑦 which have an edge between them. +Say we have two nodes 𝑢, 𝑣 which we want to force to have the same assignment in any PNE, then +we simply connect 𝑢 and 𝑥 to 𝑛𝑔1, and we connect 𝑣 and 𝑥 to 𝑛𝑔2. The gadget is demonstrated in +Figure 7. +7We will soon ensure that instances of the same variable would get the same assignment in any PNE, and thus it is sufficient +to negate the assignments of only one instance of a variable and its negation. + +M. Gilboa +9 +Fig. 6. Negation Gadget connecting 𝑢 and 𝑣. +Fig. 7. Copy Gadget connecting 𝑢 and 𝑣. +Lemma 3.8. In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through +a Copy Gadget 𝑐𝑝𝑔, 𝑢, 𝑣 must have the same assignment, and must have no productive neighbors from +𝑐𝑝𝑔. In addition, if 𝑢 = 𝑣 then there exists an assignment to the nodes of 𝑐𝑝𝑔 s.t all of them play best +response. +Proof. We first show that 𝑢 and 𝑣 must have the same assignment. This follows directly from +the fact that 𝑥 is connected to both 𝑢 and 𝑣 via a Negation Gadget. Therefore, from Lemma 3.7 +we have that 𝑥 ≠ 𝑢 and 𝑥 ≠ 𝑣, and so 𝑢 = 𝑣. Lemma 3.7 also tells us that the Negation Gadget +cannot add productive neighbors to the nodes that are connected to it in any PNE, and therefore 𝑢 +and 𝑣 have no productive neighbors from 𝑐𝑝𝑔. Lastly, we show that there exists an assignment to +the nodes of 𝑐𝑝𝑔 s.t they all play best response. From Lemma 3.7 𝑥 cannot have any productive +neighbors from 𝑛𝑔1 or 𝑛𝑔2. Therefore, if 𝑢 = 𝑣 = 0 then we can assign 𝑥 = 1,𝑦 = 0, and if 𝑢 = 𝑣 = 1 +then we can assign 𝑥 = 0,𝑦 = 1. In both cases, we assign 𝑛𝑔1 and 𝑛𝑔2 as suggested in Lemma 3.7. +One may verify that in this assignment indeed all nodes of 𝑐𝑝𝑔 play best response. +□ +Now, for each variable in the CNF formula, we connect all the literal nodes representing its +different instances via a chain of copy gadgets, thus (transitively) ensuring they are all assigned the +same value in any PNE, according to Lemma 3.8. +Given these lemmas and the graph we constructed, we can now prove Theorem 3.2. +Proof. (Theorem 3.2) The problem is in NP, since an assignment to the nodes can be easily +verified as a NTPNE by iterating over the nodes and checking whether they all play their best +response. It is left to show the problem is NP-hard. Given a ONE-IN-THREE 3SAT instance, we +construct a graph 𝐺 as described previously. If there exists a satisfying assignment to the 3SAT +problem, we can set all literal nodes according to the assignment of their matching variable, and +set all other nodes as described throughout lemmas 3.5, 3.7, 3.8, and according to those lemmas, +we get a pure Nash equilibrium. On the opposite direction, if there exists a non-trivial pure Nash +equilibrium, then by lemmas 3.6,3.4 in each clause exactly one literal node is assigned 1, and by +lemmas 3.7,3.8 we have that literal nodes have the same assignment if they represent the same +variable, and opposite ones if they represent a variable and its negation. Thus we can easily translate +the NTPNE into a satisfying ONE-IN-THREE 3SAT assignment, assigning ’True’ to variables whose +literal nodes are set to 1, and ’False’ otherwise. +□ + +u +V +t2 +t3 +b1 +b2 +b3 +b4 +b5u +x +ng1 +ng2M. Gilboa +10 +We now wish to expand this result to two slightly more general classes of patterns. Firstly, we +notice that the graph constructed throughout the proof of Theorem 3.2 is bounded8 by a maximum +degree of 6. Therefore, the proof is indifferent to entries of the pattern from index 7 onward, which +means it holds for any pattern that agrees with the first 7 entries of the 0-Or-2-Neighbors pattern. +Corollary 3.9. Let 𝑇 be a BRP such that: +• 𝑇 [0] = 𝑇 [2] = 1 +• ∀𝑘 ∈ {1, 3, 4, 5, 6} 𝑇 [𝑘] = 0 +Then NTPNE(𝑇) is NP-complete. +Secondly, according to Theorem 7 in [Gilboa and Nisan, 2022], adding 1,0 at the beginning of a +hard pattern that begins with 1 yields yet another hard pattern. Using this theorem recursively on +the patterns of Corollary 3.9, we have that the equilibrium decision problem is hard for any pattern +of the form: +𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0 +�������������������������������������� +𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ +, 0, 0, 0, ?, ?, ...]. +Corollary 3.10. Fix 𝑚 ≥ 1, and let 𝑇 be a BRP such that: +• ∀0 ≤ 𝑘 ≤ 𝑚 +(1) 𝑇 [2𝑘] = 1 +(2) 𝑇 [2𝑘 + 1] = 0 +• 𝑇 [2𝑚 + 2] = 𝑇 [2𝑚 + 3] = 𝑇 [2𝑚 + 4] = 0 +Then NTPNE(𝑇) is NP-complete. +We will see later on that this result will also be of use during the proof of Theorem 5.1. +There is one very similar class of patterns on which the proofs throughout the paper rely. This is +the class of all finite patterns that start with a finite number of 1,0, followed by 1,1, i.e. all patterns +of the form: +𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0 +�������������������������������������� +𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ +, 1, 1, ?, ?, ..., 0, 0, ...]. +The complexity of those patterns was already discussed and solved in Section 5.4 of [Gilboa and +Nisan, 2022], but was not formalized and so we state it here in the following lemma. +Lemma 3.11. Fix 𝑚 ≥ 2, and let 𝑇 be a BRP s.t: +• 𝑇 is finite +• ∀𝑘 ∈ IN 𝑠.𝑡 2𝑘 ≤ 𝑚 𝑇 [2𝑘] = 1 +• 𝑇 [1] = 0 +• ∃1 ≤ 𝑛 𝑤ℎ𝑒𝑟𝑒 2𝑛 + 1 ≤ 𝑚 + 1, 𝑠.𝑡 𝑇 [2𝑛 + 1] = 1 +Then NTPNE(𝑇) is NP-complete under Turing reduction. +Proof. The proof follows directly from Theorems 6 and 7 from [Gilboa and Nisan, 2022]9. +□ +8A literal node is connected to 4 nodes within its clause gadget, and possibly 2 nodes from copy gadgets or 1 node from a +negation gadget and 1 node from a copy gadget (assuming we connect the negation gadgets at the end of their respective +Copy-Gadget-chains). +9The reader who has read the details of Section 5.4 of [Gilboa and Nisan, 2022] may notice that in fact the use of Theorems +6 and 7 from [Gilboa and Nisan, 2022] covers a slightly more general class of patterns, but this entire class is not needed +currently, and is covered in Section 5. + +M. Gilboa +11 +4 +HARDNESS OF SEMI-SHARP PATTERNS +In this section we show hardness of semi-sharp Best-Response Patterns, beginning with a specific +sub-class of those patterns in Section 4.1, and expanding to all other semi-sharp patterns in Section +4.2. We remind the reader that semi-sharp patterns are patterns that begin with 1,0,0. +4.1 +Semi-Sharp Patterns with Isolated Odd 1 +In this section we prove that any finite, semi-sharp pattern such that there exists some ’isolated’ +1 (meaning it has a zero right before and after it) at an odd index, presents a hard equilibrium +decision problem. Those patterns can be summarized by the following form: +𝑇 = [1, 0, 0, ?, ?, ..., 0 +1 +���� +𝑜𝑑𝑑 𝑖𝑛𝑑𝑒𝑥 +, 0, ?, ?, ..., 0, 0, 0, ...] +Theorem 4.1. Let 𝑇 be a BRP which satisfies the following conditions: +• 𝑇 is finite +• 𝑇 is semi-sharp +• ∃𝑚 ≥ 1 +s.t: +(1) 𝑇 [2𝑚] = 𝑇 [2𝑚 + 2] = 0 +(2) 𝑇 [2𝑚 + 1] = 1 +Then NTPNE(𝑇) is NP-complete under Turing reduction. +Before proceeding to the proof, we introduce two gadgets and prove two lemmas regarding their +functionality. +Force-1-Gadget: The first gadget is denoted the Force-1-Gadget, and it will appear in several parts +of the graph we construct for the reduction. The goal of this gadget is to enable us to force any +node to be assigned 1 in any Nash equilibrium in a PGG defined by 𝑇. This gadget is composed +primarily of a triangle 𝑥,𝑦,𝑧, where the triangle’s nodes have also several ’Antenna’ nodes, which +are connected only to their respective node from the triangle. Specifically, 𝑥 will have 2𝑚 + 1 +Antenna nodes, and 𝑦 and 𝑧 will each have 2𝑚 Antenna nodes. Say we have some node 𝑢, whose +assignment we wish to force to be 1, then we simply connect 𝑢 to one of of the Antenna nodes of 𝑥, +denoted 𝑎. The gadget is demonstrated in Figure 8. +Add-1-Gadget: our second gadget of this proof is denoted the Add-1-Gadget, and its goal is +to enable us to assure the existence of (at least) a single productive neighbor to any node in a +Nash equilibrium of a PGG defined by 𝑇. Say we have a node 𝑣, to which we wish to add a single +productive neighbors, in any equilibrium. We construct the Add-1-Gadget as follows. We create +𝑚 + 1 nodes denoted 𝑥1, ...,𝑥𝑚+1, 𝑚 + 1 nodes denoted 𝑦1, ...𝑦𝑚+1, and an additional ’bridge’ node, +denoted 𝑏. We connect 𝑥1 and 𝑦1 to all of the other 𝑥𝑖 and 𝑦𝑖 nodes. For all 𝑖, 𝑗 ≥ 2 s.t 𝑖 ≠ 𝑗, we +create the edges (𝑥𝑖,𝑥𝑗), (𝑦𝑖,𝑦𝑗), (𝑥𝑖,𝑦𝑗) (the 𝑥𝑖,𝑦𝑖 nodes almost form a clique, except that for each +𝑖 ≥ 2 we omit the edge (𝑥𝑖,𝑦𝑖)). Additionally, For all 𝑖 ≥ 2 the bridge node 𝑏 is connected to 𝑥𝑖 and +to 𝑦𝑖. To 𝑏 we attach a Force-1-Gadget, and we also connect 𝑏 to 𝑣. The gadget is demonstrated in +Figure 9. +The following lemmas formalize the functionality of the two gadgets, beginning with the Force- +1-Gadget in Lemma 4.2. +Lemma 4.2. In any PNE in a graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.1), where 𝐺 has +a node 𝑢 that is connected to a Force-1-Gadget 𝑓 𝑔 as described, 𝑢 must be assigned 1, and its neighbor + +M. Gilboa +12 +Fig. 8. Force-1-Gadget with 𝑚 = 2, attached to 𝑢. +Fig. 9. Add-1-Gadget with 𝑚 = 2, attached to 𝑣. +from 𝑓 𝑔, 𝑎, must be assigned 0.10 Furthermore, if 𝑢 = 1 there exists an assignment to the nodes of 𝑓 𝑔 +such that they each play their best response. +Proof. First we show that 𝑢 must be assigned 1. Assume by way of contradiction that 𝑢 = 0. +Divide into the following two cases. If 𝑥 = 1, then all of its Antenna nodes must be assigned 0 +(according to 𝑇). Additionally, 𝑦 and 𝑧 must also be assigned 0, as otherwise 𝑥 wouldn’t be playing +best response, since 𝑇 is semi-sharp. Therefore, the best response of all of the Antenna nodes of 𝑦 +and 𝑧 is to play 1, which leaves 𝑦 and 𝑧 with 2𝑚 + 1 productive neighbors each, and so they are +not playing best response, in contradiction. If 𝑥 = 0, then all of its Antenna nodes must play 1. +Therefore, 𝑥 must have at least one other productive neighbor, as otherwise it would have 2𝑚 + 1 +productive neighbors and wouldn’t be playing best response; w.l.o.g assume 𝑦 = 1. Then all of +𝑦’s Antenna nodes must play 0. Therefore, 𝑧 must play 0, as otherwise 𝑦 wouldn’t be playing best +response. This means the best response for 𝑧’s Antenna nodes is to play 1, which leaves 𝑧 with +2𝑚 + 1 productive neighbors, and so it isn’t playing best response, in contradiction. We move on +to showing that 𝑎 must play 0. This follows directly from the fact that 𝑢 = 1. Since 𝑎 only has +one other neighbor (𝑥), regardless of its strategy the best response for 𝑎, according to 𝑇, would be +playing 0. It is left to show that when 𝑢 = 1 and 𝑎 = 0, there exists an assignment to the nodes +of 𝑓 𝑔 s.t they all play best response. One may verify that when we set 𝑥 = 𝑦 = 𝑧 = 0 and set all +the Antenna nodes in 𝑓 𝑔 (except for 𝑎) to 1, then all nodes of 𝑓 𝑔 play best response (specifically, +𝑥,𝑦,𝑧 would each have exactly 2𝑚 productive neighbors, which, by definition of 𝑇, means they are +playing best response). +□ +We move on to proving the following Lemma, which formalizes the functionality of the Add-1- +Gadget. +Lemma 4.3. Lemma 8 In any graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.1), where 𝐺 has +a node 𝑣 that is connected to an Add-1-Gadget 𝑎𝑔 as described, there always exists an assignment to +the nodes of 𝑎𝑔 s t they all play best response, regardless of 𝑣’s strategy. In addition, the bridge node 𝑏 +of 𝑎𝑔 must be assigned 1 in such an assignment. +Proof. The claim that 𝑏 must play 1 follows directly from the fact that it has a Force-1-Gadget +attached to it, i.e. from Lemma 4.2. Additionally, all the nodes of the Force-1-Gadget attached to 𝑏 +10The property that 𝑎 = 0 allows us to use the Force-1-Gadget without risking potentially adding productive neighbors to +the respective node. + +a +u +X +ZX1 +Y1 +X2 +Y2 +X3 +Y3 +b +Force- 1-GadgetM. Gilboa +13 +can be assigned as suggested in Lemma 4.2. It is left to show a possible assignment to the rest of the +nodes of 𝑎𝑔. We divide into cases. If 𝑣 = 0, then we set 𝑥1 = 1 and all other 𝑥𝑖,𝑦𝑖 nodes we set to 0. +If 𝑣 = 1, then we set 𝑥𝑖 = 𝑦𝑖 = 1 for all 1 ≤ 𝑖 ≤ 𝑚 + 1. One may verify that given these assignments +all nodes of 𝑎𝑔 play their best response. +□ +In addition to these two gadgets, we wish to introduce the following definition, after which we +will proceed to the proof of Theorem 4.1, which we can now prove. +Definition 4.4. Let 𝑇,𝑇 ′ be two BRPs. We say that 𝑇 ′ is shifted left by 𝑡 from 𝑇 if +∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 𝑡]. +Proof. (Theorem 4.1) Denote by 𝑇 ′ the pattern which is shifted left by 1 from T, i.e.: +∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 1]. +Notice that𝑇 ′ is flat, non-monotonic and finite, and therefore NTPNE(𝑇 ′) is NP-complete according +to Theorem 4 in [Gilboa and Nisan, 2022], which allows us to construct a Turing reduction from it. +The technique of the reduction is very similar to those of the proofs of Theorems 5,6 in [Gilboa and +Nisan, 2022]. Given any graph 𝐺 = (𝑉, 𝐸), where 𝑉 = 𝑣1, ..., 𝑣𝑛, we construct 𝑛 graphs 𝐺1, ...,𝐺𝑛, +where for each 1 ≤ 𝑖 ≤ 𝑛 the graph 𝐺𝑖 is defined as follows. The graph 𝐺𝑖 contains the original +input graph 𝐺, and in addition, we connect a unique Add-1-Gadget to each of the original nodes, +and a Force-1-Gadget only to node 𝑣𝑖. If there exists some non-trivial PNE in the PGG defined on +𝐺 by 𝑇, let 𝑣 𝑗 be some node who plays 1. Then the same NTPNE is also an NTPNE in the PGG +defined by 𝑇 ′ on 𝐺𝑗, when we assign the nodes of the additional gadget as suggested in lemmas +4.2,4.3. To see why, notice that 𝑇 ′ is shifted left by 1 from 𝑇, and the Add-1-Gadgets ensure that all +nodes have exactly one additional productive neighbor than they had in 𝐺. +In the other direction, if there exists an NTPNE in a PGG defined by 𝑇 ′ on one of the graphs 𝐺𝑗, +then by the same logic this is also a PNE in the game defined by 𝑇 on 𝐺 (ignoring the assignments +of the added nodes). Moreover, the Force-1-Gadget ensures this assignment is non-trivial even after +removing the added nodes, since 𝑣 𝑗 must play 1 in this assignment. +□ +4.2 +All Semi-Sharp Patterns +In this section we show that any finite, non-monotone, semi-sharp pattern presents a hard equilib- +rium problem. +Theorem 4.5. Let 𝑇1 be a finite, non-monotone, semi-sharp BRP. Then NTPNE(𝑇1) is NP-complete +under Turing reduction. +Before proceeding to the proof, we wish to introduce the following definition and prove two +lemmas related to it. +Definition 4.6. Let 𝑇,𝑇 ′ be two BRPs such that ∀𝑘 ∈ IN it holds that 𝑇 [𝑘] = 𝑇 ′[2𝑘]. Then we say +that 𝑇 ′ is a double-pattern of 𝑇, and 𝑇 is the half-pattern of 𝑇 ′. Notice that a pattern has a unique +half-pattern, whereas, since the definition does not restrict 𝑇 ′ in the odd indices, any pattern has +infinite double-patterns. +The first lemma is very simple and intuitive, stating that the largest index with value 1 in a half +pattern is strictly smaller than the largest index with value 1 in its original pattern. This is true +since for any index 𝑖 s.t the value of the half pattern is 1 in that index, the original pattern has a +value of 1 in index 2𝑖. +Lemma 4.7. Let 𝑇,𝑇 ′ be two finite BRPs such that 𝑇 is the half-pattern of 𝑇 ′. Denote by 𝑖 the largest +index s.t 𝑇 [𝑖] = 1 and denote by 𝑗 the largest index s.t 𝑇 ′[𝑗] = 1. Then if 𝑗 > 0 we have that 𝑖 < 𝑗. + +M. Gilboa +14 +Proof. The proof is trivially given by the definition of a half pattern, since 𝑇 ′[2𝑖] = 𝑇 [𝑖]. +□ +The next lemma is less trivial, stating the relation between hardness of a pattern and its double- +pattern. +Lemma 4.8. Let 𝑇 be a BRP such that NTPNE(𝑇) is NP-complete, and let 𝑇 ′ be a double-pattern of +𝑇. Then NTPNE(𝑇 ′) is NP-complete. +Proof. We use a specific case of the same reduction that was used to prove Theorem 4 in [Gilboa +and Nisan, 2022]. Given a graph 𝐺1 = (𝑉1, 𝐸1) as input, where 𝑉1 = 𝑣1 +1, ..., 𝑣1 +𝑛, we create another +replica of it 𝐺2 = (𝑉2, 𝐸2), where 𝑉2 = 𝑣2 +1, ..., 𝑣2 +𝑛. For each node (from both graphs), we add edges +connecting it to all replicas of its neighbors from the opposite graph. That is, the following group +of edges is added to the graph: +𝐸 = {(𝑣1 +𝑖 , 𝑣2 +𝑗)|(𝑣1 +𝑖 , 𝑣1 +𝑗) ∈ 𝐸1}. +A demonstration of the reduction can be seen in Figure 10. +Fig. 10. Example of the reduction of Lemma 4.8’s proof. +Denote by 𝑃 the PGG defined on 𝐺1 by 𝑇, and by 𝑃 ′ the PGG defined by 𝑇 ′ on 𝐺 ′ = (𝑉 ′, 𝐸′) +where 𝑉 ′ = 𝑉1 ∪ 𝑉2, 𝐸′ = 𝐸 ∪ 𝐸1 ∪ 𝐸2. We show that there exists an NTPNE in 𝑃 iff there exists +one in 𝑃 ′. If there exists an NTPNE in 𝑃, we simply give the nodes of 𝐺2 the same assignment as +those of 𝐺1. Since 𝑇 ′ is a double pattern of 𝑇, any node 𝑣 ′ ∈ 𝑉 ′ must play best response, having +exactly twice as many supporting neighbors than it had (or its replica had) in 𝑃. +In the opposite direction, if there exists an NTPNE in 𝑃 ′, notice that for all 1 ≤ 𝑖 ≤ 𝑛 it must +be that 𝑣1 +𝑖 and 𝑣2 +𝑖 have identical assignments, since they both share exactly the same neighbors, +and thus have identical best responses. Therefore, any node 𝑣 ′ ∈ 𝑉 must have an even number +of productive neighbors, half of which are in 𝑉1 and the other half in 𝑉2 (as for each productive +neighbor from 𝑉1 there is a respective productive neighbor from 𝑉2). We then simply ignore 𝐺2, and +leave the assignment of 𝐺1 as it is, and each node shall now have exactly half as many productive +neighbors as it had in the original assignment. Since 𝑇 is a half pattern of 𝑇 ′, we get an NTPNE in +𝑃.11 +□ +Given lemmas 4.7,4.8, we are now able to prove Theorem 4.5. The intuitive idea of the proof is +that we halve the pattern 𝑇1 (i.e. find its half-pattern) repeatedly, until eventually we reach some +pattern for which we already know the equilibrium problem is hard, which, as we will see, must +happen at some point. Then, by applying Lemma 4.8 recursively, we have that 𝑇1 is hard. +11In both directions of this proof, the non-triviality comes from the fact that 𝑇 [0] = 𝑇 ′[0], by definition. Therefore, a +non-trivial PNE in one domain must translate to a non-trivial one in the other. + +Vi +2 +V2 +2 +V +3M. Gilboa +15 +Proof. (Theorem 4.5) From Lemma 4.7 we have that if we halve a non-flat pattern enough times, +we will eventually reach the Best-Shot pattern: 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [0] = 1 and ∀𝑘 ≥ 1 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [𝑘] = 0. +Divide into two cases. +In the first case assume that ∀𝑘 ∈ IN it holds that𝑇1[2𝑘] = 0. In this case, we know that no matter +how many times we halve 𝑇1 into patterns 𝑇2,𝑇3, ..., for each 𝑖 we will have that 𝑇𝑖 [1] = 0, i.e. the +value in index 1 of all these half-patterns will always be 0, i.e. 𝑇𝑖 [1] = 0 for all 𝑖. Assume that we +halve𝑇1 repeatedly into patterns𝑇2,𝑇3, ...,𝑇𝑚 (where𝑇𝑖 is the half pattern of𝑇𝑖−1) such that𝑇𝑚 is the +first time that we reach the Best-Shot pattern. Observe 𝑇𝑚−1. For any even index 𝑘 ≠ 0 it must hold +that 𝑇𝑚−1[𝑘] = 0, otherwise 𝑇𝑚 would not be the Best-Shot pattern. Additionally, there must exist +at least one odd index 𝑗 s.t 𝑇𝑚−1[𝑗] = 1, since 𝑇𝑚 is the first time we reach the Best-Shot pattern. +For these two reasons, we have that 𝑇𝑚−1 satisfies the conditions of Theorem 4.1 and therefore +NTPNE(𝑇𝑚−1) is NP-complete under Turing reduction. From Lemma 4.8 (used inductively), we +have that ∀1 ≤ 𝑖 ≤ 𝑚 − 2 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically +NTPNE(𝑇1). +In the second case, assume that there exists some 𝑘 ∈ IN s.t 𝑇1[2𝑘] = 1. In that case, after at most +𝑘 halvings, we reach some pattern for which the value of index 1 is 1. Assume that we halve 𝑇1 +repeatedly into patterns 𝑇2,𝑇3, ...,𝑇𝑛 (where 𝑇𝑖 is the half pattern of 𝑇𝑖−1) such that 𝑇𝑛 is the first +time that we reach a pattern for which index 1 is 1, i.e. ∀1 ≤ 𝑖 ≤ 𝑛 − 1 𝑇𝑖 [1] = 0 and 𝑇𝑛[1] = 1. +Notice that, additionally, by definition of a half-pattern for each 𝑖 it holds that 𝑇𝑖 [0] = 1 (since +𝑇1[0] = 1). If 𝑇𝑛 is non-monotone, then by Theorem 5 in [Gilboa and Nisan, 2022] we have that +NTPNE(𝑇𝑛) is NP-complete under Turing reduction, and from Lemma 4.8 (used inductively), we +have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically +NTPNE(𝑇1). Otherwise (i.e. 𝑇𝑛 is monotone), denote by 𝑙 the largest index s.t 𝑇𝑛[𝑙] = 1, and observe +𝑇𝑛−1. By definition of double-patterns, we have that: +∀𝑗 ∈ IN 𝑇𝑛−1[2𝑗] = +� +1 +if 𝑗 ≤ 𝑙 +0 +otherwise +i.e. the value in the even indices is 1 until 2𝑙, and 0 afterwards. Since 𝑇𝑛 is defined to be the first +halving of 𝑇1 s.t its value in index 1 is 1, we have that 𝑇𝑛−1[1] = 0. However, since the definition of +a double-pattern does not restrict it in the odd indices, there might be some odd indices (strictly +larger than 1) for which the value of 𝑇𝑛−1 is 1. Divide into 3 sub-cases: +Sub-case 1: If there exists some 𝑧 ≤ 𝑙 s.t 𝑇𝑛−1[2𝑧 + 1] = 1, then by Lemma 3.11, we have that +NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction. +Sub-case 2: Otherwise, if there exists some 𝑧 > 𝑙 s.t 𝑇𝑛−1[2𝑧 + 1] = 1, then observe the pattern +𝑇 ′ +𝑛−1, which we define as the pattern shifted left by 2𝑙 from 𝑇𝑛−1 i.e.: +∀𝑗 ∈ IN 𝑇 ′ +𝑛−1[𝑗] = 𝑇𝑛−1[𝑗 + 2𝑙] +Notice that this pattern satisfies the conditions of Theorem 4.1, and therefore NTPNE(𝑇 ′ +𝑛−1) is +NP-complete under Turing reduction. Then, by applying Theorem 7 from [Gilboa and Nisan, 2022] +𝑙 times, we have that NTPNE(𝑇𝑛−1) is also NP-complete under Turing reduction. +Sub-case 3: Otherwise (i.e. there is no odd index whatsoever in which the value of 𝑇𝑛−1 is 1), then +by Corollary 3.10 we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction. +And so, in either case we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction, and +therefore from Lemma 4.8 (used inductively), we have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also +NP-complete under Turing reduction, and specifically NTPNE(𝑇1). +□ + +M. Gilboa +16 +5 +HARDNESS OF ALL SPIKED PATTERNS +There are several finite, spiked patterns that we have not yet proved hardness for, and we now have +enough tools to close the remaining gaps. We remind the reader that spiked patterns are patterns +that begin with 1,0,1. The following theorem formalizes the result of this section, and completes +the characterization of all finite patterns. +Theorem 5.1. Let𝑇 be a finite, spiked BRP. Then NTPNE(𝑇) is NP-complete under Turing reduction. +The intuitive idea of the proof is as follows. If the pattern simply alternates between 1 and 0 a +finite amount of times, followed infinite 0’s, i.e. the pattern is of the form +𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0 +�������������������������������������� +𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ +, 0, 0, 0, ...] +then the problem12 is already shown to be hard by Corollary 3.10. Otherwise, we wish to look at +the first "disturbance" where this pattern stops alternating from 1 to 0 regularly. Either the first +"disturbance" is a 1 at an odd index, i.e. the pattern is of the form +𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0 +�������������������������������������� +𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ +, 1, 1, ?, ?, ...] +or the first "disturbance" is a 0 at an even index, i.e. the pattern is of the form +𝑇 = [1, 0, 1, 0, 1, 0, ..., 1, 0 +�������������������������������������� +𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ +, 0, ?, ?, ..., 1, ?, ?, ...] +(in the latter option, after the first "disturbance" there must be some other index with value 1, since +the pattern does not fit the form of Corollary 3.10). The first option was solved in Lemma 3.11, and +the second option can be solved using our previous results, as we shall now formalize in the proof. +Proof. (Theorem 5.1) If𝑇 satisfies the conditions of Corollary 3.10 or Lemma 3.11 then NTPNE(𝑇) +is NP-complete under Turing reduction according to them. Otherwise, let 𝑘 be the smallest integer +such that 𝑇 [2𝑘] = 0. Denote by 𝑇 ′ the pattern which is shifted left by 2𝑘 − 2 from T, i.e.: +∀𝑗 ≥ 0 𝑇 ′[𝑗] = 𝑇 [𝑗 + 2𝑘 − 2] +Notice that from definition of 𝑘 (being the first even index such that 𝑇 [2𝑘] = 0) we have that for +all 𝑗 < 𝑘 it holds that 𝑇 [2𝑗] = 1. Moreover, since 𝑇 does not satisfy the conditions of Lemma 3.11 +it must hold for all 𝑗 ≤ 𝑘 that 𝑇 [2𝑗 − 1] = 0, i.e. the value of 𝑇 in the odd indices until 2𝑘 is 0 +(since otherwise𝑇 would start with a finite number of 1,0, followed by 2 consecutive 1’s, and would +satisfy the conditions of Lemma 3.11). Thus, we have that +∀𝑗 < 2𝑘 𝑇 [𝑗] = +� +1 if 𝑗 is even +0 if 𝑗 is odd +(1) +In particular, we have that𝑇 [2𝑘 − 2] = 1, 𝑇 [2𝑘 − 1] = 0, which implies that𝑇 ′[0] = 1, 𝑇 ′[1] = 0; +as 𝑇 [2𝑘] = 0 we have that 𝑇 ′[2] = 0, and thus we conclude that 𝑇 ′ is semi-sharp. In addition, since +𝑇 does not satisfy the conditions of Corollary 3.10, there must be some other index 𝑥 > 2𝑘 such +that 𝑇 [𝑥] = 1, and therefore we have that 𝑇 ′ is non-monotone. Therefore, by Theorems 4.1, 4.5, we +have that NTPNE(𝑇 ′) is NP-complete under Turing reduction. We now wish to use this in order to +prove that NTPNE(𝑇) is also hard. +12In fact, Corollary 3.10 gives a more general result, but we currently only need the private case where the pattern ends +with infinite 0’s. + +M. Gilboa +17 +From Equation 1, we can apply Theorem 7 of [Gilboa and Nisan, 2022] (𝑘 − 1) times, and we +have that NTPNE(𝑇) is NP-complete under Turing reduction. +□ +ACKNOWLEDGMENTS +I would like to thank Noam Nisan for many useful conversations throughout the work, and for +suggesting the Copy Gadget seen in the proof of Theorem 3.2. +I would like to thank Roy Gilboa for many useful conversations throughout the work and for +adjusting the Copy Gadget seen in the proof of Theorem 3.2. +I would like to thank Noam Nisan for communicating to me the solution of the monotone case by +Max Klimm, and the alternative derivation by Sigal Oren. +This project has received funding from the European Research Council (ERC) under the European +Union’s Horizon 2020 Research and Innovation Programme (grant agreement no. 740282). +REFERENCES +Yann Bramoullé and Rachel Kranton. 2007. Public Goods in Networks. 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(2021). https://doi.org/10.48550/arXiv.1911.05788 + diff --git a/5dFJT4oBgHgl3EQfkywG/content/tmp_files/load_file.txt b/5dFJT4oBgHgl3EQfkywG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..711b6dc0e294c1c37720a56d2e33e5b4ec0fe711 --- /dev/null +++ b/5dFJT4oBgHgl3EQfkywG/content/tmp_files/load_file.txt @@ -0,0 +1,788 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf,len=787 +page_content='A Characterization of Complexity in Public Goods Games MATAN GILBOA We complete the characterization of the computational complexity of equilibrium in public goods games on graphs by proving that the problem is NP-complete for every finite non-monotone best-response pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This answers the open problem of [Gilboa and Nisan, 2022], and completes the answer to a question raised by [Papadimitriou and Peng, 2021], for all finite best-response patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=" Manuscript submitted for review to the 24nd ACM Conference on Economics & Computation (EC'23)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11580v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='GT] 27 Jan 2023 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 1 1 INTRODUCTION Public goods games describe scenarios where multiple agents face a decision of whether or not to produce some "good", such that producing this good benefits not only themselves, but also other (though not necessarily all) agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Typically, we consider the good to be costly to produce, and therefore an agent might choose not to produce it, depending on the actions of the agents that affect her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This type of social scenarios can be found in various real-life examples, such as vaccination efforts (an individual pays some personal cost for being vaccinated but she and other people in her proximity gain from it) and research efforts (a research requires many resources, but the researcher benefits from the result along with other researchers in similar areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' As is common in the literature, to model this we use an undirected graph, where each node represents an agent and an edge between two nodes captures the fact that these nodes directly affect one another by their strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' As in [Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2021, Maiti and Dey, 2022, Papadimitriou and Peng, 2021, Yang and Wang, 2020, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2021], in our model the utility of an agent is completely determined by the number of productive neighbors she has, as well as by her own action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We focus on a specific version of the game which has the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Firstly, our strategy space is binary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' an agent can only choose whether or not to produce the good, rather than choose a quantity (we call an agent who produces the good a productive agent);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' secondly, our game is fully-homogeneous, meaning that all agents share the same utility function and cost of producing the good;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' and thirdly, our game is strict, which means that an agent has a single best response to any number of productive neighbors she might have (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' we do not allow indifference between the actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The game is formally defined by some fixed cost 𝑐 of producing the good, and by some "social" function 𝑋 (𝑠𝑖,𝑛𝑖), which takes into account the boolean strategy of agent 𝑖 and the number of productive neighbors she has (marked as 𝑠𝑖 and 𝑛𝑖 respectively), and outputs a number representing how much the agent gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The utility 𝑢𝑖 of agent 𝑖 is then given by the social function 𝑋 (𝑠𝑖,𝑛𝑖), reduced by the cost 𝑐 if the agent produces the good, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑢𝑖 (𝑠𝑖,𝑛𝑖) = 𝑋 (𝑠𝑖,𝑛𝑖) −𝑐 ·𝑠𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' However, since any number of productive neighbors yields a unique best response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the game is strict), we can capture the essence of the utility function and the cost using what we call (as in [Gilboa and Nisan, 2022]), a Best-Response Pattern𝑇 : IN → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We think of the Best-Response Pattern as a boolean vector in which the 𝑘𝑡ℎ entry represents the best response to exactly 𝑘 productive neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We are interested in the problem of determining the existence of a non-trivial pure Nash equilibrium in these games, which is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Equilibrium decision problem in a public goods game: For a fixed Best-Response Pattern 𝑇 : IN → {0, 1}, and with an undirected graph 𝐺 = (𝑉, 𝐸) given as input, determine whether there exists a pure non-trivial Nash equilibrium of the public goods game defined by 𝑇 on 𝐺, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' an assignment 𝑠 : 𝑉 → {0, 1} that is not all 0, such that for every 1 ≤ 𝑖 ≤ |𝑉 | we have that 𝑠𝑖 = 𝑇 [ ∑︁ 𝑗 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸 𝑠𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The first Best-Response Pattern for which this problem was studies was the so-called Best-Shot pattern (where an agent’s best response is to produce the good only if she has no productive neighbors), which was shown in [Bramoullé and Kranton, 2007] to have a pure Nash equilibrium in any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In [Bramoullé and Kranton, 2007], they also show algorithmic results for "convex" patterns, which are monotonically increasing (best response is 1 if you have at least 𝑘 productive neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The question of characterizing the complexity of this problem for all possible patterns M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 2 was first raised by [Papadimitriou and Peng, 2021], where they manage to fully answer an equivalent problem on directed graphs, showing NP-completeness for most patterns, and algorithmic results for the remaining few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The open question on undirected graphs was then partially answered in [Gilboa and Nisan, 2022], where they show NP-completeness for several classes of patterns, and a polynomial-time algorithm for one other pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' There have been several studies concerning other versions of this problem as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In [Yang and Wang, 2020], the general version of this problem (where the pattern is part of the input rather than being fixed) was shown to be NP-complete when removing the strictness assumption, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' allowing indifference between actions, such that both 0 and 1 are best responses in certain cases) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2021], NP-completeness is shown for the general version of the problem in the heterogeneous public goods game, in which the utility function varies between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In [Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2021], they show NP-completeness of the equilibrium problem when restricting the equilibrium to have at least 𝑘 productive agents, or at least some specific subset of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In [Maiti and Dey, 2022], the parameterized complexity of the equilibrium problem is studied, for a number of parameters of the graph on which the game is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Papadimitirou and Peng raised the problem of characterizing all Best-Response Patterns, and Gilboa and Nisan suggested two specific open problems regarding two specific patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One of these patterns has been recently solved by Max Klimm (personal communication) who showed that all monotonically decreasing patterns can be viewed as potential games, and thus always have a pure Nash equilibrium2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Our main contribution is completing the characterization of the equilibrium decision problem for all finite patterns, by showing that for all non-monotone patterns the problem is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Theorem: For any Best-Response Pattern that is non-monotone and finite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', has a finite number of entries with value 1), the equilibrium decision problem in a public goods game is NP-complete (under Turing reductions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The first step along this way was to prove NP-completeness for the specific open problem by [Gilboa and Nisan, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It has come to our attention that an alternative proof to this specific open problem was obtained independently and concurrently by Max Klimm and Maximilian Stahlberg (private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We note that we only focus on finite patterns, which we believe to be more applicable to real-life problems that can be modeled by this game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We believe that the characterization for all infinite patterns is of interest, and remains open, though some results can be found in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In Section 2 we introduce the formal model and some relevant definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We then set out to show hardness of all remaining patterns, dividing them into classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In Section 3 we present a solution for an open question from [Gilboa and Nisan, 2022], showing hardness of a pattern we call the 0-Or-2-Neighbors Best Response Pattern, and expanding the result to a larger sub-class of patterns that begin with 1,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In Section 4 we show hardness of all patterns beginning with 1,0,0 (where we also have a slightly more subtle division into sub-classes), and in Section 5 we show hardness of all patterns beginning with 1,0,1 that were not covered in Section 3, thus completing the characterization for all finite patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The outline of this paper is also depicted3 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 1The paper [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2020] had an earlier version [Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2021] which presented a proof for this case as well, but an error in the proof was pointed out by [Yang and Wang, 2020], who then also provided an alternative proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 2Alternatively, Sigal Oren (personal communication) observed that known results about 𝑘-Dominating and 𝑘-independent sets [Chellali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 2012] (Theorem 19) can be used to prove this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 3Some patterns which start with 1,0 were solved in [Gilboa and Nisan, 2022], though for simplicity we omit them from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Outline of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 2 MODEL AND DEFINITIONS A Public Goods Game (PGG) is defined on an undirected graph 𝐺 = (𝑉, 𝐸), 𝑉 = {𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 𝑣𝑛}, where each node represents an agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The strategy space, which is identical for all agents, is 𝑆 = {0, 1}, where 1 represent producing the good and 0 represents not producing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The utility of node 𝑣𝑖 (which is assumed to be the same for all agents) is completely determined by the number of productive neighbors 𝑣𝑖 has, as well as by 𝑣𝑖’s own strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Moreover, our model is restricted to utility functions where an agent always has a single best response to the strategies of its neighbors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' there is no indifference between actions in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, rather than defining a PGG with an explicit utility function and cost for producing the good, we can simply consider the best response of an agent for any number of productive agents in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Essentially, this can be modeled as a function 𝑇 : IN → {0, 1}, which, as in [Gilboa and Nisan, 2022], we represent in the form of a Best Response Pattern: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A Best-Response Pattern (BRP) of a PGG, denoted by 𝑇, is an infinite boolean vector in which the 𝑘𝑡ℎ entry indicates the best response for each agent 𝑣𝑖 given that exactly 𝑘 neighbors of 𝑣𝑖 (excluding 𝑣𝑖) produce the good: ∀𝑘 ≥ 0 𝑇 [𝑘] = best response to 𝑘 productive neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given a Public Goods Game defined on a graph 𝐺 = (𝑉, 𝐸) with respect to a BRP 𝑇, a strategy profile 𝑠 = (𝑠1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',𝑠𝑛) ∈ 𝑆𝑛 (where 𝑠𝑖 ∈ {0, 1} represents the strategy of node 𝑣𝑖 ∈ 𝑉 ) is a pure Nash equilibrium (PNE) if all agents play the best response to the strategies of their neighbors: ∀1 ≤ 𝑖 ≤ 𝑛 𝑠𝑖 = 𝑇 [ ∑︁ 𝑗 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑡 (𝑣𝑖,𝑣𝑗) ∈𝐸 𝑠𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' finite,non-monotonepatterns starts with 0 starts with 1,0 starts with 1,1 Solved Solved [GilboaandNisan,2022] [GilboaandNisan,2022] starts with1,0,1 starts with1,0,0 of the form: all other forms has"isolated" odd 1 doesn\'t have 1,0,1,0,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='.,1,0,0,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' "isolated"odd 1 or 1,0,1,0,1,0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',1,1,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Section 3 Section 5 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1 Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 4 In addition, if there exists 1 ≤ 𝑖 ≤ 𝑛 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑠𝑖 = 1, then 𝑠 is called a non-trivial pure Nash equilibrium (NTPNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We note that throughout the paper we also use the notation 𝑣𝑖 = 0 and 𝑣𝑖 = 1 to indicate the strategy of some node 𝑣𝑖, rather than use 𝑠𝑖 = 0 and 𝑠𝑖 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' For a fixed BRP 𝑇, the non-trivial4 pure Nash equilibrium decision problem corresponding to 𝑇, denoted by NTPNE(𝑇), is defined as follows: The input is an undirected graph 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The output is ’True’ if there exists an NTPNE in the PGG defined on 𝐺 with respect to 𝑇, and ’False’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A BRP 𝑇 is called monotonically increasing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' decreasing) if ∀𝑘 ∈ IN, 𝑇 [𝑘] ≤ 𝑇 [𝑘 + 1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇 [𝑘] ≥ 𝑇 [𝑘 + 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A BRP 𝑇 is called finite if it has a finite number of entries with value 1: ∃𝑁 ∈ IN 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑡 ∀𝑛 > 𝑁 𝑇 [𝑛] = 0 As seen in Figure 1, the only patterns for which the equilibrium decision problem remains open are patterns that begin with 1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We divide those into the two following classes of patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A BRP 𝑇 is called semi-sharp if: (1) 𝑇 [0] = 1 (2) 𝑇 [1] = 𝑇 [2] = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇 begins with 1, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A BRP 𝑇 is called spiked if: (1) 𝑇 [0] = 𝑇 [2] = 1 (2) 𝑇 [1] = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇 begins with 1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 3 HARDNESS OF THE 0-OR-2-NEIGHBORS PATTERN In this section we show that the equilibrium problem is NP-complete for the 0-Or-2-Neighbors pattern, and provide some intuition about the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This result answers an open question by Gilboa and Nisan [Gilboa and Nisan, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We then expand this to show hardness of a slightly more general class of patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the 0-Or-2-Neighbors BRP the best response is 1 only to zero or two productive neighbors, as we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The 0-Or-2-Neighbors Best Response Pattern is defined as follows: ∀𝑘 ∈ IN 𝑇 [𝑘] = � 1 if 𝑘 = 0 𝑜𝑟 𝑘 = 2 0 otherwise i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇 = [1, 0, 1, 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇 be the 0-Or-2 Neighbors BRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then NTPNE(𝑇) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 4In this paper, we only study BRPs where the best response for zero productive neighbors is 1, for which there never exists a trivial all-zero PNE (as these are the only BRPs left to solve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' However, we sometimes reduce from patterns where this is not the case, and therefore include the non-triviality restriction in our problem definition, in order to correspond with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 5 Before proving the theorem, we wish to provide basic intuition about the 0-Or-2-Neighbors BRP, by examining several simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Take for example a simple cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since𝑇 [2] = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' best response for two productive neighbors is 1), we have that any simple cycle admits a non-trivial pure Nash equilibrium, assigning 1 to all nodes (see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' However, looking at a simple path with 𝑛 nodes, we see that the all-ones assignment is never a pure Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The reason for this is that 𝑇 [1] = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' best response for one productive neighbors is 0), and so the two nodes at both edges of the path, having only one productive neighbor, do not play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Nevertheless, any simple path does admit a pure Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' To see why, let us observe the three smallest paths, of length 2,3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Notice that in a path of length two a PNE is given by the assignment 0,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' in a path of length three a PNE is given by the assignment 0,1,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' and in a path of length four a PNE is given by the assignment 1,0,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We can use these assignment to achieve a PNE in any simple path: given a simple path of length 𝑛, if 𝑛 ≡ 0 (mod 3) we use the path of length three as our basis, adding 0,1,0 to it as many times as needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' if 𝑛 ≡ 1 (mod 3) we use the path of length four as our basis, adding 0,0,1 to it as many times as needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' and if 𝑛 ≡ 2 (mod 3) we use the path of length two as our basis, adding 0,0,1 to it as many times as needed (see example in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' PNE in cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' PNE in paths of lengths 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In contrast to the graphs we have discussed so far, there are graphs in which a pure Nash equilibrium doesn’t exist for the 0-Or-2-Neighbors pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' An example of this can be seen in a graph composed of four triangles, connected as a chain where each two neighboring triangles have a single overlapping vertex, as demonstrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One may verify that no PNE exists in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This specific graph will also be of use to us during our proof5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' No PNE exists in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Having provided some intuition regarding the problem, we move on to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The reduction is from ONE-IN-THREE 3SAT, which is a well known NP-complete problem [Schaefer, 1978].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In ONE-IN-THREE 3SAT, the input is a CNF formula with 3 literals in each clause, and the 5The Negation Gadget defined throughout the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2 is constructed similarly to the graph described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 10 0 0 0M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Clause Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' goal is to determine whether there exists a boolean assignment to the variables such that in each clause exactly one of the literals is assigned True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We begin by introducing our Clause Gadget, which is a main component of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given a CNF formula, for each of its clauses we construct a 21-nodes Clause Gadget, in which three of the nodes, denoted 𝑙1,𝑙2,𝑙3 (also referred to as the literal nodes) represent the three literals of the matching clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The purpose of this gadget is to enforce the property that in any NTPNE, exactly one literal node in the gadget will be assigned 1, which easily translates to the key property of a satisfying assignment in the ONE-IN-THREE 3SAT problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The three literal nodes are connected to one another, forming a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Additionally, for each 𝑖 ∈ {1, 2, 3}, 𝑙𝑖 is connected to two other nodes 𝑥𝑖,𝑦𝑖, which are also connected to one another, forming another triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lastly, 𝑥𝑖 and 𝑦𝑖 each form yet another triangle, along with nodes 𝑎𝑖,𝑏𝑖 and 𝑐𝑖,𝑑𝑖 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We refer to 𝑥𝑖,𝑦𝑖,𝑎𝑖,𝑏𝑖,𝑐𝑖,𝑑𝑖 as the sub-gadget of 𝑙𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We note that out of the nodes of the Clause Gadget, only the literal nodes may be connected to other nodes outside of their gadget, a property on which we rely throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The structure of the Clause Gadget is demonstrated in Figure 5, where each sub-gadget is colored differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The next four lemmas lead us to the conclusion that the gadget indeed has the desired property mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if a literal node 𝑙𝑖 of 𝑐𝑔 is assigned 1 then so are its two neighbors from its respective sub-gadget, 𝑥𝑖,𝑦𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Divide into cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Case 1: If 𝑥𝑖 = 𝑦𝑖 = 0, then if 𝑎𝑖 ≠ 𝑏𝑖 (meaning only one of them is assigned 1) then 𝑥𝑖 would have two productive neighbors and would not be playing its best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' However, if 𝑎𝑖 = 𝑏𝑖 then 𝑎𝑖 and 𝑏𝑖 would not be playing their best response, and we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Case 2: If 𝑥𝑖 = 1,𝑦𝑖 = 0 (the case where 𝑥𝑖 = 0,𝑦𝑖 = 1 is, of course, symmetrical) then 𝑥𝑖 must have exactly one more productive neighbor (either 𝑎𝑖 or 𝑏𝑖) in order to be playing best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' But then that node would not be playing best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' C1 d1 a1 X1 Y1 V3 3 3 V2 a3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 7 Case 3: We are left with the option where 𝑥𝑖 = 𝑦𝑖 = 1, where it is easy to verify that all nodes of the sub-gadget of 𝑙𝑖 are playing their best response if we set 𝑎𝑖 = 𝑏𝑖 = 𝑐𝑖 = 𝑑𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any NTPNE in a graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if one of the literal nodes 𝑙𝑖 of 𝑐𝑔 is assigned 1 then the other two literal nodes of 𝑐𝑔 must be assigned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑙𝑖 = 1, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3 we have that 𝑥𝑖 = 𝑦𝑖 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, 𝑙𝑖 has two productive neighbors and cannot have any more, and so we have that the other two literal nodes must play 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if exactly one of the literal nodes of 𝑐𝑔 is assigned 1 then there exists a unique assignment to the other nodes of 𝑐𝑔 such that they all play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='g assume that 𝑙1 = 1,𝑙2 = 𝑙3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then, focusing first on the sub-gadget of 𝑙1, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3 we have that 𝑥1 = 𝑦1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑥1,𝑦1 already have two productive neighbors, they mustn’t have any others, and so it must be that 𝑎1 = 𝑏1 = 𝑐1 = 𝑑1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We move on to the sub-gadget of 𝑙2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑥2 ≠ 𝑦2 then 𝑙2 would have 2 productive neighbors and would not be playing its best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑥2 = 𝑦2 = 1 then there is no assignment to 𝑎2,𝑏2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑎2,𝑏2,𝑥2 all play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore 𝑥2 = 𝑦2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We are left only with the option of setting 𝑎2 ≠ 𝑏2 and 𝑐2 ≠ 𝑑2 (for instance 𝑎2 = 𝑐2 = 1,𝑏2 = 𝑑2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The sub-gadget of 𝑙3 is symmetrical to that of 𝑙2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One may verify that in this assignment all nodes of 𝑐𝑔 indeed play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any graph 𝐺 which includes a Clause Gadget 𝑐𝑔, if all three of the literal nodes of 𝑐𝑔 are assigned 0, and the literal nodes do not have any productive neighbors outside of 𝑐𝑔, then the assignment is not a PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Assume by way of contradiction that there exists a PNE where 𝑙1 = 𝑙2 = 𝑙3 = 0, and all three of them have no productive neighbors outside 𝑐𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It must be that the other two neighbors of 𝑙1, 𝑥1,𝑦1, are assigned with different values (otherwise 𝑙1 is not playing its best response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='g assume 𝑥1 = 1,𝑦1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Now, If the remaining neighbors of 𝑦1 (𝑐1 and 𝑑1) are both assigned with 0 or both assigned with 1, then they themselves would not be playing their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' On the other hand, if we assign them with different values then 𝑦1 would not be playing its best response, and so we have reached a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ So far, we have seen that in any PNE which includes a Clause Gadget, it must be that exactly one of the literal nodes of that gadget is assigned with 1, as long as the literal nodes don’t have productive neighbors outside of their Clause Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' As we introduce the external nodes that will be connected to the literal nodes, we will show that they all must be assigned with 0 in any PNE, and thus a literal node cannot have any productive neighbor outside of its Clause Gadget, which will finalize the property we were looking to achieve with the Clause Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Our next goal is to make sure the translation between solutions from one domain to the other is always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Specifically, we wish to ensure that in any PNE in our constructed graph, if any two literal nodes represent the same variable in the CNF formula then they will be assigned the same value, and if they represent a variable and its negation then they will be assigned opposite values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We begin with the latter, introducing our Negation Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The goal of the Negation Gadget is to force opposite assignments to two chosen nodes, in any Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The Negation Gadget is composed of 9 nodes: five ’bottom’ nodes 𝑏1,𝑏2,𝑏3,𝑏4,𝑏5, and four ’top’ nodes 𝑡1,𝑡2,𝑡3,𝑡4, and for each 𝑘 ≤ 4 we create the edges (𝑏𝑖,𝑏𝑖+1), (𝑡𝑖,𝑏𝑖) and (𝑡𝑖,𝑏𝑖+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It can intuitively be described as 6We ignore the possibility of changing between the assignments of 𝑎𝑖 and 𝑏𝑖, or 𝑐𝑖 and 𝑑𝑖 for 𝑖 ∈ {1, 2, 3}, as it does not affect anything in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 8 four triangles that are connected as a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Say we have two nodes 𝑢, 𝑣 which we want to force to have opposite assignments, we simply connect them both to node 𝑡2 of a Negation Gadget, as demonstrated in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through a Negation Gadget 𝑛𝑔, 𝑢 and 𝑣 must have different assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In addition, the node 𝑡2 of 𝑛𝑔, to which 𝑢 and 𝑣 are connected, must be assigned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We first show that 𝑢 and 𝑣 must have different assignments, dividing into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Case 1: Assume by way of contradiction that 𝑢 = 𝑣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We divide into two sub-cases, where in the first one 𝑡2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' in this case, exactly one of the two remaining neighbors of 𝑡2 must be assigned 1 in order for 𝑡2 itself to be playing best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑏2 = 0 then 𝑏3 = 1 and so, looking at 𝑡1,𝑏1 (the remaining neighbors of 𝑏2), we see that any assignment to them results either in 𝑏2 not playing best response, or in 𝑡1,𝑏1 not playing best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If, however, 𝑏3 = 0, then 𝑏2 = 1, and so, looking at 𝑡3,𝑏4 (the remaining neighbors of 𝑏3), the same logic leads us to a similar contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the second sub-case, where 𝑡2 = 1, we have that its two remaining neighbors must be assigned the same value in order for 𝑡2 itself to be playing best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑏2 = 𝑏3 = 0 then again there is no assignment to 𝑏1,𝑡1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t all of 𝑏1,𝑡1,𝑏2 play best response, and if 𝑏2 = 𝑏3 = 1 then one may verify that there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response, and so we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Case 2: Assume 𝑢 = 𝑣 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then we again divide into sub-cases according to 𝑡2’s assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑡2 = 0, it must have at least one more productive neighbor in order to play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The assignments where 𝑏2 = 𝑏3 = 1 or 𝑏2 = 0,𝑏1 = 1 are easily disqualified, seeing as there is no assignment to 𝑡1,𝑏1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑡1,𝑏1,𝑏2 all play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑏2 = 1,𝑏3 = 0 then it must hold that 𝑡3 = 𝑏4 in order for 𝑏3 to play best response, but this would mean that 𝑡3 is not playing best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑡2 = 1, then its two remaining neighbors 𝑏2,𝑏3 must be set to 0 in order for it to play best response, and then there is no assignment to 𝑏1,𝑡1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t all of 𝑡1,𝑏1,𝑏2 play best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' And so it cannot be that 𝑢 = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We move on to show that 𝑡2 must play 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Assume by way of contradiction that 𝑡2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then, seeing as exactly one of 𝑢, 𝑣 is productive, 𝑡2 must have exactly one more productive neighbor in order to play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑏2 = 1,𝑏3 = 0 we reach a contradiction as there is no assignment to 𝑡1,𝑏1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑡1,𝑏1,𝑏2 all play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑏2 = 0,𝑏3 = 1 we reach a contradiction as there is no assignment to 𝑡3,𝑡4,𝑏4,𝑏5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t all of 𝑡3,𝑡4,𝑏3,𝑏4,𝑏5 play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lastly, one may verify that in the assignment where 𝑡1 = 𝑏4 = 1,𝑏1 = 𝑏2 = 𝑏3 = 𝑏5 = 𝑡2 = 𝑡3 = 𝑡4 = 0 all nodes of the gadget play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ Now, for each variable that appears in the CNF formula, we choose one instance of it and one instance of its negation7 and connect the literal nodes representing these instances via a Negation Gadget, thus ensuring they are assigned opposite values in any PNE, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We note that this is not the only place where we use this gadget, as we will see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We move on to introduce our Copy Gadget, which we will use to force literal nodes which represent the same variable to have the same assignment in any PNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The Copy Gadget is composed of two negation gadgets 𝑛𝑔1,𝑛𝑔2, and two additional nodes 𝑥,𝑦 which have an edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Say we have two nodes 𝑢, 𝑣 which we want to force to have the same assignment in any PNE, then we simply connect 𝑢 and 𝑥 to 𝑛𝑔1, and we connect 𝑣 and 𝑥 to 𝑛𝑔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The gadget is demonstrated in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 7We will soon ensure that instances of the same variable would get the same assignment in any PNE, and thus it is sufficient to negate the assignments of only one instance of a variable and its negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Negation Gadget connecting 𝑢 and 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Copy Gadget connecting 𝑢 and 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any Nash equilibrium in a graph 𝐺 which includes two nodes 𝑢, 𝑣 connected through a Copy Gadget 𝑐𝑝𝑔, 𝑢, 𝑣 must have the same assignment, and must have no productive neighbors from 𝑐𝑝𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In addition, if 𝑢 = 𝑣 then there exists an assignment to the nodes of 𝑐𝑝𝑔 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t all of them play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We first show that 𝑢 and 𝑣 must have the same assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This follows directly from the fact that 𝑥 is connected to both 𝑢 and 𝑣 via a Negation Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7 we have that 𝑥 ≠ 𝑢 and 𝑥 ≠ 𝑣, and so 𝑢 = 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7 also tells us that the Negation Gadget cannot add productive neighbors to the nodes that are connected to it in any PNE, and therefore 𝑢 and 𝑣 have no productive neighbors from 𝑐𝑝𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lastly, we show that there exists an assignment to the nodes of 𝑐𝑝𝑔 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t they all play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7 𝑥 cannot have any productive neighbors from 𝑛𝑔1 or 𝑛𝑔2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, if 𝑢 = 𝑣 = 0 then we can assign 𝑥 = 1,𝑦 = 0, and if 𝑢 = 𝑣 = 1 then we can assign 𝑥 = 0,𝑦 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In both cases, we assign 𝑛𝑔1 and 𝑛𝑔2 as suggested in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One may verify that in this assignment indeed all nodes of 𝑐𝑝𝑔 play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ Now, for each variable in the CNF formula, we connect all the literal nodes representing its different instances via a chain of copy gadgets, thus (transitively) ensuring they are all assigned the same value in any PNE, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given these lemmas and the graph we constructed, we can now prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2) The problem is in NP, since an assignment to the nodes can be easily verified as a NTPNE by iterating over the nodes and checking whether they all play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It is left to show the problem is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given a ONE-IN-THREE 3SAT instance, we construct a graph 𝐺 as described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If there exists a satisfying assignment to the 3SAT problem, we can set all literal nodes according to the assignment of their matching variable, and set all other nodes as described throughout lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8, and according to those lemmas, we get a pure Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' On the opposite direction, if there exists a non-trivial pure Nash equilibrium, then by lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='6,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4 in each clause exactly one literal node is assigned 1, and by lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8 we have that literal nodes have the same assignment if they represent the same variable, and opposite ones if they represent a variable and its negation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Thus we can easily translate the NTPNE into a satisfying ONE-IN-THREE 3SAT assignment, assigning ’True’ to variables whose literal nodes are set to 1, and ’False’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ u V t2 t3 b1 b2 b3 b4 b5u x ng1 ng2M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 10 We now wish to expand this result to two slightly more general classes of patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Firstly, we notice that the graph constructed throughout the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2 is bounded8 by a maximum degree of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, the proof is indifferent to entries of the pattern from index 7 onward, which means it holds for any pattern that agrees with the first 7 entries of the 0-Or-2-Neighbors pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇 be a BRP such that: 𝑇 [0] = 𝑇 [2] = 1 ∀𝑘 ∈ {1, 3, 4, 5, 6} 𝑇 [𝑘] = 0 Then NTPNE(𝑇) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Secondly, according to Theorem 7 in [Gilboa and Nisan, 2022], adding 1,0 at the beginning of a hard pattern that begins with 1 yields yet another hard pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Using this theorem recursively on the patterns of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='9, we have that the equilibrium decision problem is hard for any pattern of the form: 𝑇 = [1, 0, 1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, 0 �������������������������������������� 𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ , 0, 0, 0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fix 𝑚 ≥ 1, and let 𝑇 be a BRP such that: ∀0 ≤ 𝑘 ≤ 𝑚 (1) 𝑇 [2𝑘] = 1 (2) 𝑇 [2𝑘 + 1] = 0 𝑇 [2𝑚 + 2] = 𝑇 [2𝑚 + 3] = 𝑇 [2𝑚 + 4] = 0 Then NTPNE(𝑇) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We will see later on that this result will also be of use during the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' There is one very similar class of patterns on which the proofs throughout the paper rely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This is the class of all finite patterns that start with a finite number of 1,0, followed by 1,1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' all patterns of the form: 𝑇 = [1, 0, 1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, 0 �������������������������������������� 𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ , 1, 1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The complexity of those patterns was already discussed and solved in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4 of [Gilboa and Nisan, 2022], but was not formalized and so we state it here in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fix 𝑚 ≥ 2, and let 𝑇 be a BRP s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t: 𝑇 is finite ∀𝑘 ∈ IN 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑡 2𝑘 ≤ 𝑚 𝑇 [2𝑘] = 1 𝑇 [1] = 0 ∃1 ≤ 𝑛 𝑤ℎ𝑒𝑟𝑒 2𝑛 + 1 ≤ 𝑚 + 1, 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑡 𝑇 [2𝑛 + 1] = 1 Then NTPNE(𝑇) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The proof follows directly from Theorems 6 and 7 from [Gilboa and Nisan, 2022]9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ 8A literal node is connected to 4 nodes within its clause gadget, and possibly 2 nodes from copy gadgets or 1 node from a negation gadget and 1 node from a copy gadget (assuming we connect the negation gadgets at the end of their respective Copy-Gadget-chains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 9The reader who has read the details of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4 of [Gilboa and Nisan, 2022] may notice that in fact the use of Theorems 6 and 7 from [Gilboa and Nisan, 2022] covers a slightly more general class of patterns, but this entire class is not needed currently, and is covered in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 11 4 HARDNESS OF SEMI-SHARP PATTERNS In this section we show hardness of semi-sharp Best-Response Patterns, beginning with a specific sub-class of those patterns in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1, and expanding to all other semi-sharp patterns in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We remind the reader that semi-sharp patterns are patterns that begin with 1,0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1 Semi-Sharp Patterns with Isolated Odd 1 In this section we prove that any finite, semi-sharp pattern such that there exists some ’isolated’ 1 (meaning it has a zero right before and after it) at an odd index, presents a hard equilibrium decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Those patterns can be summarized by the following form: 𝑇 = [1, 0, 0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 0 1 ���� 𝑜𝑑𝑑 𝑖𝑛𝑑𝑒𝑥 , 0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇 be a BRP which satisfies the following conditions: 𝑇 is finite 𝑇 is semi-sharp ∃𝑚 ≥ 1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t: (1) 𝑇 [2𝑚] = 𝑇 [2𝑚 + 2] = 0 (2) 𝑇 [2𝑚 + 1] = 1 Then NTPNE(𝑇) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Before proceeding to the proof, we introduce two gadgets and prove two lemmas regarding their functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Force-1-Gadget: The first gadget is denoted the Force-1-Gadget, and it will appear in several parts of the graph we construct for the reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The goal of this gadget is to enable us to force any node to be assigned 1 in any Nash equilibrium in a PGG defined by 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This gadget is composed primarily of a triangle 𝑥,𝑦,𝑧, where the triangle’s nodes have also several ’Antenna’ nodes, which are connected only to their respective node from the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Specifically, 𝑥 will have 2𝑚 + 1 Antenna nodes, and 𝑦 and 𝑧 will each have 2𝑚 Antenna nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Say we have some node 𝑢, whose assignment we wish to force to be 1, then we simply connect 𝑢 to one of of the Antenna nodes of 𝑥, denoted 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The gadget is demonstrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Add-1-Gadget: our second gadget of this proof is denoted the Add-1-Gadget, and its goal is to enable us to assure the existence of (at least) a single productive neighbor to any node in a Nash equilibrium of a PGG defined by 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Say we have a node 𝑣, to which we wish to add a single productive neighbors, in any equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We construct the Add-1-Gadget as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We create 𝑚 + 1 nodes denoted 𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',𝑥𝑚+1, 𝑚 + 1 nodes denoted 𝑦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='𝑦𝑚+1, and an additional ’bridge’ node, denoted 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We connect 𝑥1 and 𝑦1 to all of the other 𝑥𝑖 and 𝑦𝑖 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' For all 𝑖, 𝑗 ≥ 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑖 ≠ 𝑗, we create the edges (𝑥𝑖,𝑥𝑗), (𝑦𝑖,𝑦𝑗), (𝑥𝑖,𝑦𝑗) (the 𝑥𝑖,𝑦𝑖 nodes almost form a clique, except that for each 𝑖 ≥ 2 we omit the edge (𝑥𝑖,𝑦𝑖)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Additionally, For all 𝑖 ≥ 2 the bridge node 𝑏 is connected to 𝑥𝑖 and to 𝑦𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' To 𝑏 we attach a Force-1-Gadget, and we also connect 𝑏 to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The gadget is demonstrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The following lemmas formalize the functionality of the two gadgets, beginning with the Force- 1-Gadget in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In any PNE in a graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1), where 𝐺 has a node 𝑢 that is connected to a Force-1-Gadget 𝑓 𝑔 as described, 𝑢 must be assigned 1, and its neighbor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Force-1-Gadget with 𝑚 = 2, attached to 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Add-1-Gadget with 𝑚 = 2, attached to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' from 𝑓 𝑔, 𝑎, must be assigned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10 Furthermore, if 𝑢 = 1 there exists an assignment to the nodes of 𝑓 𝑔 such that they each play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' First we show that 𝑢 must be assigned 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Assume by way of contradiction that 𝑢 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Divide into the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑥 = 1, then all of its Antenna nodes must be assigned 0 (according to 𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Additionally, 𝑦 and 𝑧 must also be assigned 0, as otherwise 𝑥 wouldn’t be playing best response, since 𝑇 is semi-sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, the best response of all of the Antenna nodes of 𝑦 and 𝑧 is to play 1, which leaves 𝑦 and 𝑧 with 2𝑚 + 1 productive neighbors each, and so they are not playing best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑥 = 0, then all of its Antenna nodes must play 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, 𝑥 must have at least one other productive neighbor, as otherwise it would have 2𝑚 + 1 productive neighbors and wouldn’t be playing best response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='g assume 𝑦 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then all of 𝑦’s Antenna nodes must play 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, 𝑧 must play 0, as otherwise 𝑦 wouldn’t be playing best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This means the best response for 𝑧’s Antenna nodes is to play 1, which leaves 𝑧 with 2𝑚 + 1 productive neighbors, and so it isn’t playing best response, in contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We move on to showing that 𝑎 must play 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This follows directly from the fact that 𝑢 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑎 only has one other neighbor (𝑥), regardless of its strategy the best response for 𝑎, according to 𝑇, would be playing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It is left to show that when 𝑢 = 1 and 𝑎 = 0, there exists an assignment to the nodes of 𝑓 𝑔 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t they all play best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One may verify that when we set 𝑥 = 𝑦 = 𝑧 = 0 and set all the Antenna nodes in 𝑓 𝑔 (except for 𝑎) to 1, then all nodes of 𝑓 𝑔 play best response (specifically, 𝑥,𝑦,𝑧 would each have exactly 2𝑚 productive neighbors, which, by definition of 𝑇, means they are playing best response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ We move on to proving the following Lemma, which formalizes the functionality of the Add-1- Gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 8 In any graph 𝐺 corresponding to the BRP 𝑇 (from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1), where 𝐺 has a node 𝑣 that is connected to an Add-1-Gadget 𝑎𝑔 as described, there always exists an assignment to the nodes of 𝑎𝑔 s t they all play best response, regardless of 𝑣’s strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In addition, the bridge node 𝑏 of 𝑎𝑔 must be assigned 1 in such an assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The claim that 𝑏 must play 1 follows directly from the fact that it has a Force-1-Gadget attached to it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Additionally, all the nodes of the Force-1-Gadget attached to 𝑏 10The property that 𝑎 = 0 allows us to use the Force-1-Gadget without risking potentially adding productive neighbors to the respective node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' a u X ZX1 Y1 X2 Y2 X3 Y3 b Force- 1-GadgetM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 13 can be assigned as suggested in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' It is left to show a possible assignment to the rest of the nodes of 𝑎𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We divide into cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑣 = 0, then we set 𝑥1 = 1 and all other 𝑥𝑖,𝑦𝑖 nodes we set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑣 = 1, then we set 𝑥𝑖 = 𝑦𝑖 = 1 for all 1 ≤ 𝑖 ≤ 𝑚 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' One may verify that given these assignments all nodes of 𝑎𝑔 play their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ In addition to these two gadgets, we wish to introduce the following definition, after which we will proceed to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1, which we can now prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇,𝑇 ′ be two BRPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We say that 𝑇 ′ is shifted left by 𝑡 from 𝑇 if ∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 𝑡].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1) Denote by 𝑇 ′ the pattern which is shifted left by 1 from T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' : ∀𝑘 ≥ 0 𝑇 ′[𝑘] = 𝑇 [𝑘 + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Notice that𝑇 ′ is flat, non-monotonic and finite, and therefore NTPNE(𝑇 ′) is NP-complete according to Theorem 4 in [Gilboa and Nisan, 2022], which allows us to construct a Turing reduction from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The technique of the reduction is very similar to those of the proofs of Theorems 5,6 in [Gilboa and Nisan, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given any graph 𝐺 = (𝑉, 𝐸), where 𝑉 = 𝑣1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 𝑣𝑛, we construct 𝑛 graphs 𝐺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',𝐺𝑛, where for each 1 ≤ 𝑖 ≤ 𝑛 the graph 𝐺𝑖 is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The graph 𝐺𝑖 contains the original input graph 𝐺, and in addition, we connect a unique Add-1-Gadget to each of the original nodes, and a Force-1-Gadget only to node 𝑣𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If there exists some non-trivial PNE in the PGG defined on 𝐺 by 𝑇, let 𝑣 𝑗 be some node who plays 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then the same NTPNE is also an NTPNE in the PGG defined by 𝑇 ′ on 𝐺𝑗, when we assign the nodes of the additional gadget as suggested in lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' To see why, notice that 𝑇 ′ is shifted left by 1 from 𝑇, and the Add-1-Gadgets ensure that all nodes have exactly one additional productive neighbor than they had in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the other direction, if there exists an NTPNE in a PGG defined by 𝑇 ′ on one of the graphs 𝐺𝑗, then by the same logic this is also a PNE in the game defined by 𝑇 on 𝐺 (ignoring the assignments of the added nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Moreover, the Force-1-Gadget ensures this assignment is non-trivial even after removing the added nodes, since 𝑣 𝑗 must play 1 in this assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2 All Semi-Sharp Patterns In this section we show that any finite, non-monotone, semi-sharp pattern presents a hard equilib- rium problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇1 be a finite, non-monotone, semi-sharp BRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then NTPNE(𝑇1) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Before proceeding to the proof, we wish to introduce the following definition and prove two lemmas related to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇,𝑇 ′ be two BRPs such that ∀𝑘 ∈ IN it holds that 𝑇 [𝑘] = 𝑇 ′[2𝑘].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then we say that 𝑇 ′ is a double-pattern of 𝑇, and 𝑇 is the half-pattern of 𝑇 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Notice that a pattern has a unique half-pattern, whereas, since the definition does not restrict 𝑇 ′ in the odd indices, any pattern has infinite double-patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The first lemma is very simple and intuitive, stating that the largest index with value 1 in a half pattern is strictly smaller than the largest index with value 1 in its original pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This is true since for any index 𝑖 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t the value of the half pattern is 1 in that index, the original pattern has a value of 1 in index 2𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇,𝑇 ′ be two finite BRPs such that 𝑇 is the half-pattern of 𝑇 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Denote by 𝑖 the largest index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇 [𝑖] = 1 and denote by 𝑗 the largest index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇 ′[𝑗] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then if 𝑗 > 0 we have that 𝑖 < 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The proof is trivially given by the definition of a half pattern, since 𝑇 ′[2𝑖] = 𝑇 [𝑖].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ The next lemma is less trivial, stating the relation between hardness of a pattern and its double- pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let 𝑇 be a BRP such that NTPNE(𝑇) is NP-complete, and let 𝑇 ′ be a double-pattern of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then NTPNE(𝑇 ′) is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We use a specific case of the same reduction that was used to prove Theorem 4 in [Gilboa and Nisan, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Given a graph 𝐺1 = (𝑉1, 𝐸1) as input, where 𝑉1 = 𝑣1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 𝑣1 𝑛, we create another replica of it 𝐺2 = (𝑉2, 𝐸2), where 𝑉2 = 𝑣2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 𝑣2 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' For each node (from both graphs), we add edges connecting it to all replicas of its neighbors from the opposite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' That is, the following group of edges is added to the graph: 𝐸 = {(𝑣1 𝑖 , 𝑣2 𝑗)|(𝑣1 𝑖 , 𝑣1 𝑗) ∈ 𝐸1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' A demonstration of the reduction can be seen in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Example of the reduction of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8’s proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Denote by 𝑃 the PGG defined on 𝐺1 by 𝑇, and by 𝑃 ′ the PGG defined by 𝑇 ′ on 𝐺 ′ = (𝑉 ′, 𝐸′) where 𝑉 ′ = 𝑉1 ∪ 𝑉2, 𝐸′ = 𝐸 ∪ 𝐸1 ∪ 𝐸2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We show that there exists an NTPNE in 𝑃 iff there exists one in 𝑃 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If there exists an NTPNE in 𝑃, we simply give the nodes of 𝐺2 the same assignment as those of 𝐺1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑇 ′ is a double pattern of 𝑇, any node 𝑣 ′ ∈ 𝑉 ′ must play best response, having exactly twice as many supporting neighbors than it had (or its replica had) in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the opposite direction, if there exists an NTPNE in 𝑃 ′, notice that for all 1 ≤ 𝑖 ≤ 𝑛 it must be that 𝑣1 𝑖 and 𝑣2 𝑖 have identical assignments, since they both share exactly the same neighbors, and thus have identical best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, any node 𝑣 ′ ∈ 𝑉 must have an even number of productive neighbors, half of which are in 𝑉1 and the other half in 𝑉2 (as for each productive neighbor from 𝑉1 there is a respective productive neighbor from 𝑉2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We then simply ignore 𝐺2, and leave the assignment of 𝐺1 as it is, and each node shall now have exactly half as many productive neighbors as it had in the original assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑇 is a half pattern of 𝑇 ′, we get an NTPNE in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11 □ Given lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8, we are now able to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The intuitive idea of the proof is that we halve the pattern 𝑇1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' find its half-pattern) repeatedly, until eventually we reach some pattern for which we already know the equilibrium problem is hard, which, as we will see, must happen at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then, by applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8 recursively, we have that 𝑇1 is hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 11In both directions of this proof, the non-triviality comes from the fact that 𝑇 [0] = 𝑇 ′[0], by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, a non-trivial PNE in one domain must translate to a non-trivial one in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Vi 2 V2 2 V 3M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5) From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='7 we have that if we halve a non-flat pattern enough times, we will eventually reach the Best-Shot pattern: 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [0] = 1 and ∀𝑘 ≥ 1 𝑇𝐵𝑒𝑠𝑡−𝑆ℎ𝑜𝑡 [𝑘] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Divide into two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the first case assume that ∀𝑘 ∈ IN it holds that𝑇1[2𝑘] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In this case, we know that no matter how many times we halve 𝑇1 into patterns 𝑇2,𝑇3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', for each 𝑖 we will have that 𝑇𝑖 [1] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the value in index 1 of all these half-patterns will always be 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇𝑖 [1] = 0 for all 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Assume that we halve𝑇1 repeatedly into patterns𝑇2,𝑇3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',𝑇𝑚 (where𝑇𝑖 is the half pattern of𝑇𝑖−1) such that𝑇𝑚 is the first time that we reach the Best-Shot pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Observe 𝑇𝑚−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' For any even index 𝑘 ≠ 0 it must hold that 𝑇𝑚−1[𝑘] = 0, otherwise 𝑇𝑚 would not be the Best-Shot pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Additionally, there must exist at least one odd index 𝑗 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇𝑚−1[𝑗] = 1, since 𝑇𝑚 is the first time we reach the Best-Shot pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' For these two reasons, we have that 𝑇𝑚−1 satisfies the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1 and therefore NTPNE(𝑇𝑚−1) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8 (used inductively), we have that ∀1 ≤ 𝑖 ≤ 𝑚 − 2 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically NTPNE(𝑇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In the second case, assume that there exists some 𝑘 ∈ IN s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇1[2𝑘] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In that case, after at most 𝑘 halvings, we reach some pattern for which the value of index 1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Assume that we halve 𝑇1 repeatedly into patterns 𝑇2,𝑇3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=',𝑇𝑛 (where 𝑇𝑖 is the half pattern of 𝑇𝑖−1) such that 𝑇𝑛 is the first time that we reach a pattern for which index 1 is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' ∀1 ≤ 𝑖 ≤ 𝑛 − 1 𝑇𝑖 [1] = 0 and 𝑇𝑛[1] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Notice that, additionally, by definition of a half-pattern for each 𝑖 it holds that 𝑇𝑖 [0] = 1 (since 𝑇1[0] = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If 𝑇𝑛 is non-monotone, then by Theorem 5 in [Gilboa and Nisan, 2022] we have that NTPNE(𝑇𝑛) is NP-complete under Turing reduction, and from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8 (used inductively), we have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically NTPNE(𝑇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 𝑇𝑛 is monotone), denote by 𝑙 the largest index s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇𝑛[𝑙] = 1, and observe 𝑇𝑛−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' By definition of double-patterns, we have that: ∀𝑗 ∈ IN 𝑇𝑛−1[2𝑗] = � 1 if 𝑗 ≤ 𝑙 0 otherwise i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the value in the even indices is 1 until 2𝑙, and 0 afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Since 𝑇𝑛 is defined to be the first halving of 𝑇1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t its value in index 1 is 1, we have that 𝑇𝑛−1[1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' However, since the definition of a double-pattern does not restrict it in the odd indices, there might be some odd indices (strictly larger than 1) for which the value of 𝑇𝑛−1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Divide into 3 sub-cases: Sub-case 1: If there exists some 𝑧 ≤ 𝑙 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇𝑛−1[2𝑧 + 1] = 1, then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11, we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Sub-case 2: Otherwise, if there exists some 𝑧 > 𝑙 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='t 𝑇𝑛−1[2𝑧 + 1] = 1, then observe the pattern 𝑇 ′ 𝑛−1, which we define as the pattern shifted left by 2𝑙 from 𝑇𝑛−1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' : ∀𝑗 ∈ IN 𝑇 ′ 𝑛−1[𝑗] = 𝑇𝑛−1[𝑗 + 2𝑙] Notice that this pattern satisfies the conditions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1, and therefore NTPNE(𝑇 ′ 𝑛−1) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then, by applying Theorem 7 from [Gilboa and Nisan, 2022] 𝑙 times, we have that NTPNE(𝑇𝑛−1) is also NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Sub-case 3: Otherwise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' there is no odd index whatsoever in which the value of 𝑇𝑛−1 is 1), then by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10 we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' And so, in either case we have that NTPNE(𝑇𝑛−1) is NP-complete under Turing reduction, and therefore from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='8 (used inductively), we have that ∀1 ≤ 𝑖 ≤ 𝑛 − 1 NTPNE(𝑇𝑖) is also NP-complete under Turing reduction, and specifically NTPNE(𝑇1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 16 5 HARDNESS OF ALL SPIKED PATTERNS There are several finite, spiked patterns that we have not yet proved hardness for, and we now have enough tools to close the remaining gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We remind the reader that spiked patterns are patterns that begin with 1,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The following theorem formalizes the result of this section, and completes the characterization of all finite patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Let𝑇 be a finite, spiked BRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Then NTPNE(𝑇) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The intuitive idea of the proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' If the pattern simply alternates between 1 and 0 a finite amount of times, followed infinite 0’s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the pattern is of the form 𝑇 = [1, 0, 1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, 0 �������������������������������������� 𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ , 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='] then the problem12 is already shown to be hard by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Otherwise, we wish to look at the first "disturbance" where this pattern stops alternating from 1 to 0 regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Either the first "disturbance" is a 1 at an odd index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the pattern is of the form 𝑇 = [1, 0, 1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, 0 �������������������������������������� 𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ , 1, 1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='] or the first "disturbance" is a 0 at an even index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the pattern is of the form 𝑇 = [1, 0, 1, 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, 0 �������������������������������������� 𝑓 𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ′1,0′ , 0, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', 1, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='] (in the latter option, after the first "disturbance" there must be some other index with value 1, since the pattern does not fit the form of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' The first option was solved in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11, and the second option can be solved using our previous results, as we shall now formalize in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1) If𝑇 satisfies the conditions of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10 or Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11 then NTPNE(𝑇) is NP-complete under Turing reduction according to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Otherwise, let 𝑘 be the smallest integer such that 𝑇 [2𝑘] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Denote by 𝑇 ′ the pattern which is shifted left by 2𝑘 − 2 from T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' : ∀𝑗 ≥ 0 𝑇 ′[𝑗] = 𝑇 [𝑗 + 2𝑘 − 2] Notice that from definition of 𝑘 (being the first even index such that 𝑇 [2𝑘] = 0) we have that for all 𝑗 < 𝑘 it holds that 𝑇 [2𝑗] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Moreover, since 𝑇 does not satisfy the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11 it must hold for all 𝑗 ≤ 𝑘 that 𝑇 [2𝑗 − 1] = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' the value of 𝑇 in the odd indices until 2𝑘 is 0 (since otherwise𝑇 would start with a finite number of 1,0, followed by 2 consecutive 1’s, and would satisfy the conditions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Thus, we have that ∀𝑗 < 2𝑘 𝑇 [𝑗] = � 1 if 𝑗 is even 0 if 𝑗 is odd (1) In particular, we have that𝑇 [2𝑘 − 2] = 1, 𝑇 [2𝑘 − 1] = 0, which implies that𝑇 ′[0] = 1, 𝑇 ′[1] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' as 𝑇 [2𝑘] = 0 we have that 𝑇 ′[2] = 0, and thus we conclude that 𝑇 ′ is semi-sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In addition, since 𝑇 does not satisfy the conditions of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10, there must be some other index 𝑥 > 2𝑘 such that 𝑇 [𝑥] = 1, and therefore we have that 𝑇 ′ is non-monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Therefore, by Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='5, we have that NTPNE(𝑇 ′) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' We now wish to use this in order to prove that NTPNE(𝑇) is also hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 12In fact, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='10 gives a more general result, but we currently only need the private case where the pattern ends with infinite 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Gilboa 17 From Equation 1, we can apply Theorem 7 of [Gilboa and Nisan, 2022] (𝑘 − 1) times, and we have that NTPNE(𝑇) is NP-complete under Turing reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' □ ACKNOWLEDGMENTS I would like to thank Noam Nisan for many useful conversations throughout the work, and for suggesting the Copy Gadget seen in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' I would like to thank Roy Gilboa for many useful conversations throughout the work and for adjusting the Copy Gadget seen in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' I would like to thank Noam Nisan for communicating to me the solution of the monotone case by Max Klimm, and the alternative derivation by Sigal Oren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 740282).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' REFERENCES Yann Bramoullé and Rachel Kranton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Public Goods in Networks.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='02916 Sixie Yu, Kai Zhou, Jeffrey Brantingham, and Yevgeniy Vorobeychik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Computing Equilibria in Binary Networked Public Goods Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' 34(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Association for the Advancement of Artificial Intelligence (AAAI), New York, NY, USA, 2310–2317.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' Computing Equilibria in Binary Networked Public Goods Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFJT4oBgHgl3EQfkywG/content/2301.11580v1.pdf'} +page_content='1911.' metadata={'source': 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Gorshkov,1, 3 and Victor Galitski1, 4 +1Joint Quantum Institute, Department of Physics, +University of Maryland, College Park, MD 20742, USA +2Condensed Matter Theory Center, Department of Physics, +University of Maryland, College Park, MD 20742, USA +3Joint Center for Quantum Information and Computer Science, +NIST/University of Maryland, College Park, MD 20742, USA +4Center for Computational Quantum Physics, The Flatiron Institute, New York, NY 10010, USA +(Dated: January 13, 2023) +Quantum spin liquids are exotic phases of matter whose low-energy physics is described as the +deconfined phase of an emergent gauge theory. With recent theory proposals and an experiment +showing preliminary signs of Z2 topological order [G. Semeghini et al., Science 374, 1242 (2021)], +Rydberg atom arrays have emerged as a promising platform to realize a quantum spin liquid. In +this work, we propose a way to realize a U(1) quantum spin liquid in three spatial dimensions, +described by the deconfined phase of U(1) gauge theory in a pyrochlore lattice Rydberg atom +array. We study the ground state phase diagram of the proposed Rydberg system as a function +of experimentally relevant parameters. Within our calculation, we find that by tuning the Rabi +frequency, one can access both the confinement-deconfinement transition driven by a proliferation +of “magnetic” monopoles and the Higgs transition driven by a proliferation of “electric” charges +of the emergent gauge theory. We suggest experimental probes for distinguishing the deconfined +phase from ordered phases. This work serves as a proposal to access a confinement-deconfinement +transition in three spatial dimensions on a Rydberg-based quantum simulator. +I. +INTRODUCTION +When the classical part of a many-body Hamiltonian is +frustrated, quantum fluctuations can break the degener- +acy in interesting ways. An exotic form of such breaking +was pointed out by Anderson [1] where the ground state is +a superposition of several almost-degenerate states, and +the excitations are “fractional”[2]. Broadly, a common +feature tying together such systems called quantum spin +liquids is that, at low energies, they can be described +as lying in a deconfined phase of an emergent gauge +theory. The fractional excitations are the “charge”-like +and “flux/monopole”-like excitations of this gauge the- +ory. When these fractional excitations get confined, they +cease to be important for the low-energy physics, and the +system becomes ordered. From this point of view, tran- +sitions from a spin liquid to conventional ordered phases +are understood as a confinement-deconfinement transi- +tion, driven by a proliferation of “flux/monopole”-like ex- +citations, or a Higgs transition, driven by a proliferation +of “charge”-like excitations [3–6]. +Gauge theories and +their phase transitions are of fundamental importance in +physics [7–10]. The prospect of this physics emerging in +many-body systems provides an important motivation for +studying quantum spin liquids. They are also interesting +due to their possible role in the physics of strongly corre- +lated materials [11] and possible application in quantum +computing [12, 13]. +Traditionally, the main search space for spin liquids +has comprised of solid state systems. +While consis- +tent progress has been made [2, 14], conclusive evidence +for spin liquids is still lacking in these systems. +One +reason is that the same feature that makes spin liq- +uids interesting—being characterized by non-local order +parameters—also makes them hard to detect. +Mean- +while, over the past decade, Rydberg atom arrays have +emerged as a promising platform for engineering inter- +acting Hamiltonians [15–36]. Rydberg states have large +principal quantum number n (∼ 20 − 100), and the van +der Waals interaction between them scales as n11. The +strong tunable interactions, along with the ability to cus- +tomize the lattice of atoms, locally control qubits, and +take wavefunction snapshots, make Rydberg atom arrays +a competitive platform to explore quantum many-body +physics. Following theory proposals [37, 38], promising +signs of Z2 topological order have been observed exper- +imentally on this platform [26]. This has sparked a lot +of activity over the past two years in the general direc- +tion of proposing ways to realize exotic states on quan- +tum devices using analogue quantum simulation [39–42], +digital quantum simulation [43], and projective measure- +ments [44, 45]. +Our work is a proposal for realizing a U(1) quan- +tum spin liquid, described by the deconfined phase of +a compact U(1) gauge theory on three-dimensional Ry- +dberg atom arrays, with an eye towards accessing the +confinement-deconfinement transition. It was shown by +Polyakov [46, 47] that compact U(1) gauge theory in +2+1 dimensions is always in the confined phase in the +thermodynamic limit due to a proliferation of monopole +events. +Therefore we turn to 3+1 dimensions, where +Polyakov argued [47] for the existence of both deconfined +and confined phases separated by a transition driven +by monopole excitations. The deconfined phase consists +of gapless “photons”, gapped “monopoles” and gapped +“charge” excitations. In the early 2000s, lattice models +arXiv:2301.04657v1 [cond-mat.quant-gas] 11 Jan 2023 + +2 +of spins [48] and dimers [49] on corner-sharing polyhe- +dra were constructed that were strongly argued to realize +this phase—a U(1) spin liquid, using perturbation the- +ory, solvable limits [48] and later Quantum Monte Carlo +simulations [50, 51]. Our work is based on a spin model +with easy-axis antiferromagnetic interactions introduced +by Hermele et al. [48] on the pyrochlore lattice consisting +of corner-sharing tetrahedra (see Fig. 1). The classical +Ising limit of this model is the widely studied classical +spin ice [52–56], which has a large residual entropy at +low temperatures similar to water-ice [57]. This is be- +cause the ground states form an exponentially degener- +ate set of states obeying the “ice rule” (see Sec. II). The +quantum model in Ref. [48] has also been a subject of +intense study in the context of pyrochlore materials like +Yb2Ti2O7 and Er2Ti2O7 as potential quantum spin ice +(another name for the U(1) spin liquid) candidates [58]. +It was observed in Ref. [59] that the Hamiltonian in +Ref. [48] can be viewed as that of hard-core bosons +hopping on an optical lattice with nearest-neighbor re- +pulsion, thus extending its relevance to the cold atom +setting. Ref. [60] studied a similar model of hard-core +bosons hopping on a two-dimensional checkerboard lat- +tice. In Ref. [60], the atom’s internal state was largely +the ground state, but a dressing with Rydberg states +was used to engineer interactions between atoms. Later, +Ref. [61] showed that dimer models in two dimensions can +be implemented on configurable Rydberg arrays—where +the atoms themselves are stationary but can internally be +either in a ground state or in a Rydberg state. In this set- +ting, the atoms are driven with a laser (or a pair of lasers +making a two-photon transition) that is detuned from the +ground to Rydberg transition. The Rydberg interactions +and the detuning define a (frustrated) “classical” energy +landscape. The laser driving induces quantum fluctua- +tions controlled by the Rabi frequency, leading (pertur- +batively) to dimer moves or ring exchange terms that are +required to deconfine a gauge theory. The proposal [37] +and experiment [26] mentioned above worked in the same +setting. Our work is also based on this setting in which +the atom array is configured in a 3D pyrochlore lattice. +In Sec. II, we explain our proposal. +We show that +within a window of laser detunings, the classical land- +scape is identical to the set of ice rule obeying states. +Our Hamiltonian, when restricted to nearest-neighbor +interactions, is equivalent to the transverse-field Ising +model on the pyrochlore lattice. +In the limit of small +Rabi frequencies, it is perturbatively equivalent to the +model in [48], which was argued to have a spin liquid +ground state. Away from the perturbative limit, there is +numerical evidence for a spin liquid phase [62]. However, +once we include the long-range 1/r6 interactions beyond +nearest-neighbor, the classical landscape is no longer de- +generate, and it is a priori unclear if the spin liquid sur- +vives as the ground state. We attempt to answer this +in Sec. III by comparing the energy of an ansatz wave +function of the spin liquid with that of an ordered state. +Within our approximation, we find that by dialing up +the Rabi frequency, for fixed detuning and interaction +strength, one goes through a confinement-deconfinement +transition from an ice rule obeying ferromagnetic state +into a deconfined spin liquid phase. +Then, by fur- +ther increasing the Rabi frequency, one goes through a +Higgs transition from the spin liquid to a transverse-field- +polarized state (see Sec. III B). While the analysis till this +point focuses on the ground state, in Sec. III C, we com- +ment on the role played by dynamical state-preparation +in deciding the nature of the state prepared in experi- +ment. In Sec. IV, we present correlation functions that +distinguish the spin liquid from the confined phases, and +provide experimental protocols for measuring them. Fi- +nally, in Sec. V, we present general discussions and con- +clusions. +II. +PROPOSAL TO REALIZE A U(1) QUANTUM +SPIN LIQUID USING RYDBERG ATOMS +In this section, we describe our proposal to realize a +U(1) Quantum Spin Liquid (QSL) in Rydberg atom ar- +rays. Consider a 3D Rydberg array in which the atoms +are positioned on the sites of the pyrochlore lattice [see +Fig. 1(a)]. Each of the atoms can either be in the ground +state |g⟩ or in the Rydberg state |r⟩. In the rotating wave +approximation and in a rotating frame, the Hamiltonian +is +ˆHryd = − δ +� +i +ˆni + V +2 +� +i̸=j +Å +a +|xi − xj| +ã6 +ˆniˆnj ++ Ω +2 +� +i +(ˆbi + ˆb† +i), +(1) +where ˆbi = |gi⟩ ⟨ri|, ˆni = ˆb† +iˆbi, Ω is the Rabi frequency, δ +is the laser detuning, V is the nearest-neighbor van der +Waals interaction strength, and a is the distance between +two neighboring atoms. +The summation � +i̸=j is over +distinct sites i and j of the pyrochlore lattice (each pair +is being counted twice), and � +i is over sites i. Below, +we briefly describe the pyrochlore lattice. +The pyrochlore lattice is a face-centred cubic (FCC) +lattice with a four-site basis formed by the four vertices +of an up-pointing tetrahedron. (Since each lattice site +belongs to one up-pointing tetrahedron and one down- +pointing tetrahedron, the down-pointing tetrahedra are +formed automatically once we create the up-pointing +tetrahedra.) In Cartesian coordinates, the primitive vec- +tors of the FCC lattice are +a1 = +√ +2a(0, 1, 1), +a2 = +√ +2a(1, 0, 1), +a3 = +√ +2a(1, 1, 0). +(2) +The pyrochlore lattice sites are physically located at +r + eµ/2 [and labeled (r, µ)], where r is an FCC lattice + +3 +Figure 1: (a) The pyrochlore lattice. White circles denote atoms in the ground state, while black circles denote atoms +in the Rydberg state. The configuration shown satisfies n += 2 on each tetrahedron. The label x is used to denote +the sites of the pyrochlore lattice. (b) The diamond lattice. It is the bipartite lattice formed by the centers of the +tetrahedra marked by green (A sublattice) and blue (B sublattice) dots. eµ for µ ∈ {0, 1, 2, 3} label the vectors joining +an A site to its neighboring B sites. The label r is used to denote the sites of the diamond lattice. (c) The red links +are the edges of the lattice dual to the diamond lattice shown in (b). This lattice is also a diamond lattice, and we +refer to it as the “dual diamond lattice” in this paper to distinguish it from the “diamond lattice” in (b). The sites of +the dual diamond lattice are centers of the “polyhedra” formed by four puckered hexagons of the diamond lattice. uµ +for µ ∈ {0, 1, 2, 3} label the vectors joining an A site to its neighboring B sites on the dual diamond lattice. The label +r [notice the difference in the font as compared to r in (b)] is used to denote the sites of the dual diamond lattice. +vector, and the vectors eµ for µ ∈ {0, 1, 2, 3} are defined +as [see Fig. 1(b)] +e0 = +a +√ +2(1, 1, 1) = 1 +4(a1 + a2 + a3), +e1 = +a +√ +2(1, −1, −1), +e2 = +a +√ +2(−1, 1, −1), +e3 = +a +√ +2(−1, −1, 1). +(3) +We map the two levels of the atoms to spins-1/2s: +|g⟩ → |↓⟩, |r⟩ → |↑⟩, ˆni → ˆSz +i + 1/2 and ˆbi + ˆb† +i → 2 ˆSx +i . +The term ˆniˆnj therefore maps to an ˆSz +i ˆSz +j interaction in +addition to a Zeeman term ˆSz +i . Written in terms of spins, +the Hamiltonian, up to an additive constant, is +ˆHryd = − h +� +i +ˆSz +i + V +2 +� +i̸=j +Å +a +|xi − xj| +ã6 +ˆSz +i ˆSz +j ++ Ω +� +i +ˆSx +i , +(4) +where +h = δ − V +2 +� +i̸=0 +Å +a +|xi − x0| +ã6 +, +(5) +and is independent of the choice of x0 for an infinite lat- +tice. Evaluating this sum numerically for the pyrochlore +lattice, we obtain h = δ − 3.46V . +It is useful to sep- +arate the total Hamiltonian, Eq. (4), into three parts, +ˆHryd = ˆH0 + ˆHΩ + ˆHLR, where +ˆH0 =V +2 +� +⟨i,j⟩ +ˆSz +i ˆSz +j − h +� +i +ˆSz +i , +ˆHΩ =Ω +� +i +ˆSx +i , and ˆHLR = V +2 +� +i̸=j +′ Å +a +|xi − xj| +ã6 +ˆSz +i ˆSz +j , +(6) +where � +⟨i,j⟩ is over nearest-neighbor pairs and �′ +i̸=j in +ˆHLR is over the remaining pairs that are not nearest- +neighbor (in both � and �′, each pair is counted twice). +The coefficients of the second, third, and fourth nearest- +neighbor interactions are V/27, V/64, and V/125, respec- +tively. +Since these are small in comparison to V , we +will drop ˆHLR for the rest of this section because do- +ing so allows us to connect to some previously known +results [48, 51, 63]. We will study the effect of the long- +range van der Waals interaction ˆHLR in Sec. III. +Since the pyrochlore lattice is made of corner-sharing +tetrahedra and since all edges in a tetrahedron are of +equal length, we see that ˆH0 can be written up to an +additive constant as (for convenience, in the expression +below, we switch back to the hard-core boson notation) +ˆH0 = V +2 +� +r +�ˆn +r − ρ�2 , +(7) +where the sum is over all tetrahedra, ρ = 1 +2 +�4 + h +V +� = +1 +2 +�0.54 + δ +V +�, and ˆn +r = � +i∈ +r ˆni denotes the total +number of atoms in the excited state on a given tetrahe- +dron +r. Minimizing ˆH0 to obtain the classical ground + +4 +Figure 2: Mapping between Rydberg array configura- +tions and dimer configurations. A Rydberg atom (black +dot) is mapped to the presence of a dimer (orange bar), +while a ground state atom (white dot) is mapped to the +absence of a dimer. (a), (b), and (c) show example dimer +configurations corresponding to n += 1, 2, and 3, respec- +tively. In each case, n +many dimers touch the center of +each tetrahedron (the centers of the tetrahedra form the +diamond lattice). +state imposes a constraint on n +for each tetrahedron +depending on the value of ρ: +n += +� +� +� +� +� +0 +if ρ < 1/2, +floor �ρ + 1 +2 +� +if 1/2 < ρ < 7/2, +4 +if 7/2 < ρ. +(8) +The cases n += 0 and n += 4 are trivial, in that the +classical ground state is unique. +However, in the re- +maining cases, namely n += 1, 2, 3, the classical ground +state manifold is degenerate with exponentially (in sys- +tem size) many states in it. In the case n += 2, the num- +ber of configurations satisfying this constraint is approxi- +mately (3/2)Ntetrahedra (where Ntetrahedra is the number of +tetrahedra) [64]. This is based on an argument similar to +the one given by Pauling to explain the residual entropy +of water-ice at zero temperature [57]. From now onward, +we will refer to the condition n += 2 as the “ice rule”. +An ice rule obeying configuration is shown in Fig. 1(a). +In these non-trivial cases, the configurations with fixed +n +can be mapped to configurations of dimers on the +bipartite diamond lattice formed by the centers of tetra- +hedra of the pyrochlore lattice [Fig. 1(b)], with exactly +n +many dimers touching each diamond site (see Fig. 2). +The A and B sites of the diamond lattice are located at +n and n+e0, respectively, where n is an FCC lattice vec- +tor. For later use in this paper, we also show the lattice +dual to this diamond lattice in Fig. 1(c) (also a diamond +lattice, which we call the “dual diamond lattice”). An +atom in the Rydberg state on site i is mapped to a dimer +on the corresponding link of the diamond lattice, while +an atom in the ground state is mapped to no dimer. Such +dimer models have been studied extensively in both two +and three dimensions [48, 65–67]. +In the limit Ω ≪ V , ˆHΩ leads to quantum fluctua- +tions that break the exponential degeneracy of the low- +energy manifold. +We will study this effect perturba- +tively in the following section (Sec. II A). Classically, the +energy gap between the degenerate ground state space +and the lowest excited states corresponding to two tetra- +hedra violating Eq. (8) by either +1 or −1 is 2V × +min ({ρ + 1/2}, 1 − {ρ + 1/2}). Here, {x} ≡ x − floor(x) +is the fractional part of x. It should be noted that, in the +borderline cases when ρ = m + 1/2 with m ∈ {0, 1, 2, 3}, +the energy gap closes and our perturbative analysis can- +not be used. We assume going forward that ρ is away +from these borderline values. +A. +Perturbation theory +We work in the limit Ω ≪ V and treat ˆHΩ as a per- +turbation over ˆH0, ignoring for now ˆHLR whose effects +will be considered later in Sec. III. We calculate the ef- +fective Hamiltonian within the ground state manifold of +ˆH0 using the Schrieffer-Wolff formulation of perturbation +theory. For simplicity, we present the calculation of the +effective Hamiltonian only for n += 2 here. The only +difference between these three cases will be the Hilbert +space on which the Hamiltonian acts. +Calculating, at +kth order in perturbation theory, the matrix element of +the effective Hamiltonian between two states |n⟩ and |m⟩ +lying in the degenerate manifold involves starting from +|m⟩, applying the perturbation k times, and reaching the +state |n⟩. Since ˆHΩ changes the particle number by ±1, +the corrections at all odd orders are zero. Hence, we need +to consider only the corrections at even orders. +Acting with +Ω +2 (ˆbi + ˆb† +i) on an ice rule obeying state +creates two excited tetrahedra (whose common site is +i), which violate the constraint n += 2. Therefore, the +only second-order process that takes us back to the ice +manifold (the degenerate manifold of the ice rule obey- +ing states) is the one in which two excited tetrahedra are +created and annihilated, as illustrated in Figs. 3(a) and +(b). +Since such processes are present for all the states of +the ice manifold, they contribute only a constant energy +shift and can be ignored. The same is true for the fourth- +order processes. Now, the pyrochlore lattice has hexago- +nal plaquettes, some of which are shown in Fig. 4. This +allows for non-trivial processes to exist at sixth order. +In fact, non-trivial ring exchange over hexagonal plaque- +ttes of the pyrochlore lattice is obtained by the process +shown in Figs. 3(a)–(g) (some sixth-order processes also +result in a constant energy shift which we neglect). A +flippable configuration—one in which atoms on a hexag- +onal plaquette are alternately in the ground and Rydberg +states—is mapped to the complementary flippable con- +figuration by the ring exchange process as illustrated in +Fig. 3(h). Thus, the effective Hamiltonian consists of ring +exchange terms: +ˆHeff = −Jring(ρ) +� +� +�� +� � +�� + H.c., +(9) +where Jring(ρ) = γ(ρ)Ω6/V 5, the sum is over all hexag- +onal plaquettes of the pyrochlore lattice, and γ(ρ) is a + +5 +Figure 3: (a) and (b) constitute a virtual process at sec- +ond order in perturbation theory in Ω/V . Starting from +(a) which is a configuration that satisfies n += 2 on all +sites, ˆb1 + ˆb† +1 is applied giving (b). To complete the sec- +ond order process, ˆb1 + ˆb† +1 is applied to (b) giving back +(a). +Tetrahedra for which n +̸= 2 are shaded in red. +Sub-figures (a)–(g) constitute a sixth-order process in the +perturbation theory that contributes to the ring exchange +term in the effective Hamiltonian, Eq. (9). Starting from +(a), the perturbation ˆbi + ˆb† +i is applied sequentially on +sites i = 1, 2, . . . , 6. At the end of the six steps, a config- +uration with n += 2 is obtained as shown in (g). Note +that the configuration of the atoms on the hexagon is +flipped in (g) as compared to (a) thereby producing the +effect of a ring exchange. +Other sixth-order processes +where the perturbation is not applied sequentially also +contribute to Eq. (9), but are not shown here. (h) Ring +exchange process which appears in the effective Hamilto- +nian Eq. (9). A flippable configuration is mapped to the +complimentary flippable configuration. +dimensionless number obtained by summing over virtual +processes and is plotted as a function of ρ in Fig. 5 . We +note that, when ρ is an integer, the value of γ(ρ) is 63/16 +and is the same as the one appearing in Refs. [68, 69]. +Although the effective Hamiltonian was derived here as- +suming n += 2, the effective Hamiltonian we obtain for +n += 1, 3 is also given by Eq. (9). +In terms of dimers on the diamond lattice, the effective +Hamiltonian Eq. (9) corresponds to a kinetic energy of +the dimers. It is well known that dimer models can be +made exactly solvable by adding a potential energy VRK +for the dimers and tuning to a special point VRK = Jring +called the Rokhsar-Kivelson (RK) point [65]. The Hamil- +Figure 4: Shaded in red are the four nonequivalent hexag- +onal plaquettes of the pyrochlore lattice. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0 +2 +4 +6 +8 +10 +12 +ρ +γ(ρ) +Figure 5: Plot showing the variation of γ(ρ) (which is +the proportionality constant in Jring(ρ) = γ(ρ)Ω6/V 5) as +a function of ρ. For ρ = 0.5, 1.5, 2.5, and 3.5, the energy +gap between the low-energy and the high-energy sectors +closes and γ(ρ) diverges. +tonian with such a potential energy term takes the form +ˆHdimer = − Jring(ρ) +� +� +�� +� � +�� + H.c. +(10) ++ VRK +� +� +�� +� � +�� + +�� +� � +�� . +The Rydberg system we are interested in [Eq. (9)] is +obtained from Eq. (10) by setting VRK = 0. +B. +U(1) quantum spin liquid—relation to +Hermele-Fisher-Balents [48] +The Hamiltonian in Eq. (10) was also derived by Her- +mele, Fisher and Balents in Ref. [48] starting from the +Heisenberg model on the pyrochlore lattice and taking +the easy-axis limit where the Hamiltonian is +ˆHeasy-axis = 1 +2 +� +⟨i,j⟩ +î +Jz ˆSz +i ˆSz +j + J⊥ +Ä ˆSx +i ˆSx +j + ˆSy +i ˆSy +j +äó +, +(11) +where Jz ≫ J⊥ > 0. When J⊥ = 0, the ground state is +exponentially degenerate with Sz = 0 on each tetrahe- +dron, which is equivalent to n += 2. The J⊥ term was + +6 +treated as a perturbation over the Jz term, and at third +order, a ring exchange term identical to Eq. (9) was ob- +tained. Written in terms of the spins, the ring-exchange +term is +ˆHeff = −Jring +� +� +ˆS+ +1 ˆS− +2 ˆS+ +3 ˆS− +4 ˆS+ +5 ˆS− +6 + H.c., +(12) +where the sum is over hexagonal plaquettes of the py- +rochlore lattice. The RK potential term was added by +hand in Ref. [48] giving Eq. (10). +Hermele et al. then go to the quantum rotor variables +nrr′ ∈ Z and θrr′ ∈ [−π, π), which live on the links rr′ of +the diamond lattice (equivalently, sites of the pyrochlore +lattice) and satisfy the canonical commutation relations +[ˆnrr′, ˆθrr′] = i: +ˆSz → ˆn − 1 +2, +ˆS± → e±iˆθ. +(13) +The constraint n = 0 or 1 is imposed by adding a term +to the Hamiltonian that energetically penalizes states vi- +olating this constraint: +ˆHeff =U +2 +� +⟨r,r′⟩ +Å +ˆnrr′ − 1 +2 +ã2 +(14) +− 2Jring +� +� +cos +Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 +ä +, +where the first sum is over all the links of the diamond lat- +tice and the second one is over the hexagonal plaquettes +of the pyrochlore lattice. In the limit U → ∞, Eq. (14) +reduces to the effective Hamiltonian Eq. (12). +The local constraint, Sz +r = 0 for each tetrahedron, +gives a gauge structure to the effective Hamiltonian +where the gauge transformations are generated by ˆSz +r. +The presence of this local symmetry motivated Hermele +et al. to write Eq. (14) as a lattice U(1) gauge theory. +The electric field and the vector potential were defined +as +ˆerr′ = ± +Å +ˆnrr′ − 1 +2 +ã +, +ˆarr′ = ±ˆθrr′. +(15) +The positive (negative) sign is chosen if r belongs to A +(B) sublattice of the diamond lattice. The Hamiltonian +written in terms of the electric field and the vector po- +tential takes the form of a compact U(1) lattice gauge +theory [46, 70]: +ˆHeff = U +2 +� +⟨r,r′⟩ +ˆe2 +rr′ − 2Jring +� +� +cos ((curl ˆa)�) , +(16) +where the second summation is over hexagonal plaquettes +of the diamond lattice and +(curl ˆa)� = +� +r,r′∈� +ˆarr′, +(17) +where � +r,r′∈� is a sum along the directed bonds of a +hexagonal plaquette of the diamond lattice. The adjec- +tive “compact” refers to the vector potential ˆarr′ being +an angular variable. +There is an important difference +between the above gauge theory and the compact U(1) +gauge theory studied by Polyakov [46, 47, 71]—the gauge +theory obtained by Hermele et al. is an odd gauge theory, +i.e., electric fields are half-integers, err′ ∈ Z + 1/2, while +the gauge theory studied by Polyakov was an even gauge +theory, i.e., the electric fields were integers, err′ ∈ Z. Be- +cause of this difference, the phases of the two theories +differ. +The phases of a gauge theory can be characterized by +the interaction between two externally added opposite +electric charges separated by a distance R. +If the po- +tential between charges goes to zero (or increases as at +most log R in 2 + 1D) as R → ∞, then the gauge theory +is in the deconfined phase. +On the other hand, if the +potential increases linearly with R or faster, then these +opposite charges cannot be separated, and the gauge the- +ory is in the confined phase. In the limit U → ∞, the +even gauge theory was shown to be in the confined phase +in Refs. [46, 70], while the odd gauge theory can be in +either the confined phase or the deconfined phase [48]. +This can be understood intuitively as follows. +In the even gauge theory, in the limit U → ∞, the +electric fields are forced to be 0, err′ = 0, to minimize +the energy in the absence of any external charges. How- +ever, in the presence of two opposite external charges, the +Gauss’s law requires that the electric field can no longer +be zero everywhere. The spreading of the electric field is, +however, penalized by the term U +2 +� +⟨r,r′⟩ ˆe2 +rr′. This forces +the electric field to be nonzero only in a narrow tube join- +ing the two charges, leading to a linearly rising potential +between the two charges. Thus, in the limit U → ∞, +the even gauge theory is in a confined phase, and there +is no deconfined phase in this limit. This confinement of +charges has been shown in Refs. [46, 47, 70, 72]. +On the other hand, in an odd gauge theory, in the +limit U → ∞, the electric field can take two values, +err′ = ±1/2. +This results in frustration, i.e., allows +for many configurations of the electric field, so that the +ground state in this limit is non-trivial. When two exter- +nal charges are introduced, the electric field is not neces- +sarily confined in a string between the charges, but can +spread in space similar to the familiar Coulomb-law field +lines of a non-compact U(1) gauge theory. This suggests +that it is possible for the odd gauge theory to be in the +deconfined phase even in the U → ∞ limit. In fact, the +odd gauge theory on the pyrochlore lattice (14) is indeed +in the deconfined phase in the U → ∞ limit [50, 51, 63]. +Hermele et al. have shown that the dimer model with +the Hamiltonian Eq. (10) is described by the deconfined +phase of the underlying compact U(1) gauge theory close +to the RK point (for VRK smaller than Jring but close +to Jring). This phase is the U(1) quantum spin liquid. +It has three types of emergent excitations—gapless pho- +tons, gapped magnetic monopoles and gapped fractional- + +7 +Figure 6: For ρ ∈ (3/2, 5/2), corresponding to n += 2, +the system is in the U(1) spin liquid phase at VRK = +0 [51]. +On the other hand, for ρ ∈ (1/2, 3/2) and +ρ ∈ (5/2, 7/2), corresponding to n += 1 and 3, respec- +tively, the system is in an ordered phase at VRK = 0 [63]. +Note that for ρ = 1/2, 3/2, and 5/2, the perturbation +theory described in Sec. II A does not apply, and we can- +not comment on the phase of the system. +ized electric charges, also called as spinons. The spinons +are the tetrahedra which violate the constraint on n , +Eq. (8). +C. +Previous numerical work +In this section, we summarize some of the known work +on the dimer model with the Hamiltonian Eq. (10) and +on the nearest-neighbor transverse-field Ising model on +the pyrochlore lattice. +Using quantum Monte Carlo simulations, Refs. [51] +and [63] studied the range of VRK [see Eq. (10)] over +which the U(1) spin liquid exists. They found that the +spin liquid is present in the range −0.5Jring < VRK < +Jring for the dimer model with n += 2 and in the +range 0.77Jring < VRK < Jring for the dimer model with +n += 1. The dimer model with n += 3 is equivalent to +the one with n += 1 by a particle-hole transformation. +These numerical results are summarized in Fig. 6. +While a theory proposal to realize the RK potential ex- +ists [73], the RK potential is a six-body term for the py- +rochlore lattice and is difficult to engineer experimentally. +Thus, we focus on the case where VRK = 0. From Fig. 6, +we see that to obtain a spin liquid phase for VRK = 0, one +must have n += 2, which corresponds to 3/2 < ρ < 5/2. +In the cases n += 1 and 3, the system is in an ordered +state when VRK = 0. Hence, in conclusion, assuming the +long-range interactions ˆHLR can be ignored, we expect +that, in the limit Ω ≪ V , the Rydberg system will be in +a U(1) quantum spin liquid phase for 3/2 < ρ < 5/2. +When ρ = 2, or equivalently h = 0, and the long- +range interactions ˆHLR are ignored, the Hamiltonian of +the system ˆH0+ ˆHΩ in Eq. (6) is the transverse field Ising +model on the pyrochlore lattice. For Ω ≪ V , we know +from the perturbative analysis of Sec. II A and Ref. [63] +that the system is in the U(1) quantum spin liquid phase. +For large Ω/V , where perturbation theory cannot be ap- +plied, Ref. [62] found using quantum Monte Carlo cal- +culations that the U(1) spin liquid exists in the region +Ω < 0.55(5)V , while, for Ω > 0.55(5)V , the system is in +a transverse-field-polarized (TFP) phase, which extends +to Ω/V → ∞ where the ground state is polarized in the +x-direction. This transition was also studied in Ref. [74] +using perturbation theory, where a transition was found +at Ω ≈ 0.6V . +The effects of adding a third nearest-neighbor inter- +action, V3NN, to the dimer model were considered in +Ref. [75]. It was found that the quantum spin liquid tran- +sitioned into an ordered state (antiferromagnet [76]) at +V3NN ≈ Jring. Thus non-nearest-neighbor interactions +can destabilize the quantum spin liquid. +In fact, in a +2D model with neutral atoms located on the bonds of a +kagome lattice (same as the sites of a ruby lattice), a spin +liquid ground state was found if the interactions were +short-ranged using DMRG on cylinders [26, 37]. How- +ever, with the full long-range van der Waals interactions, +the spin liquid ceased to exist [26, 37]. Thus it is impor- +tant to consider the effects of long-range interactions. In +the following section, we will study the phase diagram +of Hamiltonian (6) in the presence of long-range interac- +tions, using approximate methods. +III. +PHASE DIAGRAM—APPROXIMATE +METHODS +The goal of this section is to study the ground state +phase diagram of Hamiltonian (4) for δ = 3.46V (which +corresponds to ρ = 2) including long-range interactions +ˆHLR. +A. +Confinement-deconfinement transition—Monte +Carlo assisted perturbation theory +Consider the full Hamiltonian ˆH = ˆH0 + ˆHΩ + ˆHLR +from Eq. (6) in the case ρ = 2 [see Eq. (7)]: +ˆH0 =V +2 +� +r +Ñ +� +i∈ +r +ˆSz +i +é2 +, +ˆHΩ =Ω +� +i +ˆSx +i , and ˆHLR = V +2 +� +i̸=j +′ Å +a +|xi − xj| +ã6 +ˆSz +i ˆSz +j . +(18) +The long-range interaction ˆHLR splits the exponential +degeneracy of the ice manifold, and selects one config- +uration diagonal in the ˆSz basis as the ground state of +ˆH0 + ˆHLR, which we call the “ordered state”. On the +other hand, ˆHΩ prefers superpositions of ice rule obey- +ing states, the U(1) quantum spin liquid (QSL) being +one such superposition. Further, we also note that quan- +tum fluctuations around the “ordered state” due to ˆHΩ +may also lead to a change in its energy relative to the + +3 +n2 +n1 +n8 +QSL. It is this competition between kinetic energy and +long-range interactions that we will study in this section. +We first show that the ground state in the classical +limit Ω = 0 is the zero-momentum state satisfying the +ice rule which we call the “ice ferromagnet”. We assume +that, as one increases Ω, there is no phase transition to a +different ordered state before the putative transition to a +QSL. In order to determine whether a QSL phase exists +and, if yes, at what Ω the transition to the QSL occurs, +one needs to compare the energies of ansatz wavefunc- +tions of the QSL and the ordered state. When Ω ̸= 0, +such wavefunctions would necessarily involve configura- +tions that violate the ice rule. We incorporate the effect +of nonzero Ω on the wavefunction using perturbation the- +ory. Our strategy is as follows. We treat ˆH1 ≡ ˆHΩ+ ˆHLR, +i.e., both the laser driving term and the long-range inter- +actions, as a perturbation to ˆH0 (unlike Sec. II A, where +we dropped ˆHLR). We perturbatively find an effective +Hamiltonian ˆHeff acting on the low-energy ice manifold. +We then compare the expectation value of ˆHeff in candi- +date wavefunctions that live entirely in this low-energy +space. Since a QSL wavefunction is a linear superposi- +tion of exponentially (in system size) many ice rule obey- +ing states, we calculate +¨ ˆHeff +∂ +numerically using classical +Monte Carlo sampling. +1. +Expression for ˆHeff +We perform a Schrieffer-Wolff transformation +ˆ˜H = ˆUS ˆH ˆU † +S = ˆUS +Ä ˆH0 + ˆHΩ + ˆHLR +ä ˆU † +S, +(19) +for a unitary ˆUS = e ˆS, where ˆS is an anti-hermitian oper- +ator chosen to make ˆ˜H block-diagonal in the (degenerate) +eigenbasis of ˆH0, i.e., +ˆ˜H = ˆP ˆ˜H ˆP + (1 − ˆP) ˆ˜H(1 − ˆP), +(20) +where ˆP projects onto the ice manifold. In the remainder +of this paper, we will restrict ourselves to the low-energy +sector and therefore only consider the ˆHeff ≡ ˆP ˆ˜H ˆP term +above. We calculate ˆHeff perturbatively in ˆH1 = ˆHΩ + +ˆHLR (see Appendix B of Ref. [77] for general expressions +of ˆHeff). +As we saw in Sec. II A, if we consider only +ˆHΩ as the perturbation, then the first non-trivial term +appearing in ˆHeff is −Jring +� +� +�� +� � +�� + H.c., where +Jring = 63 +16 +Ω6 +V 5 + Θ +Å Ω8 +V 7 +ã +. +(21) +Since we are performing perturbation theory in two op- +erators ˆHΩ and ˆHLR, each of them comes with its own +small parameter. Since the perturbative expansion will +involve polynomials in these two small parameters, there +is some arbitrariness in deciding how to compare the two +parameters relative to each other and thus in how to +truncate the expansion. In our calculation, we follow an +operational scheme of keeping all the terms up to sixth +order in ˆHΩ + ˆHLR. Following this truncation scheme, +we get (up to additive constants) +ˆHeff ≈ − Jring +� +� +�� +� � +�� + H.c. ++ +Å +1 − Ω2 +V 2 − 61 +18 +Ω4 +V 4 +ã +ˆHLR +− Ω2 +V +Ä ˆW (2) +LR + ˆW (3) +LR + ˆW (4) +LR +ä +− Ω4 +V 3 +Å152 +27 +ˆW (2) +LR − ˆL(2) +LR + ˆ +M (2) +LR +ã +, +(22) +where +ˆW (2) +LR ≡ 1 +4 +� +j +� +k1̸=j +k2̸=j +vj,k1vj,k2 ˆSz +k1 ˆSz +k2, +(23) +ˆL(2) +LR ≡ 109 +432 +� +j1̸=k1 +j2̸=k2 +δ⟨j1,j2⟩vj1,k1vj2,k2 ˆSz +k1 ˆSz +k2, +(24) +ˆ +M (2) +LR ≡ 20 +27 +� +j1̸=k1 +j2̸=k2 +δ⟨j1,j2⟩vj1,k1vj2,k2 ˆSz +j1 ˆSz +k1 ˆSz +j2 ˆSz +k2, +(25) +ˆW (3) +LR ≡ 1 +2 +� +j +� +k1̸=j +k2̸=j +k3̸=j +vj,k1vj,k2vj,k3 ˆSz +k1 ˆSz +k2 ˆSz +k3 ˆSz +j , +(26) +ˆW (4) +LR ≡ 1 +4 +� +j +� +k1̸=j +k2̸=j +k3̸=j +k4̸=j +vj,k1vj,k2vj,k3vj,k4 ˆSz +k1 ˆSz +k2 ˆSz +k3 ˆSz +k4, +(27) +and vi,j ≡ +� +a6 +|xi−xj|6 if xi, xj are not nearest neighbors, +0 +otherwise. +(28) +In the above equations, δ⟨i,j⟩ enforces i and j to be near- +est neighbors. +The expectation value of the Hamiltonian (6) in a given +state |Ψ⟩ is +⟨Ψ| ˆH|Ψ⟩ = +Ä +⟨Ψ| ˆU † +S +ä Ä ˆUS ˆH ˆU † +S +ä Ä ˆUS |Ψ⟩ +ä +. +(29) +Suppose ˆUS |Ψ⟩ (i.e., |Ψ⟩ transformed by the Schrieffer- +Wolff transformation) lies entirely in the ice manifold, +then using Eq. (19), we get +⟨Ψ| ˆH|Ψ⟩ = +Ä +⟨Ψ| ˆU † +S +ä ˆHeff +Ä ˆUS |Ψ⟩ +ä +. +(30) +For the ground state, |Ψg⟩ of the full Hamiltonian ˆH, +ˆUS |Ψg⟩ lies entirely in the ice manifold. Thus, we pick +an ansatz wavefunction for ˆUS |Ψ⟩ that also lies entirely +in the ice manifold and compute the expectation value of +ˆHeff in our ansatz state to get the energy. Before describ- +ing our ansatz states in Sec. III A 3, we first consider the +limit Ω = 0 in the next section. + +9 +2. +Classical ground state of the long-range Hamiltonian +Here, we will find the ground state selected by long- +range interactions in the limit Ω = 0 where there are no +quantum fluctuations. The Hamiltonian is ˆHcl = ˆH0 + +ˆHLR. We find the ground state by going to the Fourier +space. Since the pyrochlore lattice is an FCC lattice with +a four-site basis, we use the notation ˆSz +r,µ for spins where +r is an FCC lattice vector and µ ∈ {0, 1, 2, 3} labels the +sites within the basis. The spin ˆSz +r,µ is physically located +at r+eµ/2 where eµ are the vectors joining a diamond A +site to a neighboring diamond B site. (See Fig. 1(b) for +the precise definition.) As we are considering the classical +limit in this section, we drop hats on quantities which +would otherwise be operators. The Fourier transform of +Sz +r,µ is +Sz +r,µ = +1 +√Nu.c. +� +k +eik·rSz +k,µ, +(31) +where Nu.c. is the number of FCC unit cells. Substituting +this in Hcl, we get +Hcl = +� +µ,ν,k +Vµν,kSz +k,µSz +−k,ν, +(32) +where k is a vector in the Brillouin zone of the FCC +lattice and Vµν,k is the Fourier transform of the van der +Waals potential: +Vµν,k = V +2 +� +r +eik·r +Å +a +|r + (eµ − eν)/2| +ã6 +. +(33) +Diagonalizing the matrix Vµν,k for each k gives +Hcl = +� +µ,k +εk,µ|S +′z +k,µ|2, +(34) +where S +′z +k,µ is related to Sz +k,ν through a multiplication by +a unitary matrix Uµν,k which diagonalizes Vµν,k: S +′z +k,µ = +� +ν Uµν,kSz +k,ν. Recall that Sz +r,µ is either +1/2 or −1/2. +This imposes the following constraint: +� +k,µ +|S +′z +k,µ|2 = +� +k,µ +|Sz +k,µ|2 = +� +r,µ +�Sz +r,µ +�2 = Nu.c.. +(35) +From Eq. (34), the energy can be interpreted as a +weighted sum of εk,µ with the corresponding weights be- +ing |S +′z +k,µ|2. Because of the constraint in Eq. (35), the +energy is minimized by having the full weight on the +smallest εk,µ and no weight on the rest of the εk,µ. This +holds provided that such a configuration of S +′z +k,µ in the +momentum space corresponds to some configuration in +the real space where Sz +r,µ are ±1/2. +Calculating the Fourier transform of the long-range po- +tential, Eq. (33), and its eigenvalues εk,µ, we find that the +Figure 7: An ice ferromagnet state. +It is an ice rule +obeying state (i.e., n += 2 on every tetrahedron) with +k = 0. All the up-pointing tetrahedra are copies of each +other. The same is true for the down-pointing tetrahedra. +There are six (4C2) such states, and together they make +up the ground subspace of ˆHcl. +minimum of εk,µ occurs for k = 0 and is triply degener- +ate. In particular, +Vµν,k=0 = +Öv1 v2 v2 v2 +v2 v1 v2 v2 +v2 v2 v1 v2 +v2 v2 v2 v1 +è +, +(36) +where v1 = 0.113V and v2 = 1.12V . Its eigenvalues are +ε0,0 = 3.46V and ε0,1 = ε0,2 = ε0,3 = −1.004V . The +unitary that diagonalizes the above matrix also relates +S +′z +0,µ to Sz +0,ν as +à +S +′z +0,0 +S +′z +0,1 +S +′z +0,2 +S +′z +0,3 +í += 1 +2 +à +1 +1 +1 +1 +1 +1 +−1 −1 +1 −1 +1 +−1 +1 −1 −1 +1 +í à +Sz +0,0 +Sz +0,1 +Sz +0,2 +Sz +0,3 +í +. +(37) +Since ε0,1, ε0,2 and ε0,3 are the minimum eigenvalues, the +energy is minimized by having all the weight distributed +between S +′z +0,1, S +′z +0,2 and S +′z +0,3 and no weight on the remain- +ing S +′z +k,µ, that is, S +′z +k̸=0,µ = 0 and S +′z +0,0 = 0. There indeed +exist states satisfying these two conditions. The first con- +dition, S +′z +k̸=0,µ = 0, implies that the ground state is a +k = 0 state, while the second condition, S +′z +0,0 = 0, im- +plies that the ground state satisfies the ice rule (so that +the total spin, which is S +′z +0,0 is 0), see Eq. (37). There +are six such states, and we refer to them as the “ice fer- +romagnet” or “ice FM” states. One of these is shown in +Fig. 7. +3. +Ansatz wavefunctions for the ordered state and for the +quantum spin liquid +We now assume that, as one increases Ω starting from +Ω = 0, the ground state remains adiabatically connected +to the ice ferromagnet derived in the previous section till + +10 +the point where it undergoes the putative phase transi- +tion to the QSL. Therefore, our ansatz for the ordered +state is +|Ψord⟩ = ˆU † +S |ΨIFM⟩ , +(38) +where |ΨIFM⟩, a product state in the ˆSz basis, is the +k = 0 ice ferromagnet defined Sec. III A 2. This configu- +ration is given by Sz +r,µ = 1 +2εµ (independent of r), where +(ε0, ε1, ε2, ε3) ≡ (1, 1, −1, −1). We note that there are +six such choices for εµ that satisfy the ice rule. We pick +one such choice, but our calculations are not sensitive +to which one we pick. +|ΨIFM⟩ lives entirely in the ice +manifold. Left-multiplication by ˆU † +S takes it back to the +original Hilbert space with ice rule violations. +Our ansatz wave function for the spin liquid state is +|ΨQSL⟩ = ˆU † +S |ΨRK⟩ , +(39) +where |ΨRK⟩ is a uniform superposition of all dimer cov- +erings [65] of the diamond lattice (with n += 2). |ΨRK⟩ +lives in the ice manifold. Like before, we left-multiply it +by ˆU † +S to take it back to the original Hilbert space. The +justification for our choice is the following. |ΨRK⟩ is the +ground state of the dimer model at the RK point [see +Eq. (10)]. When the RK potential is zero, |ΨRK⟩ has an +energy expectation value of −4Nu.c.Jringnflip, where nflip +is the average fraction of flippable hexagons in the RK +wavefunction. We find numerically that nflip = 0.1757 +(also calculated in Ref. [48]). Therefore, the energy of +|ΨRK⟩ is −0.7028JringNu.c. which is not too far from the +ground state energy of the dimer model (12) found in +Ref. [51] to be −0.756JringNu.c.. Even though |ΨRK⟩ has +slightly higher energy, it has the advantage of being sim- +pler to sample by classical Monte Carlo. This explains +our choice. +For comparison, we will also calculate the energy of +a different ordered state |Ψ′ +ord⟩ = ˆU † +S |ΨIAFM⟩ that we +call an ice antiferromagnet. Here |ΨIAFM⟩ is an ice rule +obeying state with ordering wave vector k = π (b1 + b2), +where b1, b2 and b3 are primitive reciprocal lattice vec- +tors of the FCC lattice satisfying ai ·bj = δij. This state +is known elsewhere in literature as the 2π(001) state (this +nomenclature uses an enlarged cubic unit cell of the FCC +lattice) [76, 78, 79]. +4. +Numerical results—energy expectation values and phase +diagram +We now describe our computation of the expectation +value of +ˆHeff [see Eq. (22)] in |ΨRK⟩, |ΨIFM⟩ and in +|ΨIAFM⟩. +While the expectation value in |ΨIFM⟩ and +|ΨIAFM⟩ can be computed straightforwardly, the expec- +tation value in |ΨRK⟩ requires classical Monte Carlo sam- +pling. We use a system with 8×8×8 unit cells (i.e., con- +taining 2048 pyrochlore sites) with periodic boundary +conditions in the a1, a2, and a3 directions. We restrict +our sampling to sectors in which the total electric flux +Figure 8: ⟨ ˆHeff⟩ in |ΨRK⟩, |ΨIFM⟩, and |ΨIAFM⟩ calcu- +lated by inserting the values in Table I in Eq. (22). +piercing through any 2D torus cross-section (as defined +in Sec. IV B of Ref. [48]) is 0. Our sampling is done us- +ing loop moves as described in Refs. [48, 78, 79] – in each +Monte Carlo run, we perform 512 × 500, 000 loop moves. +We calculate ¯nflip, HLR, W (2) +LR and L(2) +LR after every 512 +loop moves, i.e., we take 500,000 data points. We cal- +culate M (2) +LR, W (3) +LR, and W (4) +LR after every 512 × 10, 000 +loop moves, i.e. +we take 50 data points. +We repeat +this procedure for 9 independent runs in order to cal- +culate the uncertainties. Our results are summarized in +Table I. With these values at hand, we calculate the ex- +pectation value of ˆHeff using Eq. (22) in |ΨRK⟩, |ΨIFM⟩, +and |ΨIAFM⟩, and the result is plotted in Fig. 8. As we +turn on Ω, the transition point ΩC can be determined +within our approximation as the Ω for which the energy +of the ice ferromagnet becomes higher than that of the +RK wavefunction, as calculated using Eq. (22). We find +ΩC = 0.43927(1)V . +(40) +There is an important question on whether our use +of perturbation theory is justified. +First, we argue +that treating ˆHLR perturbatively is justified. +¶ ˆHLR +© +, +¶ ˆW (2) +LR, ˆL(2) +LR, ˆ +M (2) +LR +© +, +¶ ˆW (3) +LR +© +, and +¶ ˆW (4) +LR +© +are sets of op- +erators that are first, second, third, and fourth order re- +spectively in ˆHLR. As we can see from Table I, the ex- +pectation values of these operators in |Ψ⟩RK drops by an +order of magnitude each time one goes one order higher in +ˆHLR. Next, is perturbation theory in ˆHΩ justified, given +that our calculated ΩC is outside the Ω ≪ V regime? +We observe that the leading contribution to Jring that we +dropped, 33833 +2592 +(ΩC)8 +V 7 += 0.018V +[74], is smaller than the +one we kept, 63 +16 +(ΩC)6 +V 5 += 0.028V . If we had kept higher +order contributions to Jring, it would only decrease the +energy of the QSL relative to the ice ferromagnet and +ice antiferromagnet. Further, the energy of the QSL that +we present is a conservative estimate since we used the +RK wavefunction which has higher energy than the true +ground state of Hamiltonian (9). This gives us hope that + +11 +Operator +|ΨRK⟩ +|ΨIFM⟩ +|ΨIAFM⟩ +ˆR +0.70288(4)Nu.c. +0 +0 +ˆHLR +2.6037(1) × 10−2Nu.c. +−0.4002 × 10−2Nu.c. +3.8722 × 10−2Nu.c. +ˆ +W (2) +LR +1.11778(1) × 10−3Nu.c. +0.01642 × 10−3Nu.c. +1.4994 × 10−3Nu.c. +ˆL(2) +LR +−2.7467(3) × 10−4Nu.c. +−0.0829 × 10−4Nu.c. +−7.5662 × 10−4Nu.c. +ˆ +M (2) +LR +2.96(3) × 10−3Nu.c. +0.073 × 10−3Nu.c. +6.66 × 10−3Nu.c. +ˆ +W (3) +LR +5.25(4) × 10−5Nu.c. +−0.00665 × 10−5Nu.c. +5.81 × 10−5Nu.c. +ˆ +W (4) +LR +−3.57(2) × 10−6Nu.c. +−0.0309 × 10−6Nu.c. +−5.35 × 10−6Nu.c. +Table I: The expectation values of the operators in the left column in ansatz wavefunctions |ΨRK⟩, |ΨIFM⟩ and |ΨIAFM⟩ +respectively. The operator ˆR is defined as ˆR = � +� +�� +� � +�� + H.c. In the RK wavefunction, ⟨ΨRK| ˆR|ΨRK⟩ = +4¯nflipNu.c.. To calculate expectation values in |ΨRK⟩, we have used classical Monte Carlo sampling. +our result obtained using perturbation theory is qualita- +tively correct. +Within our approximation, for Ω < ΩC, the ground +state is an ice ferromagnet, an ordered state satisfying +the ice rule. +For Ω > ΩC but also close to ΩC, the +ground state is in the QSL phase, i.e., the deconfined +phase of a U(1) gauge theory. From the point of view of +the QSL, the ordered ice ferromagnet state is obtained +when monopole excitations of the spin liquid proliferate, +and the monopole-antimonopole string operator, to be +defined in Sec. IV B, Eq. (53), acquires an expectation +value. As a consequence of this, the fractional “electric +charges”, or spinons, get confined [46, 47]. The monopole +creation operator (see Sec. IV B and Ref. [48]) is diagonal +in the ˆSz basis, and acts in the sector that obeys the ice +rule. It is thus plausible that the confined phase is in- +deed the ice ferromagnet. While our calculation provides +microscopic intuition for this transition, we emphasize +that, to prove the existence of, locate and characterize +this transition accurately, one needs to do a more careful +quantum Monte Carlo calculation. +B. +Large Ω—Higgs transition +From the Hamiltonian in Eq. (6), it is clear that, in +the limit Ω ≫ V , the ground state is a transverse-field- +polarized (TFP) state, i.e., a product state of (|g⟩ − |r⟩)i +at each site i. Thus, as Ω is increased away from ΩC, the +system should eventually go through a phase transition +from the putative QSL phase into the TFP phase. From +the point of view of the QSL, this is a Higgs transition +because the operator ˆSx that acquires expectation value +in the TFP phase creates a pair of “electric”-charge ex- +citations in the spin liquid. The perturbation theory in +Ω/V that we performed in Sec. III A relies on the ability +to go to a basis where the Hilbert space decouples into ice +rule obeying and ice rule disobeying sectors separated by +an energy gap of V . But the ground state in the Ω ≫ V +limit (TFP) straddles both of these sectors. So we do not +expect perturbation theory in Ω/V to capture the phase +Figure 9: Approximate ground state phase diagram of +the Hamiltonian in Eq. (6). The ground state for Ω = 0 +was calculated exactly to be an ice ferromagnet (ice +FM) in Sec. III A 2. We assume that, as Ω is increased, +no phase transition occurs to a different ordered state. +The transition point from the ice ferromagnet (confined +phase) to the QSL (deconfined phase) at ΩC ≈ 0.44V +is obtained by comparing energies of ansatz wavefunc- +tions in the effective Hamiltonian obtained using pertur- +bation theory in ˆHΩ and ˆHLR. For the Higgs transition +to the TFP phase, we make an approximation by drop- +ping ˆHLR, in which case ΩH was calculated in Ref. [62] +to be 0.55(5). +transition into the TFP phase that contains the Ω → ∞ +ground state. Hence, we will present an indirect reason- +ing below. +In the Ω ≪ V limit, ˆHLR was important, +since it was the dominant term splitting the degeneracy +in the ice manifold. On the other hand, in the vicinity of +the putative Higgs transition, ˆHLR may not be as impor- +tant since the largest term in ˆHLR has magnitude V/27, +and as justified above using Table I, the effect of ˆHLR is +indeed perturbative. Therefore, we drop ˆHLR as a zeroth- +order approximation for calculating the Higgs transition +point. The resulting Hamiltonian is the transverse field +Ising model on the pyrochlore lattice. Refs. [62] and [74] +studied this model and found the transition point ΩH to +be at ΩH = 0.55(5)V and 0.6V respectively. This leads +us to expect that, in the window 0.44 < Ω < 0.55, the +ground state may be a QSL, leading us to sketch the +phase diagram shown in Fig. 9. Within our approxima- +tion, ΩC < ΩH and there is a window where the QSL + +12 +is the ground state. However, the introduction of ˆHLR +may result in a lowering of the energy of the TFP state +relative to the QSL. Calculating this effect and verifying +that this does not bring down ΩH far enough to destroy +the QSL phase requires a more careful calculation which +is beyond the scope of this work. +In the remainder of this section, we provide some intu- +ition for the Higgs transition by performing a gauge mean +field theory (gMFT) calculation introduced in Ref. [80]. +1. +Gauge mean field theory—Higgs transition +The main idea of this approach is to first recast the +microscopic Hamiltonian as an exact gauge theory by +introducing ancillary degrees of freedom followed by a +mean-field decoupling of the interactions. +This theory +involves bosonic charges hopping in the presence of a +fluctuating gauge field whose mean-field value is chosen +self-consistently. If this mean-field gauge-field configura- +tion is such that the hopping amplitudes of the bosonic +charges is 0, then the theory is in a confined phase. If +not, the theory is in the deconfined phase as long as the +bosons do not condense. If the bosonic charges condense, +then the theory is in a Higgs phase, which is adiabatically +connected to the TFP state. +Concretely, the construction is as follows. For r ∈ A, +where A is a sublattice of the diamond lattice, +ˆS+ +r→r+eµ = ˆΦ† +rˆs+ +r→r+eµ ˆΦr+eµ, +(41) +where +ˆS+ +r→r+eµ +≡ +ˆS+ +r+eµ/2 += +ˆS+ +r,µ (and similarly +ˆs+ +r→r+eµ ≡ ˆs+ +r+eµ/2 = ˆs+ +r,µ) lives on a bond of the dia- +mond lattice connecting sites r and r + eµ (recall that +centers of the bonds of the diamond lattice are sites of +the pyrochlore lattice). ˆsz is also a spin-1/2 operator and +has eigenvalues ±1/2. Here, ˆΦ† +r serves as a raising oper- +ator for ˆQ +r ≡ ηr(ˆn +r − 2), where ηr = 1 for r ∈ A and +ηr = −1 for r ∈ B. For convenience, we drop the sym- +bol +from now on. ˆQr and ˆΦ† +r satisfy the commutation +relation: +î ˆQr, ˆΦ† +r +ó += ˆΦ† +r. Note that ˆΦr is not a canonical +boson but a rotor satisfying +ˆΦ† +r ˆΦr = 1. +(42) +To recover the original spin Hilbert space, one imposes +the constraint that the total gauge charge at r is +ˆQr = ηr +� +µ +ˆsz +r+ηeµ/2. +(43) +Rewriting the Hamiltonian (6) in terms of the fictitious +variables, ˆQr, ˆΦr and ˆsr,µ we get +ˆH =V +2 +� +r∈A,B +ˆQ2 +r − Ω +2 +� +(r∈A),µ +ÄˆΦ† +rˆs+ +r→r+eµ ˆΦr+eµ + H.c. +ä ++ 1 +2 +� +r,r′∈A +� +µ,ν +Vµν(r − r′)ˆsz +rµˆsz +r′,ν, +(44) +where Vµν(r−r′) = V +Ä +a +r−r′+eµ/2−eν/2 +ä6 whenever (r, µ) +and (r′, ν) are distinct and are not nearest-neighbors. +Vµν(r − r′) is 0 otherwise. +Following Ref. [80], we perform the zeroth-order mean- +field decoupling: ˆΦ† ˆΦˆs → ˆΦ† ˆΦ ⟨ˆs⟩ + +¨ˆΦ† ˆΦ +∂ +ˆs − +¨ˆΦ† ˆΦ +∂ +⟨ˆs⟩ +and ˆsˆs → ˆs ⟨ˆs⟩ + ⟨ˆs⟩ˆs − ⟨ˆs⟩ ⟨ˆs⟩ (where ˆs could either be +ˆs+, ˆs−, or ˆsz). +Upon doing so, the Hamiltonian de- +couples into a Hamiltonian of bosons hopping on the +diamond lattice and a Hamiltonian of spins in an ex- +ternal field, which itself is set self-consistently by the +Green’s function of the bosons. Before solving the re- +sulting theory, one needs to enforce the constraints (42) +and (43) using Lagrange multipliers λr and vr, respec- +tively. Within the mean-field theory, it is assumed that +these Lagrange multipliers take a spatially homogeneous +value at the saddle point. We then find the minimum +value of ΩMF +H +such that, for any Ω > ΩMF +H , it is pos- +sible to self-consistently choose λ only by macroscopi- +cally occupying a boson mode. This ΩMF +H +marks the lo- +cation of the Bose-Einstein-condensation transition (or +Higgs transition within the mean-field theory). We find +ΩMF +H +≈ 0.7V . In Appendix A, we present more details +of this calculation. An artifact of this technique is that, +although we include long-range interactions in our cal- +culation, they do not play any role at the saddle point +near the Higgs transition. Therefore, the final steps and +result of our calculation are identical to the ones carried +out in [68]. +In Appendix A, we also point out a major limitation +of this technique in the small-Ω limit that may not have +been appreciated in previous literature on gauge mean +field theory. +C. +Comments on dynamical state preparation +So far, we have focused on the nature of the ground +state of Hamiltonian (6) as a function of Ω/V . +How- +ever, what is often experimentally relevant is the nature +of the state prepared by a ramping of parameters in a +finite amount of time. This issue was studied in Ref. [81] +for a Z2 gauge theory in the context of the experiment +in Ref. [26]. Since the main ideas of Ref. [81] are gen- +eral enough, here we will present an adaptation of the +conclusions of Ref. [81] to our setting. +The excitations of a U(1) QSL are gapless “photons”, +magnetic monopoles, and “electric charges” (spinons). +The transition of a QSL to an ice ferromagnet is driven +by the condensation of monopoles, while the transition to +the TFP phase is driven by the condensation of spinons. +The gapless “photons” are not directly involved in these +transitions. Also, a state with “photon” modes excited +on top of a QSL state is still in the deconfined phase of +the U(1) gauge theory. This allows us to ignore “pho- +tons” in this section. Since the confined phase, ice fer- +romagnet has an extensive number of monopoles, we +use the difference per unit cell between the energies of + +13 +Figure 10: A qualitative sketch of the energy scales (per +unit cell) in our problem. For Ω > ΩC, the ground state is +a U(1) QSL. Ice ferromagnet is the ordered state obtained +when monopoles proliferate, i.e., the ice ferromagnet has +an extensive number of monopoles. +We therefore use +the energy difference per unit cell between the QSL and +the ice ferromagnet at Ω = 0, obtained in Table I, as a +proxy for the monopole energy scale. This scale ∼ 0.03V +is much smaller than the spinon energy scale (“electric +charge”), which is ∼ V. +the QSL and ice ferromagnet states as a proxy for the +monopole energy scale. +At Ω = 0, this difference is +�¨ ˆHLR +∂ +QSL − +¨ ˆHLR +∂ +IFM +� +/Nu.c. ≈ 0.03V (see Table I), +which is much smaller than the spinon energy scale (see +Fig. 10 for a sketch). Suppose one starts with an initial +state (for a small ϵ ∼ Ω/V ) +��Ψ(t=0) +� = ⊗i (|g⟩i + ϵ |r⟩i) , +(45) +which is the ground state in the limit of large negative +δ/V and small Ω/V . +As shown in Sec. II, the clas- +sical ground state lies in the ice manifold when δ ∈ +(2.46V , 4.46V ). Now suppose that δ is ramped up from +its initial large negative value to a value in this range +such that the ramp is adiabatic with respect to the spinon +gap V , but is sudden with respect to the monopole scale +∼ 0.03V , while keeping Ω/V ≪ 1. Using arguments in +Ref. [81], this protocol will not prepare the ground state, +which, from Fig. 9, is an ice ferromagnet. Instead, it will +(approximately) project out violations of the ice rule (due +to adiabaticity with respect to the spinon scale) from the +initial state +��Ψ(t=0) +�. The resulting final state is +|Ψfinal⟩ ≈ ˆP {⊗i (|g⟩i + ϵ |r⟩i)} = |ΨRK⟩ , +(46) +where ˆP is the projector onto the ice manifold. The pro- +jected wavefunction is an equal-weight superposition of +all coverings, which is simply the RK wavefunction and +which lies in the QSL phase [48]. There is one catch to the +above argument—the spinon gap closes during the above +ramp. So it is impossible to be sudden with respect to the +monopole scale and yet be strictly adiabatic with respect +to the spinon gap throughout the ramp. For a short du- +ration (while the ramp is going through the spinon gap +closing), adiabaticity with respect to the spinon gap will +be violated. By the Kibble-Zurek mechanism, the result- +ing state is composed of finite-size puddles of QSL-like re- +gions with a nonzero density of spinons interspersed [81– +84]. +Thus, in summary, there are two different ways +in which one can prepare a U(1) QSL-like state in ex- +periment and study a confinement-deconfinement transi- +tion1. +1. Ω/V ≪ 1: +Perform a ramp of δ starting from +a large negative value and ending in the range +(2.46V , 4.46V ) for a fixed Ω/V +≪ 1 such that +the ramp is adiabatic with respect to V (spinon +gap) but sudden with respect to the monopole +scale (∼ 0.03V ). +Even though the ground state +is not a QSL for these parameters, this procedure +would create puddles of QSL-like regions by the +argument in Ref. [81]. +To see a deconfinement- +confinement transition, the ramp of δ should be +slowed down and, once it is adiabatic with respect +to the monopole gap, an ordered, i.e. confined state +will be prepared. +2. Adiabatic: Perform a ramp of δ starting from +a large negative value and ending in the range +(2.46V , 4.46V ) and a ramp in Ω starting from +Ω/V ≪ 1 and ending in a final value Ωf, such +that both ramps are adiabatic with respect to the +monopole scale always. The two ramps can be per- +formed simultaneously, or such that the ramp in δ +precedes the ramp in Ω. This would approximately +create the ground state of Hamiltonian (6). As the +final value Ωf goes through ΩC (ΩH), the nature +of the final state prepared this way goes through a +confinement-deconfinement (Higgs) transition. +Once a state is prepared by either of the above schemes, +one needs to devise measurements that can tell whether +the state is in the confined phase or in the deconfined +phase. We address this in the following section. +IV. +DIAGNOSIS OF THE QUANTUM SPIN +LIQUID +Access to wavefunction snapshots in the ˆSz basis, com- +bined with access to unitary evolution, allows one to use +the Rydberg-atom platform to measure non-local observ- +ables, a feature generally unavailable in traditional con- +densed matter systems. In this section, we describe some +1 We note that the confinement-deconfinement transition of U(1) +gauge theory in 3+1D is strictly speaking, a ground state tran- +sition [71, 85]. Therefore, in this paper, when we use the phrase +confinement-deconfinement transition, we mean signatures of +this transition in a finite-size state prepared in finite time. + +14 +Figure 11: Notation for the plaquette correlators. P and +P ′ are two hexagonal plaquettes of the pyrochlore lattice. +r, r′, r + uµ, and r′ + uν are the sites of the dual diamond +lattice. uµ and uν are vectors perpendicular to P and +P ′. +measurable correlators which can be used to observe the +signatures of a quantum spin liquid state. In this section, +we assume that the detuning is chosen such that ρ = 2. +A. +Plaquette-plaquette correlators +The plaquette operators are off-diagonal in the ˆSz ba- +sis and map one dimer covering to another. They are +important in distinguishing a coherent quantum super- +position from a classical admixture of states. We define +two plaquette operators ˆXP and ˆYP for a hexagonal pla- +quette P of the pyrochlore lattice as +ˆXP = +Ä ˆS+ +1 ˆS− +2 ˆS+ +3 ˆS− +4 ˆS+ +5 ˆS− +6 + H.c. +ä +, +ˆYP = −i ˆS+ +1 ˆS− +2 ˆS+ +3 ˆS− +4 ˆS+ +5 ˆS− +6 + H.c., +(47) +where 1, 2, . . . , 6 denote the sites around a plaquette P. +Either of the two correlators, ⟨ ˆXP ˆXP ′⟩ and ⟨ ˆYP ˆYP ′⟩, of +the plaquette operators on two plaquettes P and P ′ can +distinguish a QSL phase from other phases including a +classical spin ice (see Table II). We compare the two cor- +relators and provide protocols to measure them. We as- +sume throughout that the two plaquettes P and P ′ do +not have any sites in common. +Using the mapping between the spins and the effective +U(1) gauge theory from Eq. (13), we see that the opera- +tors ˆXP and ˆYP are equal to (twice) the cosine and the +sine of the magnetic field, respectively: +ˆXP = 2 cos +Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 +ä += 2 cos +Ĉbr,µ +ä +, +ˆYP = 2 sin +Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 +ä += 2 sin +Ĉbr,µ +ä +, +(48) +where r belongs to the dual diamond lattice [see +Fig. 1(c)], and µ ∈ {0, 1, 2, 3} labels the direction of mag- +netic field. ˆbr,µ is along uµ, which are vectors joining an A +site of the dual diamond lattice to its neighboring B sites. +These vectors are perpendicular to the plaquettes of the +pyrochlore lattice, see Fig. 11. The effective theory in the +deconfined phase (QSL) is Maxwell electromagnetism. +In the 3 + 1D continuum Maxwell electromagnetism, +the correlators of the Cartesian components of the mag- +netic field ˆbr,i for i ∈ {x, y, z} can be expressed as Gaus- +sian integrals which evaluate to [48] +⟨ˆb0,iˆbR,j⟩0 ∝ 1 +R4 +Å +2RiRj +R2 +− δij +ã +≡ CB +ij(R), +(49) +where ⟨·⟩0 is the expectation value with respect to the +Maxwell action. +The correlators of the magnetic field operators ˆbr,µ for +µ ∈ {0, 1, 2, 3} on the pyrochlore plaquettes in the decon- +fined phase are obtained from Eq. (49) by taking compo- +nents of the Cartesian magnetic field along the vectors +uµ. The result is +⟨ ˆXP ˆXP ′⟩ − 4 ∝ 1 +R8 +� +�� +k,l +(uµ)k(uν)l +Å +2RlRk +R2 +− δk,l +ã� +� +2 +, +⟨ ˆYP ˆYP ′⟩ ∝ 1 +R4 +� +�� +k,l +(uµ)k(uν)l +Å +2RlRk +R2 +− δk,l +ã� +� , +(50) +where the summation is over k, l ∈ {x, y, z}, R = r − r′, +and R is assumed to be large compared to the monopole +correlation length. The factors inside the square brack- +ets are geometric factors, which depend on the direction +of the vectors uµ, uν, and R, but are independent of the +distance R between the two plaquettes. Ref. [48] also sep- +arately studied the correlators precisely at the RK point +(which sits at the phase boundary between deconfined +and confined phases) where the effective field theory dif- +fers from the regular Maxwell theory. In the RK wave- +function, while the behavior of the plaquette correlators +differs from Eq. (50), it is still a power law with a slower +decay [48]. We note that, if the experimentally prepared +state is close to an RK wavefunction (see discussion in +Sec. III C), then this distinction will be important. +Outside of the deconfined phase (QSL) of the com- +pact U(1) gauge theory, Maxwell electromagnetism is no +longer the effective theory, and the behavior of the corre- +lators is different. The ice ferromagnet state is an ordered +state with the spins pointing in the z-direction, thus the +two plaquette correlators are expected to decay exponen- +tially with R. The TFP phase has ⟨ ˆS+⟩ ̸= 0, and hence +the plaquette X correlator approaches a nonzero constant +at large R, while the plaquette Y correlator is 0 (or decays +exponentially or faster in R). Lastly, in a classical spin +liquid, which consists of an incoherent mixture of expo- +nentially many dimer coverings, the plaquette correlators +decay at least exponentially (see Table II). +Since the plaquette correlators involve off-diagonal op- +erators, they cannot be read out directly from the snap- +shots of a Rydberg-atom array. However, we show that +they can be measured by evolving the system under a +modified Hamiltonian for a specific time duration fol- +lowed by measurement of a diagonal operator [26, 37]. +We describe the protocols to measure both plaquette X +and plaquette Y correlators in the sections below. + +15 +Correlator +Confined (Ice FM) +Deconfined (QSL) +Higgs (TFP) +Classical Spin Ice +⟨ ˆ +XP ˆ +XP ′⟩ − 4 +Exp. or faster decay +1/R8 +Nonzero const. +Exp. or faster decay +⟨ ˆYP ˆYP ′⟩ +Exp. or faster decay +1/R4 +Exp. or faster decay +Exp. or faster decay +¨ ˆ +M† ˆ +M(r1 → r2) +∂ +Nonzero const. +Exp. decay +Exp. decay +Exp. or faster decay +χE +C +Nonzero const.a +Exp. or faster decay +Nonzero const. +Exp. or faster decay +χM +C +Nonzero const. +Exp. or faster decay +Nonzero const.a +Exp. or faster decay +⟨ ˆSz +ri ˆSz +r′j⟩ +Nonzero const. +1/R4 +Exp. decay +1/R4 +a Distinguishing this non-zero constant from zero for χE +C in the confined phase (Ice FM) and for χM +C +in the Higgs phase (TFP) may be +practically challenging. +Table II: Behavior of various correlators. +ˆXP and ˆYP are plaquette operators defined in Eq. (47). +ˆ +M† ˆ +M(r1 → r2) +is a monopole string operator defined in Eq. (53). χE +C and χM +C +are BFFM order parameters defined in Eq. (60) and +Eq. (64), respectively. In this table, we have omitted the form factors multiplying 1/R4 and 1/R8 that are provided +in Eqs. (50) and (65). +1. +Plaquette X correlator +For a state |Ψ0⟩ completely within the ice manifold, +the expectation value of ˆXP ˆXP ′ is the same as that of a +product of ˆSx operators on the 12 sites of P and P ′, that +is, +⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩ = ⟨Ψ0| +12 +� +i=1 +2 ˆSx +i |Ψ0⟩, +(51) +where the sites i = 1, 2, . . . , 6 are on the plaquette P and +sites i = 7, 8, . . . , 12 are on the plaquette P ′. This can +be seen by writing 2 ˆSx +i = ˆS+ +i + ˆS− +i and noticing that the +only terms that preserve the ice rule are ring exchanges +over P and P ′. When the remaining terms act on a state +in the ice manifold, they either take the state outside of +the ice manifold or annihilate it. Thus the expectation +value of these remaining operators in |Ψ0⟩ is zero. For ex- +ample, ˆS+ +1 ˆS+ +2 ˆS+ +3 ˆS− +4 ˆS− +5 ˆS− +6 ˆS+ +7 ˆS− +8 ˆS+ +9 ˆS− +10 ˆS+ +11 ˆS− +12 acting on a +state in the ice manifold would either annihilate this state +or give a state that violates the ice rule on four of the +tetrahedra surrounding P. The protocol to measure the +plaquette X correlator is as follows. +We abruptly change the phase and the amplitude of the +Rabi frequency, so that the new Hamiltonian is ˆHY ≈ +ΩY +� +i ˆSy +i with ΩY ≫ V . +(Achieving ΩY ≫ V may +require working with atom spacings that are sufficiently +large and/or with Rydberg principal quantum numbers +that are sufficiently low, but not low enough to make +Rydberg lifetime a problem.) +It is assumed that this +change of the Hamiltonian is done sufficiently rapidly so +that the sudden approximation is valid and the state of +the system does not change. We evolve the system under +ˆHY for a time tY = π/(2ΩY ), which amounts to a π/2 +pulse about the y-axis, and then measure all atoms in +the {|g⟩ , |r⟩} basis. Thanks to Eq. (51), this allows us to +compute the plaquette X correlator. +Eq. (51) assumes that the state |Ψ0⟩ is in the ice man- +ifold. However, the ground state |Ψg⟩ of the system is +not completely in the ice manifold. The error introduced +because of assuming |Ψg⟩ to be in the ice manifold is +of sixth order in Ω/V , as we show in Appendix B. The +intuitive reason for this is that six factors of ˆHΩ are re- +quired to give a state that can have nonzero overlap with +�12 +i=1 2 ˆSx +i |Ψ0⟩. +We note that, for Ω/V ≫ 1 (i.e. in the TFP phase), the +experimentally measured quantity ⟨Ψg| �12 +i=1 2 ˆSx +i |Ψg⟩ +is +not +the +same +as +the +plaquette +X +correlator +⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, but their behaviors are nevertheless the +same, i.e., both approach a nonzero constant as the dis- +tance between the plaquettes increases. +The plaquette X correlator behaves as ⟨ ˆXP ˆXP ′⟩ − 4 ∝ +1/R8 (the angular dependence is not shown here) in the +QSL phase. Although it is a power law, the decay is very +rapid, and it might be practically difficult to distinguish +it from an exponential decay. This issue is less promi- +nent in the plaquette Y correlator, and we now provide +a protocol to measure it. +2. +Plaquette Y correlator +This protocol relies on the fact that, for a state |Ψ0⟩ +within the ice manifold, we have an identity similar to +Eq. (51): +⟨Ψ0| ˆYP ˆYP ′|Ψ0⟩ = ⟨Ψ0| +6 +� +i=1 +(2 ˆSx +2i−1)(2 ˆSy +2i)|Ψ0⟩. +(52) +Recall that sites 1, 2, . . . , 6 are on P, +while sites +7, 8, . . . , 12 are on P ′. This protocol is the same as the +protocol for measuring the plaquette X correlator, except +that now the π/2 pulses on sites 2i for i = 1, 2, . . . , 6 are +about the x-axis on the Bloch sphere while the π/2 pulses +on sites 2i − 1 for i = 1, 2, . . . , 6 are around the y-axis. +After applying these π/2 pulses, �12 +i=1 2 ˆSz +i is measured +by taking a snapshot of the array. Note that, for any +set of non-overlapping plaquettes, we can simultaneously +measure Y correlators between all pairs of plaquettes in +this set. +This protocol requires control over individual sites and +has an error of order (Ω/V )6 (the estimation of the er- + +16 +ror is similar to the one for the plaquette X correlator +done in Appendix B). The advantage of the plaquette Y +correlator over the plaquette X correlator is that the for- +mer decays as 1/R4, which is slower than the decay of +the plaquette X correlator and can be easier to observe +experimentally. +B. +Monopole-monopole correlator +In the deconfined phase, +monopoles are gapped. +Therefore, the expectation value of an (equal-time) op- +erator that creates a string with a monopole and an- +timonopole at its endpoints should decay exponentially +with the length of the string. +On the other hand, in +the confined phase, monopoles are condensed, and hence +the expectation value should approach a nonzero con- +stant as the length of the string increases. In the contin- +uum, the following operator inserts a string that creates +a monopole at r1 and an antimonopole at r2 [48]: +ˆ +M† ˆ +M(r1 → r2) ∼ ei +� +d3r′A(r′)·ˆe(r′). +(53) +Here A(r′) is a classical (non single-valued) vector poten- +tial such that the flux φΣ of B = ∇×A through a closed +surface Σ is +φΣ ≡ +� +Σ +B · dS = 2πqQΣ, +(54) +where QΣ = 1 when Σ encloses r1 and not r2, QΣ = −1 +when Σ encloses r2 and not r1, and QΣ = 0 otherwise. q +is an integer and denotes the “charge” of the monopole +string. For simplicity, we will set q = 1 in this section. +We clarify that B and φΣ are classical numbers and are +different from ˆb and ˆΦΣ which are operators. ˆb ≡ ∇׈a, +for gauge-field (operator) ˆa, and ˆΦΣ is defined as +ˆΦΣ ≡ +� +Σ +ˆb · dS = 2π ˆm, +(55) +where ˆm takes integer eigenvalues. +The form of the +monopole string operator is chosen so that it increases +the flux through Σ by 2πQΣ, i.e., +îˆΦΣ, ˆ +M† ˆ +M(r1 → r2) +ó += 2πQΣ ˆ +M† ˆ +M(r1 → r2). +(56) +We now adapt this operator to the Rydberg setting. +Consider the diamond lattice formed by the centers of +tetrahedra of the pyrochlore lattice, Fig. 1(b). +Unlike +the continuum, it is now important to specify that the +endpoints of the monopole string r1 and r2 belong to +the dual diamond lattice [see Fig. 1(c)], whose sites are +centers of “polyhedra” made of four puckered-hexagonal +“plaquettes” of the diamond lattice2, see Fig. 12(a). Let +2 In terms of the original pyrochlore lattice, the vertices of the +dual diamond lattice are centers of the truncated tetrahedra [see +Fig. 12(b)] which fill the voids between the tetrahedra. +Figure 12: (a) The “polyhedron” formed by four puck- +ered hexagons of the diamond lattice is shown in orange. +The centers of these “polyhedra” form the dual diamond +lattice. (b) The center of the “polyhedron” in (a) is also +the center of a truncated tetrahedron (shown in red) of +the pyrochlore lattice. +x ≡ r + eµ/2 be a site on the pyrochlore lattice, where +r is an A-site of the diamond lattice. Ax ≡ Ar,r+eµ is +the discrete version of A integrated (Fig. 1(b) shows the +vectors eµ). along the line pointing from the center of +an A tetrahedron (centred at r) to the B tetrahedron +(centred at r+eµ) such that the two tetrahedra touch at +x. +Ax is required to satisfy the discrete version of +Eq. (54), and hence depends on r1, r2, the “magnetic +field” configuration B and the gauge choice for Ax. For +the pyrochlore lattice, we have +ˆ +M† ˆ +M(r1 → r2) = ei � +x∈pyrochlore Ax(ˆnx−1/2). +(57) +This operator is purely diagonal in the ˆnx basis (i.e., in +the ˆSz-basis). So, experimentally, one can calculate this +phase for each snapshot and average over shots. +Theoretically, one expects +��� +¨ ˆ +M† ˆ +M(r1 → r2) +∂��� ∼ +® +e−|r2−r1|/λ, +deconfined phase, +constant, +confined phase, +(58) +where λ is a correlation length that depends on the +monopole gap and the “photon” velocity. +In Fig. 13, +we provide an example of one configuration of the classi- +cal numbers Ax that defines a monopole string operator. +Below, we comment on the freedom in choosing Ax. +1. +Choice of A +The classical numbers Ax should of course obey the +constraint that the flux of ∇×A through a closed surface +Σ is 2πQΣ, as mentioned above. However, one still has a +freedom in the choice of A in the following two respects: +1. Freedom in the arrangement of the field lines of +∇ × A. For example, they can be confined to a +thin tube connecting r1 and r2, or be spread out +according to Coulomb’s law, or be something in +between. Different such arrangements, due to their +different energy costs, would differ in sub-leading +corrections to the exponentially decaying behavior, + +17 +Figure 13: An example of the monopole string operator +ˆ +M† ˆ +M(r1 → r2) for which we provide Ax explicitly. In our +example, the string carries 2π flux through a tube with a width of 7 puckered hexagons of the diamond lattice. (a) A +schematic of the tube running along the z-direction. The diamond lattice (whose vertices are centers of tetrahedra +of the pyrochlore lattice) can be seen as ABC stacking of layers of “honeycomb” lattices made of chair-like puckered +hexagons. The tube consists of three types of layers shown in yellow, orange, and cyan. Each layer is made of 7 +puckered hexagons. To convey a sketch, we depict such a layer by a big hexagon with some thickness. (b) A side view +of the stack showing three of its layers, which then need to be repeated in the z direction to get the entire string. For +bonds x with arrows, the value of Ax is written next to the bond. For bonds x without arrows, Ax = 0. (c) Top view +of three of the layers of the stack. It can be seen from all three sub-figures (a)-(c) that the flux through any closed +surface Σ that completely encloses an integer number of layers, such that the bottom layer is included but not the +top, is 2π. However, if Σ partially encloses a layer, then ΦΣ is 0. This difficulty in defining arbitrary integer multiples +of 2π flux through a volume enclosed by a finite number of plaquettes has been observed before [48]. Therefore in +our construction, r1 and r2 have to be seen as being smeared across 7 points of the dual diamond lattice below the +bottom layer and above the top layer respectively, in order to be consistent with Eq. (54). +but the leading behavior would be unchanged. In +Fig. 13, we provide a choice of A, such that the +monopole string is localized to a thin tube. +2. For a fixed choice of field lines, we still have a +gauge choice for A. Consider a gauge transforma- +tion Ar,r+eµ → Ar,r+eµ + λr+eµ − λr, where λr is + +18 +an r-dependent real number. It results in +ˆ +M† ˆ +M(r1 → r2) → ˆ +M† ˆ +M(r1 → r2)e−i � +r λrηr(ˆn r−2), +(59) +where ηr = 1 for r ∈ A and ηr = −1 for r ∈ B. In +the Ω/V ≪ 1 limit, we have ˆn +r = 2, so the ex- +pectation value is invariant under the gauge trans- +formation. Away from this limit, a gauge transfor- +mation on Ar,r+eµ generically results in a physical +transformation on the monopole string operator. +However, as long as the external field h = 0 [h +is defined in Eq. (5)], by particle-hole symmetry, +we have �ˆn +r +� = 2, and �(ˆn +r − 2)2� is bounded. +Hence we do not expect the gauge transformation +on Ar,r+eµ to qualitatively change the behavior of +Eq. (58). +C. +BFFM order parameter +It is known that the confined and deconfined phases of +a gauge theory without matter fields can be distinguished +by the scaling of the Wilson loops WL = +¨ +ei +� +L Aµdxµ∂ +, +where Aµ is the gauge field and L is a closed loop. In the +deconfined phase, the Wilson loop follows the perimeter +law, WL ∝ e−Perimeter of L, while in the confined phase, +it follows the area law, WL ∝ e−Area of L. However, in +the presence of matter fields (which are generically al- +ways present), the Wilson loop follows the perimeter law +in both phases [86, 87], and it cannot be used to distin- +guish them. +The Bricmont-Fr¨olich-Fredenhagen-Marcu +(BFFM) order parameter is useful in such cases and has +a different behavior in the two phases [26, 37, 88–93]. The +BFFM order parameter, denoted here by χE +C , is defined +as +χE +C = +���ei � +C ˆarr′ + H.c.��� +»���ei � +L ˆarr′ + H.c.���, +(60) +where C is an open curve and L is the closed loop formed +by combining C with its mirror image about a plane that +intersects C only at its end points. This order parameter +detects long-range order in the “electric charge”-creation +string. In the Higgs phase, “electric charges” are con- +densed, and hence χE +C approaches a nonzero constant. +In the deconfined phase, the numerator in Eq. (60) (cal- +culated on an open curve) decays to zero faster than the +denominator (calculated on a closed loop, giving the Wil- +son loop), as the length of C is increased. Therefore, in +the deconfined phase, χE +C goes to 0 as the length of C is in- +creased. In the confined phase, it was argued in Ref. [89] +that while both the numerator and the denominator go +to zero as the length of C is increased, the limit of their +ratio approaches a constant. However distinguishing this +constant from zero in finite systems for finite length of C +may be difficult. Below we explain how to measure χE +C . +Using the mapping from spin operators to gauge fields, +Eqs. (13) and (15), we see that +ei � +C ˆarr′ ≃ ˆS+ +1 ˆS− +2 ˆS+ +3 · · · , +(61) +where the product of ˆS+ and ˆS− operators is over the +sites on the curve C. The denominator in χE +C has a similar +expression in terms of spin operators. +From the point of view of measurement, it is more +convenient to consider another quantity, which has the +same behavior as χE +C in the three phases, defined as: +˜χE +C ≡ +��� +¨� +i∈C ˆSx +i +∂��� +…��� +¨� +i∈L ˆSx +i +∂��� +, +(62) +In the transverse-field-polarized (Higgs) phase, ˜χE +C ap- +proaches a nonzero constant, just like χE +C . Now, we argue +that even in the QSL and confined phases, ˜χE +C and χE +C +have the same behavior. For a state |Ψ⟩ that dominantly +lies in the ice manifold, with corrections from outside the +ice manifold being of order Ω/V (such as the ground state +|Ψg⟩), we have +⟨Ψ| ˆS+ +1 ˆS− +2 ˆS+ +3 · · · + H.c.|Ψ⟩ += ⟨Ψ|(2 ˆSx +1 )(2 ˆSx +2 )(2 ˆSx +3 ) · · · |Ψ⟩ + Θ �(Ω/V )L� , +(63) +where L is the number of sites on C. The correction is +of order (Ω/V )L by an argument similar to the one used +to show that the error is sixth order in the protocol to +measure the plaquette X correlator (see Appendix B). +Thus, for small Ω/V , χE +C and ˜χE +C are equal up to order +(Ω/V )L. +The numerator and the denominator of ˜χE +C can be mea- +sured by applying π/2 pulses about the y-axis and mea- +suring, from the snapshots, products of ˆSz along C and +L. This procedure is similar to the protocol to measure +the plaquette X correlator, described in Sec. IV A 1. +The operator ei � +C ˆarr′ creates two opposite “electric +charges” at the endpoints of C. So a magnetic analogue +of χE +C can also be defined, where the numerator is the +expectation value of the operator that creates a monopole +and an antimonopole at the endpoints of C. Such an order +parameter, χM +C , detects long-range order in the monopole +string operator and is given by +χM +C = +� +ˆ +M† ˆ +M +� +r1 +C−→ r2 +�� +…� +ˆ +M† ˆ +M +� +r1 +L +−→ r1 +��, +(64) +where +ˆ +M† ˆ +M(r1 +C−→ r2) inserts a monopole-antimonopole +string along C and was defined in Eq. (57). In this section, +we use the notation where the path of the monopole- +antimonopole string is explicitly written in the argument +of ˆ +M† ˆ +M. Since this operator is diagonal in the ˆSz basis, +it can be measured straightforwardly from the snapshots +of the Rydberg-atom array. + +19 +In the confined phase, monopoles are condensed, so χM +C +should be a nonzero constant. In the deconfined phase, +by the argument of Ref. [89], the numerator of Eq. (64) +decays to zero faster than the denominator as the length +of C increases. Therefore, in the deconfined phase, χM +C +goes to zero as the length of C increases. In the Higgs +phase, even though there is no long-range order in the +monopole string and both the numerator and denomina- +tor go to zero, by the argument in Ref. [89], the ratio +(i.e. χM +C ) approaches a nonzero constant as the length +of C increases. But distinguishing this non-zero constant +from zero in finite-size numerics and experiment may be +challenging (similar to the situation for χE +C in the con- +fined phase). +The behavior of the BFFM order parameters in various +phases is summarized in Table II. +Before proceeding, we note that our protocols to mea- +sure the plaquette correlators and the BFFM order pa- +rameter χE +C work in the limit Ω/V ≪ 1, which is outside +the window in which the ground state of Hamiltonian (6) +is a QSL. However, we explained in Sec. III C that it is +possible to dynamically prepare finite puddles of QSL re- +gions even in the Ω/V ≪ 1 limit when the ground state +is not a QSL. Our protocols can then be applicable. +D. +Two-point ˆSz correlator +Consider two spins ˆSz +r,µ and ˆSz +r′,ν located on the sites +r + eµ/2 and r′ + eν/2, where r and r′ are the centers +of two up-pointing tetrahedra and µ, ν ∈ {0, 1, 2, 3} label +the sites of the tetrahedra (see Fig. 14). From the map- +ping of spins to gauge theory, Eqs. (13) and (15), it can +be seen that the two-point correlator of these two spins +⟨ ˆSz +r,µ ˆSz +r′,ν⟩ is the same as the two-point correlator of the +electric field. +The effective theory in the deconfined phase is the +Maxwell electromagnetism. +In the 3 + 1D continuum +Maxwell electromagnetism, the correlator of the Carte- +sian components of the electric field ˆer,i for i ∈ {x, y, z} +can be expressed as Gaussian integral which evaluate +to [48] +⟨ˆe0,iˆeR,j⟩0 ∝ 1 +R4 +Å +2RiRj +R2 +− δij +ã +, +(65) +where ⟨·⟩0 denotes expectation value with respect to the +Maxwell action. +The correlator of the electric field operators ˆer,µ for +µ ∈ {0, 1, 2, 3} along the links of the diamond lattice +are obtained from Eq. (65) by taking components of the +Cartesian electric field along the vectors eµ. The result +is +⟨ ˆSz +r,µ ˆSz +r′,ν⟩ = +� +k,l∈{x,y,z} +(eµ)k(eν)l⟨ˆer,kˆer′,l⟩0, +(66) +In the confined phase (ice ferromagnet), which is pri- +marily diagonal in the ˆSz basis, this correlator should +Figure 14: Notation for the two-point ˆSz correlator. r +and r′ are the positions of the centers of the tetrahe- +dra. +eµ are the vectors joining the center of an up- +pointing tetrahedron to the centers of its neighboring +down-pointing tetrahedra. +approach a constant for large R = |r − r′|. On the other +hand, in the TFP phase, which is primarily a product +state in the ˆSx basis, this correlator decays at least ex- +ponentially with R (see Table II). Since ˆSz +r,µ ˆSz +r′,ν is a +diagonal operator, its correlator can be measured exper- +imentally by capturing snapshots of the Rydberg-atom +array and averaging over them. +V. +DISCUSSION +In this work, we have presented a proposal to prepare +and detect the deconfined phase of the U(1) gauge the- +ory in 3+1 dimensions on a Rydberg atom simulator. We +first showed that laser-driven neutral atoms trapped in a +pyrochlore lattice using optical tweezer arrays naturally +realise a U(1) quantum spin liquid as the ground state +when the laser detuning lies in a specified window and +the interactions between Rydberg atoms are restricted +to nearest-neighbor. We then studied the effect of van +der Waals interactions beyond nearest-neighbor. In the +classical limit obtained by dropping the Rabi frequency +term, we showed that long-range interactions break the +degeneracy to select an ice ferromagnet as the ground +state. +We then studied the competition between the +long-ranged interactions that prefer an ordered state and +quantum fluctuations that prefer a QSL state, by cal- +culating the energies in ansatz wavefunctions using per- +turbation theory. We found that, for Rabi frequencies +greater than ΩC ≈ 0.44V , the ground state is a QSL +within our approximation. +When Ω is increased fur- +ther, we argued that the QSL goes into a transverse- +field-polarized state via a Higgs transition. +While we +have focused on the ground state, we also commented +on the effect of dynamical state preparation in deciding +the nature of the prepared state. We then provided ex- +perimental protocols for measuring the plaquette correla- +tors, Bricmont-Fr¨olich-Fredenhagen-Marcu order param- +eters, the monopole-monopole correlator, and the “elec- +tric field” correlator that can distinguish a QSL phase +from ordered phases. +Our ground state phase diagram is the result of an +approximate calculation. +While it is possible that the + +20 +Figure 15: A lattice made of corner-sharing tetrahedra +different from the pyrochlore lattice. The lattice consists +of ABAB . . . stacking of the blue (A) and the orange (B) +layers. A configuration satisfying n += 2 is shown here. +true phase diagram differs from what we found, we note +that there are other knobs one can tune to get a de- +sired phase diagram. Dressed states created from multi- +ple Rydberg and possibly ground levels can be used to +customize the interactions away from the isotropic 1/r6 +form we considered in this paper [94–99]. Designing a +dressing scheme compatible with the symmetries of the +pyrochlore lattice and exploring the resulting phase di- +agrams is an interesting direction for future work. We +also note that our proposal requires two Rydberg excita- +tions per tetrahedron, meaning that it lies outside of the +Rydberg-blockade regime and is therefore sensitive to im- +perfections and thermal fluctuations in nearest-neighbor +spacing. It will therefore be useful to extend our pro- +posal to the blockade regime of one excitation per tetra- +hedron. While previous numerical work on dimer models +have required a nonzero RK potential (6-body term) to +achieve this, it will be worthwhile to study if one can +engineer long-range Rydberg interactions that stabilize a +spin-liquid in the blockade regime. +One can also look for other lattices that could real- +ize a U(1) QSL ground state. One such possibility is a +lattice of corner-sharing tetrahedra where all up-pointing +tetrahedra (and separately all down-pointing tetrahedra) +form a hexagonal close-packed lattice shown in Fig. 15. +If only nearest-neighbor interactions are considered be- +tween atoms positioned on the sites of this lattice, then, +by perturbation theory in Ω/V for a particular range of +detunings, one gets ring exchange terms similar to the +ones obtained in Sec. II A, and the system maps onto a +dimer model. It is not known if this dimer model is in the +QSL phase when the RK potential is zero and long range +van der Waals interactions are included. Another open +problem is to construct lattices where a dimer model can +be realized within the blockade regime without the RK +potential. +Next, we note that, formally, a distinction between the +confined and deconfined phases exists only in the thermo- +dynamic limit. Experimentally, there are two finiteness +effects that can be important. +First, a realistic three +dimensional Rydberg array will likely have a relatively +small linear dimension. Some of the correlators presented +in Sec. IV require asymptotic behavior in distance to dis- +tinguish different phases. Second, as found in Ref. [81] +and mentioned in Sec. III C, a finite-time state prepara- +tion scheme would generically prepare puddles of spin- +liquid regions as opposed to an entire spin liquid. It is +therefore necessary to quantitatively study how the be- +havior of the correlators is modified under these condi- +tions. +We also note that, to translate field-theory observables +into microscopic variables, we relied on the perturbative +limit of small Ω/V . However in the phase diagram that +we found, the region where the spin liquid is a ground +state does not satisfy Ω/V ≪ 1. +Understanding how +the field-theory operators (e.g. plaquette, monopole, and +electric-field operators) get renormalized away from the +perturbative limit is important both from fundamental +and practical standpoints. +Our work is a proposal to prepare a gapless U(1) spin +liquid using unitary evolution. An interesting research di- +rection would be to come up with schemes that also use +projective measurements to expedite the state prepara- +tion along the lines of Refs. [44, 100]. One can also ex- +plore how other exotic phases of matter such as fractons +and 3+1D topological order can potentially be realized +on a Rydberg simulator. +ACKNOWLEDGMENTS +We thank Nikita Astrakhantsev, Peter Lunts, Nathan +Schine, Alexander Schuckert, Dayal Singh, and Ashvin +Vishwanath for discussions. 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III B 1 and performing +the mean-field decoupling, we get +ˆHMF = ˆHΦ + ˆHs + ˆHc, where +ˆHΦ =V +2 +� +r∈A,B +ˆQ2 +r − Ω +2 +� +(r∈A),µ +ÄˆΦ† +r ˆΦr+eµ +�ˆs+ +r,µ +� + H.c. +ä +, +ˆHs = − Ω +2 +� +(r∈A),µ +ĨˆΦ† +r ˆΦr+eµ +∂ +ˆs+ +r,µ + H.c. +ä ++ +� +(r∈A),µ +ˆsz +r,µ +� +(r′∈A),ν +�Vµν(r − r′) �ˆsz +r′,ν +�� , +ˆHc =Ω +2 +� +(r∈A),µ +ĨˆΦ† +r ˆΦr+eµ +∂ �ˆs+ +r,µ +� + H.c. +ä +− 1 +2 +� +(r∈A),µ +ˆsz +r,µ +� +(r′∈A),ν +�Vµν(r − r′) �ˆsz +r′,ν +�� . +(A1) +ˆHc +is +a +constant, +and +Vµν(r − r′) +was +de- +fined +in +Sec. +III B 1. +ˆHs +above +is +of +the +form +− � +(r∈A),µ +�hx +r,µˆsx +r,µ + hz +r,µˆsz +r,µ +�, where +hx +r,µ = Ω +¨ˆΦ† +r ˆΦr+eµ +∂ +, +hz +r,µ = − +� +(r′∈A),ν +�Vµν(r − r′) �ˆsz +r′,ν +�� , +(A2) +and +¨ˆΦ† +r ˆΦr+eµ +∂ +is calculated in the ground state of ˆHΦ, +which in turn depends on ⟨ˆs+⟩. (We have implicitly as- +sumed here that +¨ˆΦ† +r ˆΦr+eµ +∂ +is real, which we will show +can be assumed self-consistently.) This implies that, in +the ground state, +�ˆsi +r,µ +� = +hi +r,µ +2|hr,µ| for i = x, z. +(A3) +Our goal is to self-consistently minimize the ground-state +energy of the mean-field Hamiltonian subject to the con- +straints in Eqs. (42) and (43). We showed in Sec. III A 2 +that the ordered ground state at Ω = 0 has momentum + +24 +k = 0. +Also, the TFP state in the large-Ω limit is a +k = 0 state. So we start with a mean-field ansatz with +full translation symmetry (similar to Ref. [80]): +�s+ +r,µ +� = 1 +2 cos θ, +�sz +r,µ +� = 1 +2εµ sin θ, +(A4) +where εµ = 1, 1, −1, −1 for µ = 0, 1, 2, 3, respectively. To +solve the matter sector, it is convenient to deal with the +Lagrangian instead of the Hamiltonian. The imaginary- +time Lagrangian for the matter sector is +L = +1 +2V +� +r∈A,B +|(∂τ − ivr)Φr|2 +− Ω cos θ +4 +� +(r∈A),µ +(Φ∗ +rΦr+eµeiar,µ + c.c.) +− i +� +r∈A,B +� +ηrvr +�� +µ +sz +r+ηreµ/2 +� ++ ˜λr(|Φr|2 − 1) +� +, +(A5) +where the Lagrange multiplier ˜λr (which gets integrated +over) enforces the constraint |Φr|2 = 1. The Lagrange +multiplier vr enforces the constraint (43). To zeroth or- +der, we ignore the gauge fluctuation ar,µ. The matter +Lagrangian alone, despite being quadratic in the rotor +variables, is nevertheless interacting because a quadratic +term in rotor operators is nonlinear in terms of canonical +bosons (in other words, it is a cosine term in the phase +of the rotor.) In order to make progress, Ref. [80] as- +sumes that, at the saddle point, ˜λr takes on a spatially +uniform and purely imaginary value iλ, and also implic- +itly assumes that vr is 0 at the saddle point. Here, we +will follow suit while acknowledging that these approxi- +mations are uncontrolled. Making these simplifications, +we obtain +L = 1 +2V +� +r +|∂τΦr|2 − Ω cos θ +4 +� +(r∈A),µ +�Φ∗ +rΦr+eµ + c.c.� ++ λ +� +r +(|Φr|2 − 1). +(A6) +The constraints now simplify to +�Φ† +rΦr +� = 1, +(A7) +hx = Ω �Φ† +rΦr+eµ +� . +(A8) +Now, we have a quadratic Lagrangian, which we solve +by Fourier transformation. Our Fourier transformation +convention is (for α ∈ {A, B}) +Φr,α(τ) = T +� +ωn +� +k∈BZ +Φk,α(ωn)ei(k·r−ωnτ), +(A9) +where T is the temperature, ωn are Matsubara frequen- +cies and we eventually take the limit T → 0. Eq. (A6) +becomes +L = T +� +k,ωn +�Φ∗ +k,A(ωn) Φ∗ +k,B(ωn)� G−1 +k (ωn) +ÅΦk,A(ωn) +Φk,B(ωn) +ã +, +(A10) +where +G−1 +k (ωn) = +Ç +ω2 +n +2V + λ +− Ω cos θ +4 +fk +− Ω cos θ +4 +f ∗ +k +ω2 +n +2V + λ +å +. +(A11) +Here, +fk = 1 + e−ik1 + e−ik2 + e−ik3, +(A12) +where k ≡ k1b1 + k2b2 + k3b3, and b1, b2 and b3 are +reciprocal lattice vectors of the FCC lattice satisfying +ai · bj = δij. +Upon inverting the above matrix, we find that the +eigenvalues of Gk(ωn) are +2V +ω2n+(ω± +k (λ,θ)) +2 , where the dis- +persion of the two bosonic bands is +ω± +k (λ, θ) = +  +2V +Å +λ ± Ω cos θ +4 +|fk| +ã +. +(A13) +As long as the spinon dispersion is gapped, spinons will +not condense. From the dispersion above, we see that the +dispersion becomes gapless when λ = Ω cos θ. However, +as we will see below, for fixed θ and Ω, λ is determined +by the constraint in Eq. (A7). Therefore the condition +λ = Ω cos θ is met for a specific Ω = ΩMF +H , which we +will calculate below. Before that, will go through a few +intermediate steps. First, the matrix form of Gk(ωn) is +(assuming Ω > 0) +Gk(ωn) = V +Ñ +1 +ω2n+(ω+ +k )2 + +1 +ω2n+(ω− +k )2 +gk +� +1 +ω2n+(ω+ +k )2 − +1 +ω2n+(ω− +k )2 +� +g∗ +k +� +1 +ω2 +n+(ω+ +k )2 − +1 +ω2 +n+(ω− +k )2 +� +1 +ω2 +n+(ω+ +k )2 + +1 +ω2 +n+(ω− +k )2 +é +, +(A14) +where +gk = +® +− fk +|fk| +when 0 ≤ θ < π/2, +0 +when θ = π/2. +(A15) +With the Green’s function in hand, we are now ready + +25 +to impose the constraints, Eq. (A7) and Eq. (A8). First, +we calculate equal-time correlation functions of Φ (by +performing the Matsubara sum on the Green’s function). +Using these, the constraints in Eq. (A7) and Eq. (A8) +become, respectively, +F1(λ, θ) ≡ +V +2Nu.c. +� +k +Ç +1 +��ω+ +k +�� + +1 +��ω− +k +�� +å += 1, +(A16) +ΩF2(λ, θ) ≡ Ω +V +2Nu.c. +� +k +gk +Ç +1 +��ω− +k +�� − +1 +��ω+ +k +�� +å += hx. +(A17) +Next, by imposing Eq. (A3) with the help of Eq. (A2), +we get +hz = −B sin θ +2 +, where B = sin θ +2 +� +(r′∈A),ν +V0,ν(−r′)εν. +(A18) +For a given θ, Eq. (A16) determines λ. We see that there +are three self-consistent solutions for θ: +θ = +� +� +� +� +� +0, +π/2, +cos−1 Ä 2ΩF2(λ,θ) +B +ä +. +(A19) +Within gMFT (gauge mean field theory), these three so- +lutions correspond to a QSL, a “Coulomb ferromagnet” +(spin liquid with nonzero ice ferromagnetic order param- +eter), and an ice ferromagnet, respectively [80]. For a +fixed parameter Ω, the true solution depends on which of +the three solutions above has lower energy with respect +to the mean-field Hamiltonian (A1). Suppose that, for +large enough Ω, one is in the QSL phase, i.e., θ = 0 and +¨ˆΦr +∂ += 0. +Now, the bosons will condense when their +dispersion becomes gapless, i.e., λ = Ω. Using constraint +(A16), we find that this transition point is ΩH +MF ≈ 0.7V , +as also found in Ref. [68]. For Ω > ΩH +MF, the ground state +is in the TFP phase. +Having identified the Higgs transition point, we now +attempt to identify the confinement-deconfinement tran- +sition for low Ω, i.e., find Ω at which θ = 0 becomes the +lowest-energy saddle-point. Using Eq. (A1), we get the +following expression for the mean-field energy: +EMF = K − Nu.c. +Å +2ΩF2(λ, θ) cos θ + B +2 sin2 θ +ã +, (A20) +where K is the total kinetic energy of the bosons and can +be calculated to be +K = 1 +2 +� +k +�ω+ +k + ω− +k +� . +(A21) +In Fig. 16, we plot the energy EMF for θ = 0 (QSL) +and θ = π/2 (ice ferromagnet), and find a transition at +Ω ≈ 0.13V . (The third solution for θ becomes the lowest- +energy solution only in a minuscule window around Ω ≈ +Figure 16: The energy per unit cell (in units of V ) of +saddle points θ = 0 (QSL) and θ = π/2 (ice ferromagnet) +given by Eq. (A20) up to an overall additive constant +that is the same for θ = 0 and θ = π/2. We also plot +− Ω2 +4V 2 arising from trivial spin-flip pairs: this plot almost +overlaps with the energy of the θ = 0 state. +0.13V , so we ignore it.) However, we will now argue that +this result is misleading. +In gMFT, the energy reduction in the QSL phase with +respect to the ordered phase (ice ferromagnet) arises from +the minimization of kinetic energy of the bosonic charges +ˆΦr that are allowed to hop. When θ = 0, the hopping +coefficient is maximized, while, for θ = π/2, the hopping +coefficient is 0. However, microscopically, this hopping +corresponds to a single spin-flip. A pair of spin-flips at +the same site leads to a constant reduction of energy +coming from second order perturbation theory, given by +−Ω2Nu.c./V . It is constant in the sense that this reduc- +tion is obtained for any state including the QSL and the +ice ferromagnet. +The mean-field calculation, however, +unfairly assigns this reduction to the QSL but not to the +ordered state. In fact, in Fig. 16, we have also plotted +−Ω2/(4V ) (the factor of 1/4 can perhaps be attributed +to using spin-1/2 and classical spins at the same time). +As can be seen, this plot almost completely overlaps with +the energy of the QSL calculated within gMFT. So it is +clear that, within gMFT, the difference between the ener- +gies of the QSL and the confined phase is quadratic in Ω +to leading order even though we know from perturbation +theory that the leading order term should be proportional +to Ω6. Hence, gMFT cannot be used in the vicinity of +the confinement-deconfinement transition unless gauge- +fluctuations are properly taken into consideration. +Appendix B: Corrections in the measurement +protocol of the plaquette X correlator +In the protocol described in Sec. IV A 1, we equated +the expectation value of �12 +i=1(2 ˆSx +i ) with ˆXP ˆXP ′ in the +ground state (more generally, the state prepared in ex- + +26 +periment) assuming that the state is supported entirely +on the ice manifold. +Corrections arise because this is +satisfied only approximately. However, we show in this +appendix that these corrections are of sixth order in Ω/V . +Let |Ψg⟩ be the ground state of the system. This im- +plies that |Ψ0⟩ = ˆUS |Ψg⟩ is in the ice manifold, where ˆUS +is the unitary operator that implements the Schrieffer- +Wolff transformation (see Sec. III A 1). +First we show +that the first order correction in Ω/V is zero in the limit +Ω ≪ V . +The quantity measured in an experiment implementing +the protocol of Sec. IV A 1, assuming that the experiment +prepares the ground state, is +Cexp +X += ⟨Ψg|ei ˆ +HYtY +12 +� +i=1 +(2 ˆSz +i )e−i ˆ +HYtY|Ψg⟩ +≈ ⟨Ψ0| ˆUS ˆA ˆU † +S|Ψ0⟩, +where, ˆA = �12 +i=1 2 ˆSx +i . Now, +ˆUS ˆA ˆU † +S = ˆA + +î ˆS, ˆA +ó ++ 1 +2! +î ˆS, +î ˆS, ˆA +óó ++ 1 +3! +î ˆS, +î ˆS, +î ˆS, ˆA +óóó ++ · · · . +(B1) +Since |Ψ0⟩ is in the ice manifold, the expectation value +of the first term above is the required plaquette X cor- +relator, ⟨Ψ0| ˆA|Ψ0⟩ = ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, as explained in +Sec. IV A 1. +The expectation value of the remaining +terms is the error. +ˆS can be written as �∞ +i=1 ˆSi where +ˆSi is of order (Ω/V )i. First, we will show that the first +order error, E1 = ⟨Ψ0| +î ˆS1, ˆA +ó +|Ψ0⟩, is zero using the ex- +pressions for ˆS up to first order in Ω/V from Appendix +B of [40]: +ˆS1 = ˆP ˆHΩ ˆD − ˆD ˆHΩ ˆP, where ˆD = − +ˆ1 − ˆP +ˆH0 − E0 +. +(B2) +Here, ˆP is the projector into the ice manifold, E0 is the +energy of the ice manifold, and ˆH0 and ˆHΩ are given in +Eq. (6). Plugging in the expression for ˆS1 in E1 gives +E1 = ⟨ ˆP ˆHΩ ˆD ˆA − ˆA ˆP ˆHΩ ˆD − ˆD ˆHΩ ˆP ˆA + ˆA ˆD ˆHΩ ˆP⟩0 += ⟨Ψ0| ˆHΩ ˆD ˆA|Ψ0⟩ + ⟨Ψ0| ˆA ˆD ˆHΩ|Ψ0⟩. +(B3) +We have used ˆD |Ψ0⟩ = 0 and ˆP |Ψ0⟩ = |Ψ0⟩ above, and +the expectation values are taken in |Ψ0⟩. Since ˆA flips the +12 spins on P and P ′, while ˆHΩ flips only one spin, it is +not possible to go back to the ice manifold after applying +ˆHΩ ˆD ˆA on |Ψ0⟩. Thus ⟨ ˆHΩ ˆD ˆA⟩0 = 0, implying E1 = 0. +The corrections at order i involves expectation value +in |Ψ0⟩ of operators that consist of i factors of ˆHΩ and +one factor of ˆA. For example, the second-order error is +E2 = ⟨Ψ0| +î ˆS2, ˆA +ó ++ 1 +2! +î ˆS1, +î ˆS1, ˆA +óó +|Ψ0⟩ , +(B4) +where +ˆS2 =P ˆHΩ ˆD ˆHΩ ˆD − ˆD ˆHΩ ˆD ˆHΩ ˆP − ˆP ˆHΩ ˆP ˆHΩ ˆD2 ++ ˆD2 ˆHΩ ˆP ˆHΩ ˆP. +(B5) +It can be seen that, at order i = 1, 2, 3, 4, and 5, the +error will be zero because ˆA would flip 12 spins, while +i factors of ˆHΩ would flip only i spins. At sixth order, +however, the error can be nonzero if |Ψ0⟩ has support on +a configuration where one of the two plaquettes, say P ′, +is flippable. In such a situation, ˆA will map P ′ to the +complementary flippable configuration and will also flip +the spins on P. The six factors of ˆHΩ can bring P back +to the original configuration, giving a state within the ice +manifold which could have nonzero overlap with |Ψ0⟩. +In conclusion, we find that the experimentally mea- +sured correlator, Cexp +X , is the same as the theoretically +needed correlator, ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, with corrections of +order (Ω/V )6: +Cexp +X += ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩ + Θ �(Ω/V )6� . +(B6) + diff --git a/9tE3T4oBgHgl3EQfrArn/content/tmp_files/load_file.txt b/9tE3T4oBgHgl3EQfrArn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7c73a61d2cc5139371fd11e71e831eacf6a3fb7 --- /dev/null +++ b/9tE3T4oBgHgl3EQfrArn/content/tmp_files/load_file.txt @@ -0,0 +1,2053 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf,len=2052 +page_content='Quantum spin ice in three-dimensional Rydberg atom arrays Jeet Shah,1, 2 Gautam Nambiar,1 Alexey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Gorshkov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3 and Victor Galitski1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 4 1Joint Quantum Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' MD 20742,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' USA 2Condensed Matter Theory Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' MD 20742,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' USA 3Joint Center for Quantum Information and Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' NIST/University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' MD 20742,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' USA 4Center for Computational Quantum Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' NY 10010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' USA (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 2023) Quantum spin liquids are exotic phases of matter whose low-energy physics is described as the deconfined phase of an emergent gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' With recent theory proposals and an experiment showing preliminary signs of Z2 topological order [G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Semeghini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', Science 374, 1242 (2021)], Rydberg atom arrays have emerged as a promising platform to realize a quantum spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this work, we propose a way to realize a U(1) quantum spin liquid in three spatial dimensions, described by the deconfined phase of U(1) gauge theory in a pyrochlore lattice Rydberg atom array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We study the ground state phase diagram of the proposed Rydberg system as a function of experimentally relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Within our calculation, we find that by tuning the Rabi frequency, one can access both the confinement-deconfinement transition driven by a proliferation of “magnetic” monopoles and the Higgs transition driven by a proliferation of “electric” charges of the emergent gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We suggest experimental probes for distinguishing the deconfined phase from ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This work serves as a proposal to access a confinement-deconfinement transition in three spatial dimensions on a Rydberg-based quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' INTRODUCTION When the classical part of a many-body Hamiltonian is frustrated, quantum fluctuations can break the degener- acy in interesting ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' An exotic form of such breaking was pointed out by Anderson [1] where the ground state is a superposition of several almost-degenerate states, and the excitations are “fractional”[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Broadly, a common feature tying together such systems called quantum spin liquids is that, at low energies, they can be described as lying in a deconfined phase of an emergent gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The fractional excitations are the “charge”-like and “flux/monopole”-like excitations of this gauge the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When these fractional excitations get confined, they cease to be important for the low-energy physics, and the system becomes ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From this point of view, tran- sitions from a spin liquid to conventional ordered phases are understood as a confinement-deconfinement transi- tion, driven by a proliferation of “flux/monopole”-like ex- citations, or a Higgs transition, driven by a proliferation of “charge”-like excitations [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Gauge theories and their phase transitions are of fundamental importance in physics [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The prospect of this physics emerging in many-body systems provides an important motivation for studying quantum spin liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' They are also interesting due to their possible role in the physics of strongly corre- lated materials [11] and possible application in quantum computing [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Traditionally, the main search space for spin liquids has comprised of solid state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While consis- tent progress has been made [2, 14], conclusive evidence for spin liquids is still lacking in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' One reason is that the same feature that makes spin liq- uids interesting—being characterized by non-local order parameters—also makes them hard to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Mean- while, over the past decade, Rydberg atom arrays have emerged as a promising platform for engineering inter- acting Hamiltonians [15–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Rydberg states have large principal quantum number n (∼ 20 − 100), and the van der Waals interaction between them scales as n11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The strong tunable interactions, along with the ability to cus- tomize the lattice of atoms, locally control qubits, and take wavefunction snapshots, make Rydberg atom arrays a competitive platform to explore quantum many-body physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Following theory proposals [37, 38], promising signs of Z2 topological order have been observed exper- imentally on this platform [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This has sparked a lot of activity over the past two years in the general direc- tion of proposing ways to realize exotic states on quan- tum devices using analogue quantum simulation [39–42], digital quantum simulation [43], and projective measure- ments [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our work is a proposal for realizing a U(1) quan- tum spin liquid, described by the deconfined phase of a compact U(1) gauge theory on three-dimensional Ry- dberg atom arrays, with an eye towards accessing the confinement-deconfinement transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It was shown by Polyakov [46, 47] that compact U(1) gauge theory in 2+1 dimensions is always in the confined phase in the thermodynamic limit due to a proliferation of monopole events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore we turn to 3+1 dimensions, where Polyakov argued [47] for the existence of both deconfined and confined phases separated by a transition driven by monopole excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The deconfined phase consists of gapless “photons”, gapped “monopoles” and gapped “charge” excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the early 2000s, lattice models arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='04657v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='quant-gas] 11 Jan 2023 2 of spins [48] and dimers [49] on corner-sharing polyhe- dra were constructed that were strongly argued to realize this phase—a U(1) spin liquid, using perturbation the- ory, solvable limits [48] and later Quantum Monte Carlo simulations [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our work is based on a spin model with easy-axis antiferromagnetic interactions introduced by Hermele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] on the pyrochlore lattice consisting of corner-sharing tetrahedra (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The classical Ising limit of this model is the widely studied classical spin ice [52–56], which has a large residual entropy at low temperatures similar to water-ice [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This is be- cause the ground states form an exponentially degener- ate set of states obeying the “ice rule” (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The quantum model in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] has also been a subject of intense study in the context of pyrochlore materials like Yb2Ti2O7 and Er2Ti2O7 as potential quantum spin ice (another name for the U(1) spin liquid) candidates [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It was observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [59] that the Hamiltonian in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] can be viewed as that of hard-core bosons hopping on an optical lattice with nearest-neighbor re- pulsion, thus extending its relevance to the cold atom setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [60] studied a similar model of hard-core bosons hopping on a two-dimensional checkerboard lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [60], the atom’s internal state was largely the ground state, but a dressing with Rydberg states was used to engineer interactions between atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Later, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [61] showed that dimer models in two dimensions can be implemented on configurable Rydberg arrays—where the atoms themselves are stationary but can internally be either in a ground state or in a Rydberg state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this set- ting, the atoms are driven with a laser (or a pair of lasers making a two-photon transition) that is detuned from the ground to Rydberg transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Rydberg interactions and the detuning define a (frustrated) “classical” energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The laser driving induces quantum fluctua- tions controlled by the Rabi frequency, leading (pertur- batively) to dimer moves or ring exchange terms that are required to deconfine a gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The proposal [37] and experiment [26] mentioned above worked in the same setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our work is also based on this setting in which the atom array is configured in a 3D pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II, we explain our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We show that within a window of laser detunings, the classical land- scape is identical to the set of ice rule obeying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our Hamiltonian, when restricted to nearest-neighbor interactions, is equivalent to the transverse-field Ising model on the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the limit of small Rabi frequencies, it is perturbatively equivalent to the model in [48], which was argued to have a spin liquid ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Away from the perturbative limit, there is numerical evidence for a spin liquid phase [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, once we include the long-range 1/r6 interactions beyond nearest-neighbor, the classical landscape is no longer de- generate, and it is a priori unclear if the spin liquid sur- vives as the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We attempt to answer this in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III by comparing the energy of an ansatz wave function of the spin liquid with that of an ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Within our approximation, we find that by dialing up the Rabi frequency, for fixed detuning and interaction strength, one goes through a confinement-deconfinement transition from an ice rule obeying ferromagnetic state into a deconfined spin liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Then, by fur- ther increasing the Rabi frequency, one goes through a Higgs transition from the spin liquid to a transverse-field- polarized state (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While the analysis till this point focuses on the ground state, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III C, we com- ment on the role played by dynamical state-preparation in deciding the nature of the state prepared in experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV, we present correlation functions that distinguish the spin liquid from the confined phases, and provide experimental protocols for measuring them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Fi- nally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' V, we present general discussions and con- clusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' PROPOSAL TO REALIZE A U(1) QUANTUM SPIN LIQUID USING RYDBERG ATOMS In this section, we describe our proposal to realize a U(1) Quantum Spin Liquid (QSL) in Rydberg atom ar- rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Consider a 3D Rydberg array in which the atoms are positioned on the sites of the pyrochlore lattice [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Each of the atoms can either be in the ground state |g⟩ or in the Rydberg state |r⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the rotating wave approximation and in a rotating frame, the Hamiltonian is ˆHryd = − δ � i ˆni + V 2 � i̸=j Å a |xi − xj| ã6 ˆniˆnj + Ω 2 � i (ˆbi + ˆb† i), (1) where ˆbi = |gi⟩ ⟨ri|, ˆni = ˆb† iˆbi, Ω is the Rabi frequency, δ is the laser detuning, V is the nearest-neighbor van der Waals interaction strength, and a is the distance between two neighboring atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The summation � i̸=j is over distinct sites i and j of the pyrochlore lattice (each pair is being counted twice), and � i is over sites i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Below, we briefly describe the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The pyrochlore lattice is a face-centred cubic (FCC) lattice with a four-site basis formed by the four vertices of an up-pointing tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (Since each lattice site belongs to one up-pointing tetrahedron and one down- pointing tetrahedron, the down-pointing tetrahedra are formed automatically once we create the up-pointing tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') In Cartesian coordinates, the primitive vec- tors of the FCC lattice are a1 = √ 2a(0, 1, 1), a2 = √ 2a(1, 0, 1), a3 = √ 2a(1, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (2) The pyrochlore lattice sites are physically located at r + eµ/2 [and labeled (r, µ)], where r is an FCC lattice 3 Figure 1: (a) The pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' White circles denote atoms in the ground state, while black circles denote atoms in the Rydberg state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The configuration shown satisfies n = 2 on each tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The label x is used to denote the sites of the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (b) The diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is the bipartite lattice formed by the centers of the tetrahedra marked by green (A sublattice) and blue (B sublattice) dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' eµ for µ ∈ {0, 1, 2, 3} label the vectors joining an A site to its neighboring B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The label r is used to denote the sites of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (c) The red links are the edges of the lattice dual to the diamond lattice shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This lattice is also a diamond lattice, and we refer to it as the “dual diamond lattice” in this paper to distinguish it from the “diamond lattice” in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The sites of the dual diamond lattice are centers of the “polyhedra” formed by four puckered hexagons of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' uµ for µ ∈ {0, 1, 2, 3} label the vectors joining an A site to its neighboring B sites on the dual diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The label r [notice the difference in the font as compared to r in (b)] is used to denote the sites of the dual diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' vector, and the vectors eµ for µ ∈ {0, 1, 2, 3} are defined as [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(b)] e0 = a √ 2(1, 1, 1) = 1 4(a1 + a2 + a3), e1 = a √ 2(1, −1, −1), e2 = a √ 2(−1, 1, −1), e3 = a √ 2(−1, −1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (3) We map the two levels of the atoms to spins-1/2s: |g⟩ → |↓⟩, |r⟩ → |↑⟩, ˆni → ˆSz i + 1/2 and ˆbi + ˆb† i → 2 ˆSx i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The term ˆniˆnj therefore maps to an ˆSz i ˆSz j interaction in addition to a Zeeman term ˆSz i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Written in terms of spins, the Hamiltonian, up to an additive constant, is ˆHryd = − h � i ˆSz i + V 2 � i̸=j Å a |xi − xj| ã6 ˆSz i ˆSz j + Ω � i ˆSx i , (4) where h = δ − V 2 � i̸=0 Å a |xi − x0| ã6 , (5) and is independent of the choice of x0 for an infinite lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Evaluating this sum numerically for the pyrochlore lattice, we obtain h = δ − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is useful to sep- arate the total Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (4), into three parts, ˆHryd = ˆH0 + ˆHΩ + ˆHLR, where ˆH0 =V 2 � ⟨i,j⟩ ˆSz i ˆSz j − h � i ˆSz i , ˆHΩ =Ω � i ˆSx i , and ˆHLR = V 2 � i̸=j ′ Å a |xi − xj| ã6 ˆSz i ˆSz j , (6) where � ⟨i,j⟩ is over nearest-neighbor pairs and �′ i̸=j in ˆHLR is over the remaining pairs that are not nearest- neighbor (in both � and �′, each pair is counted twice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The coefficients of the second, third, and fourth nearest- neighbor interactions are V/27, V/64, and V/125, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since these are small in comparison to V , we will drop ˆHLR for the rest of this section because do- ing so allows us to connect to some previously known results [48, 51, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We will study the effect of the long- range van der Waals interaction ˆHLR in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the pyrochlore lattice is made of corner-sharing tetrahedra and since all edges in a tetrahedron are of equal length, we see that ˆH0 can be written up to an additive constant as (for convenience, in the expression below, we switch back to the hard-core boson notation) ˆH0 = V 2 � r �ˆn r − ρ�2 , (7) where the sum is over all tetrahedra, ρ = 1 2 �4 + h V � = 1 2 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='54 + δ V �, and ˆn r = � i∈ r ˆni denotes the total number of atoms in the excited state on a given tetrahe- dron r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Minimizing ˆH0 to obtain the classical ground 4 Figure 2: Mapping between Rydberg array configura- tions and dimer configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A Rydberg atom (black dot) is mapped to the presence of a dimer (orange bar), while a ground state atom (white dot) is mapped to the absence of a dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (a), (b), and (c) show example dimer configurations corresponding to n = 1, 2, and 3, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In each case, n many dimers touch the center of each tetrahedron (the centers of the tetrahedra form the diamond lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' state imposes a constraint on n for each tetrahedron depending on the value of ρ: n = � � � � � 0 if ρ < 1/2, floor �ρ + 1 2 � if 1/2 < ρ < 7/2, 4 if 7/2 < ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (8) The cases n = 0 and n = 4 are trivial, in that the classical ground state is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, in the re- maining cases, namely n = 1, 2, 3, the classical ground state manifold is degenerate with exponentially (in sys- tem size) many states in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the case n = 2, the num- ber of configurations satisfying this constraint is approxi- mately (3/2)Ntetrahedra (where Ntetrahedra is the number of tetrahedra) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This is based on an argument similar to the one given by Pauling to explain the residual entropy of water-ice at zero temperature [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From now onward, we will refer to the condition n = 2 as the “ice rule”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' An ice rule obeying configuration is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In these non-trivial cases, the configurations with fixed n can be mapped to configurations of dimers on the bipartite diamond lattice formed by the centers of tetra- hedra of the pyrochlore lattice [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(b)], with exactly n many dimers touching each diamond site (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The A and B sites of the diamond lattice are located at n and n+e0, respectively, where n is an FCC lattice vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For later use in this paper, we also show the lattice dual to this diamond lattice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(c) (also a diamond lattice, which we call the “dual diamond lattice”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' An atom in the Rydberg state on site i is mapped to a dimer on the corresponding link of the diamond lattice, while an atom in the ground state is mapped to no dimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Such dimer models have been studied extensively in both two and three dimensions [48, 65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the limit Ω ≪ V , ˆHΩ leads to quantum fluctua- tions that break the exponential degeneracy of the low- energy manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We will study this effect perturba- tively in the following section (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Classically, the energy gap between the degenerate ground state space and the lowest excited states corresponding to two tetra- hedra violating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (8) by either +1 or −1 is 2V × min ({ρ + 1/2}, 1 − {ρ + 1/2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Here, {x} ≡ x − floor(x) is the fractional part of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It should be noted that, in the borderline cases when ρ = m + 1/2 with m ∈ {0, 1, 2, 3}, the energy gap closes and our perturbative analysis can- not be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We assume going forward that ρ is away from these borderline values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Perturbation theory We work in the limit Ω ≪ V and treat ˆHΩ as a per- turbation over ˆH0, ignoring for now ˆHLR whose effects will be considered later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We calculate the ef- fective Hamiltonian within the ground state manifold of ˆH0 using the Schrieffer-Wolff formulation of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For simplicity, we present the calculation of the effective Hamiltonian only for n = 2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The only difference between these three cases will be the Hilbert space on which the Hamiltonian acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Calculating, at kth order in perturbation theory, the matrix element of the effective Hamiltonian between two states |n⟩ and |m⟩ lying in the degenerate manifold involves starting from |m⟩, applying the perturbation k times, and reaching the state |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since ˆHΩ changes the particle number by ±1, the corrections at all odd orders are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hence, we need to consider only the corrections at even orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Acting with Ω 2 (ˆbi + ˆb† i) on an ice rule obeying state creates two excited tetrahedra (whose common site is i), which violate the constraint n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, the only second-order process that takes us back to the ice manifold (the degenerate manifold of the ice rule obey- ing states) is the one in which two excited tetrahedra are created and annihilated, as illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since such processes are present for all the states of the ice manifold, they contribute only a constant energy shift and can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The same is true for the fourth- order processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Now, the pyrochlore lattice has hexago- nal plaquettes, some of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This allows for non-trivial processes to exist at sixth order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In fact, non-trivial ring exchange over hexagonal plaque- ttes of the pyrochlore lattice is obtained by the process shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3(a)–(g) (some sixth-order processes also result in a constant energy shift which we neglect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A flippable configuration—one in which atoms on a hexag- onal plaquette are alternately in the ground and Rydberg states—is mapped to the complementary flippable con- figuration by the ring exchange process as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, the effective Hamiltonian consists of ring exchange terms: ˆHeff = −Jring(ρ) � � �� � � �� + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', (9) where Jring(ρ) = γ(ρ)Ω6/V 5, the sum is over all hexag- onal plaquettes of the pyrochlore lattice, and γ(ρ) is a 5 Figure 3: (a) and (b) constitute a virtual process at sec- ond order in perturbation theory in Ω/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Starting from (a) which is a configuration that satisfies n = 2 on all sites, ˆb1 + ˆb† 1 is applied giving (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' To complete the sec- ond order process, ˆb1 + ˆb† 1 is applied to (b) giving back (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Tetrahedra for which n ̸= 2 are shaded in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Sub-figures (a)–(g) constitute a sixth-order process in the perturbation theory that contributes to the ring exchange term in the effective Hamiltonian, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Starting from (a), the perturbation ˆbi + ˆb† i is applied sequentially on sites i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' At the end of the six steps, a config- uration with n = 2 is obtained as shown in (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Note that the configuration of the atoms on the hexagon is flipped in (g) as compared to (a) thereby producing the effect of a ring exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Other sixth-order processes where the perturbation is not applied sequentially also contribute to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9), but are not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (h) Ring exchange process which appears in the effective Hamilto- nian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A flippable configuration is mapped to the complimentary flippable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' dimensionless number obtained by summing over virtual processes and is plotted as a function of ρ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We note that, when ρ is an integer, the value of γ(ρ) is 63/16 and is the same as the one appearing in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Although the effective Hamiltonian was derived here as- suming n = 2, the effective Hamiltonian we obtain for n = 1, 3 is also given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In terms of dimers on the diamond lattice, the effective Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9) corresponds to a kinetic energy of the dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is well known that dimer models can be made exactly solvable by adding a potential energy VRK for the dimers and tuning to a special point VRK = Jring called the Rokhsar-Kivelson (RK) point [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Hamil- Figure 4: Shaded in red are the four nonequivalent hexag- onal plaquettes of the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5 0 2 4 6 8 10 12 ρ γ(ρ) Figure 5: Plot showing the variation of γ(ρ) (which is the proportionality constant in Jring(ρ) = γ(ρ)Ω6/V 5) as a function of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5, the energy gap between the low-energy and the high-energy sectors closes and γ(ρ) diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' tonian with such a potential energy term takes the form ˆHdimer = − Jring(ρ) � � �� � � �� + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10) + VRK � � �� � � �� + �� � � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Rydberg system we are interested in [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9)] is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10) by setting VRK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' U(1) quantum spin liquid—relation to Hermele-Fisher-Balents [48] The Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10) was also derived by Her- mele, Fisher and Balents in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] starting from the Heisenberg model on the pyrochlore lattice and taking the easy-axis limit where the Hamiltonian is ˆHeasy-axis = 1 2 � ⟨i,j⟩ î Jz ˆSz i ˆSz j + J⊥ Ä ˆSx i ˆSx j + ˆSy i ˆSy j äó , (11) where Jz ≫ J⊥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When J⊥ = 0, the ground state is exponentially degenerate with Sz = 0 on each tetrahe- dron, which is equivalent to n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The J⊥ term was 6 treated as a perturbation over the Jz term, and at third order, a ring exchange term identical to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (9) was ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Written in terms of the spins, the ring-exchange term is ˆHeff = −Jring � � ˆS+ 1 ˆS− 2 ˆS+ 3 ˆS− 4 ˆS+ 5 ˆS− 6 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', (12) where the sum is over hexagonal plaquettes of the py- rochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The RK potential term was added by hand in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] giving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hermele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' then go to the quantum rotor variables nrr′ ∈ Z and θrr′ ∈ [−π, π), which live on the links rr′ of the diamond lattice (equivalently, sites of the pyrochlore lattice) and satisfy the canonical commutation relations [ˆnrr′, ˆθrr′] = i: ˆSz → ˆn − 1 2, ˆS± → e±iˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (13) The constraint n = 0 or 1 is imposed by adding a term to the Hamiltonian that energetically penalizes states vi- olating this constraint: ˆHeff =U 2 � ⟨r,r′⟩ Å ˆnrr′ − 1 2 ã2 (14) − 2Jring � � cos Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 ä , where the first sum is over all the links of the diamond lat- tice and the second one is over the hexagonal plaquettes of the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the limit U → ∞, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (14) reduces to the effective Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The local constraint, Sz r = 0 for each tetrahedron, gives a gauge structure to the effective Hamiltonian where the gauge transformations are generated by ˆSz r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The presence of this local symmetry motivated Hermele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (14) as a lattice U(1) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The electric field and the vector potential were defined as ˆerr′ = ± Å ˆnrr′ − 1 2 ã , ˆarr′ = ±ˆθrr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (15) The positive (negative) sign is chosen if r belongs to A (B) sublattice of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Hamiltonian written in terms of the electric field and the vector po- tential takes the form of a compact U(1) lattice gauge theory [46, 70]: ˆHeff = U 2 � ⟨r,r′⟩ ˆe2 rr′ − 2Jring � � cos ((curl ˆa)�) , (16) where the second summation is over hexagonal plaquettes of the diamond lattice and (curl ˆa)� = � r,r′∈� ˆarr′, (17) where � r,r′∈� is a sum along the directed bonds of a hexagonal plaquette of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The adjec- tive “compact” refers to the vector potential ˆarr′ being an angular variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' There is an important difference between the above gauge theory and the compact U(1) gauge theory studied by Polyakov [46, 47, 71]—the gauge theory obtained by Hermele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' is an odd gauge theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', electric fields are half-integers, err′ ∈ Z + 1/2, while the gauge theory studied by Polyakov was an even gauge theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', the electric fields were integers, err′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Be- cause of this difference, the phases of the two theories differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The phases of a gauge theory can be characterized by the interaction between two externally added opposite electric charges separated by a distance R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If the po- tential between charges goes to zero (or increases as at most log R in 2 + 1D) as R → ∞, then the gauge theory is in the deconfined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, if the potential increases linearly with R or faster, then these opposite charges cannot be separated, and the gauge the- ory is in the confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the limit U → ∞, the even gauge theory was shown to be in the confined phase in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [46, 70], while the odd gauge theory can be in either the confined phase or the deconfined phase [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This can be understood intuitively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the even gauge theory, in the limit U → ∞, the electric fields are forced to be 0, err′ = 0, to minimize the energy in the absence of any external charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' How- ever, in the presence of two opposite external charges, the Gauss’s law requires that the electric field can no longer be zero everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The spreading of the electric field is, however, penalized by the term U 2 � ⟨r,r′⟩ ˆe2 rr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This forces the electric field to be nonzero only in a narrow tube join- ing the two charges, leading to a linearly rising potential between the two charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, in the limit U → ∞, the even gauge theory is in a confined phase, and there is no deconfined phase in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This confinement of charges has been shown in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [46, 47, 70, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, in an odd gauge theory, in the limit U → ∞, the electric field can take two values, err′ = ±1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This results in frustration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', allows for many configurations of the electric field, so that the ground state in this limit is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When two exter- nal charges are introduced, the electric field is not neces- sarily confined in a string between the charges, but can spread in space similar to the familiar Coulomb-law field lines of a non-compact U(1) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This suggests that it is possible for the odd gauge theory to be in the deconfined phase even in the U → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In fact, the odd gauge theory on the pyrochlore lattice (14) is indeed in the deconfined phase in the U → ∞ limit [50, 51, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hermele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' have shown that the dimer model with the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10) is described by the deconfined phase of the underlying compact U(1) gauge theory close to the RK point (for VRK smaller than Jring but close to Jring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This phase is the U(1) quantum spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It has three types of emergent excitations—gapless pho- tons, gapped magnetic monopoles and gapped fractional- 7 Figure 6: For ρ ∈ (3/2, 5/2), corresponding to n = 2, the system is in the U(1) spin liquid phase at VRK = 0 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, for ρ ∈ (1/2, 3/2) and ρ ∈ (5/2, 7/2), corresponding to n = 1 and 3, respec- tively, the system is in an ordered phase at VRK = 0 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Note that for ρ = 1/2, 3/2, and 5/2, the perturbation theory described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A does not apply, and we can- not comment on the phase of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ized electric charges, also called as spinons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The spinons are the tetrahedra which violate the constraint on n , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Previous numerical work In this section, we summarize some of the known work on the dimer model with the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10) and on the nearest-neighbor transverse-field Ising model on the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using quantum Monte Carlo simulations, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [51] and [63] studied the range of VRK [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10)] over which the U(1) spin liquid exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' They found that the spin liquid is present in the range −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5Jring < VRK < Jring for the dimer model with n = 2 and in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='77Jring < VRK < Jring for the dimer model with n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The dimer model with n = 3 is equivalent to the one with n = 1 by a particle-hole transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' These numerical results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While a theory proposal to realize the RK potential ex- ists [73], the RK potential is a six-body term for the py- rochlore lattice and is difficult to engineer experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, we focus on the case where VRK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 6, we see that to obtain a spin liquid phase for VRK = 0, one must have n = 2, which corresponds to 3/2 < ρ < 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the cases n = 1 and 3, the system is in an ordered state when VRK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hence, in conclusion, assuming the long-range interactions ˆHLR can be ignored, we expect that, in the limit Ω ≪ V , the Rydberg system will be in a U(1) quantum spin liquid phase for 3/2 < ρ < 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When ρ = 2, or equivalently h = 0, and the long- range interactions ˆHLR are ignored, the Hamiltonian of the system ˆH0+ ˆHΩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (6) is the transverse field Ising model on the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For Ω ≪ V , we know from the perturbative analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [63] that the system is in the U(1) quantum spin liquid phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For large Ω/V , where perturbation theory cannot be ap- plied, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [62] found using quantum Monte Carlo cal- culations that the U(1) spin liquid exists in the region Ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='55(5)V , while, for Ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='55(5)V , the system is in a transverse-field-polarized (TFP) phase, which extends to Ω/V → ∞ where the ground state is polarized in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This transition was also studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [74] using perturbation theory, where a transition was found at Ω ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='6V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The effects of adding a third nearest-neighbor inter- action, V3NN, to the dimer model were considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It was found that the quantum spin liquid tran- sitioned into an ordered state (antiferromagnet [76]) at V3NN ≈ Jring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus non-nearest-neighbor interactions can destabilize the quantum spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In fact, in a 2D model with neutral atoms located on the bonds of a kagome lattice (same as the sites of a ruby lattice), a spin liquid ground state was found if the interactions were short-ranged using DMRG on cylinders [26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' How- ever, with the full long-range van der Waals interactions, the spin liquid ceased to exist [26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus it is impor- tant to consider the effects of long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the following section, we will study the phase diagram of Hamiltonian (6) in the presence of long-range interac- tions, using approximate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' PHASE DIAGRAM—APPROXIMATE METHODS The goal of this section is to study the ground state phase diagram of Hamiltonian (4) for δ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V (which corresponds to ρ = 2) including long-range interactions ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Confinement-deconfinement transition—Monte Carlo assisted perturbation theory Consider the full Hamiltonian ˆH = ˆH0 + ˆHΩ + ˆHLR from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (6) in the case ρ = 2 [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (7)]: ˆH0 =V 2 � r Ñ � i∈ r ˆSz i é2 , ˆHΩ =Ω � i ˆSx i , and ˆHLR = V 2 � i̸=j ′ Å a |xi − xj| ã6 ˆSz i ˆSz j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (18) The long-range interaction ˆHLR splits the exponential degeneracy of the ice manifold, and selects one config- uration diagonal in the ˆSz basis as the ground state of ˆH0 + ˆHLR, which we call the “ordered state”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, ˆHΩ prefers superpositions of ice rule obey- ing states, the U(1) quantum spin liquid (QSL) being one such superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Further, we also note that quan- tum fluctuations around the “ordered state” due to ˆHΩ may also lead to a change in its energy relative to the 3 n2 n1 n8 QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is this competition between kinetic energy and long-range interactions that we will study in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We first show that the ground state in the classical limit Ω = 0 is the zero-momentum state satisfying the ice rule which we call the “ice ferromagnet”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We assume that, as one increases Ω, there is no phase transition to a different ordered state before the putative transition to a QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In order to determine whether a QSL phase exists and, if yes, at what Ω the transition to the QSL occurs, one needs to compare the energies of ansatz wavefunc- tions of the QSL and the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When Ω ̸= 0, such wavefunctions would necessarily involve configura- tions that violate the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We incorporate the effect of nonzero Ω on the wavefunction using perturbation the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We treat ˆH1 ≡ ˆHΩ+ ˆHLR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', both the laser driving term and the long-range inter- actions, as a perturbation to ˆH0 (unlike Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A, where we dropped ˆHLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We perturbatively find an effective Hamiltonian ˆHeff acting on the low-energy ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We then compare the expectation value of ˆHeff in candi- date wavefunctions that live entirely in this low-energy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since a QSL wavefunction is a linear superposi- tion of exponentially (in system size) many ice rule obey- ing states, we calculate ¨ ˆHeff ∂ numerically using classical Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Expression for ˆHeff We perform a Schrieffer-Wolff transformation ˆ˜H = ˆUS ˆH ˆU † S = ˆUS Ä ˆH0 + ˆHΩ + ˆHLR ä ˆU † S, (19) for a unitary ˆUS = e ˆS, where ˆS is an anti-hermitian oper- ator chosen to make ˆ˜H block-diagonal in the (degenerate) eigenbasis of ˆH0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', ˆ˜H = ˆP ˆ˜H ˆP + (1 − ˆP) ˆ˜H(1 − ˆP), (20) where ˆP projects onto the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the remainder of this paper, we will restrict ourselves to the low-energy sector and therefore only consider the ˆHeff ≡ ˆP ˆ˜H ˆP term above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We calculate ˆHeff perturbatively in ˆH1 = ˆHΩ + ˆHLR (see Appendix B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [77] for general expressions of ˆHeff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As we saw in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A, if we consider only ˆHΩ as the perturbation, then the first non-trivial term appearing in ˆHeff is −Jring � � �� � � �� + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', where Jring = 63 16 Ω6 V 5 + Θ Å Ω8 V 7 ã .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (21) Since we are performing perturbation theory in two op- erators ˆHΩ and ˆHLR, each of them comes with its own small parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the perturbative expansion will involve polynomials in these two small parameters, there is some arbitrariness in deciding how to compare the two parameters relative to each other and thus in how to truncate the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In our calculation, we follow an operational scheme of keeping all the terms up to sixth order in ˆHΩ + ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Following this truncation scheme, we get (up to additive constants) ˆHeff ≈ − Jring � � �� � � �� + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' + Å 1 − Ω2 V 2 − 61 18 Ω4 V 4 ã ˆHLR − Ω2 V Ä ˆW (2) LR + ˆW (3) LR + ˆW (4) LR ä − Ω4 V 3 Å152 27 ˆW (2) LR − ˆL(2) LR + ˆ M (2) LR ã ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (22) where ˆW (2) LR ≡ 1 4 � j � k1̸=j k2̸=j vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k1vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k2 ˆSz k1 ˆSz k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (23) ˆL(2) LR ≡ 109 432 � j1̸=k1 j2̸=k2 δ⟨j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='j2⟩vj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k1vj2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k2 ˆSz k1 ˆSz k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (24) ˆ M (2) LR ≡ 20 27 � j1̸=k1 j2̸=k2 δ⟨j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='j2⟩vj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k1vj2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k2 ˆSz j1 ˆSz k1 ˆSz j2 ˆSz k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (25) ˆW (3) LR ≡ 1 2 � j � k1̸=j k2̸=j k3̸=j vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k1vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k2vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k3 ˆSz k1 ˆSz k2 ˆSz k3 ˆSz j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (26) ˆW (4) LR ≡ 1 4 � j � k1̸=j k2̸=j k3̸=j k4̸=j vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k1vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k2vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k3vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='k4 ˆSz k1 ˆSz k2 ˆSz k3 ˆSz k4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (27) and vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='j ≡ � a6 |xi−xj|6 if xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' xj are not nearest neighbors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (28) In the above equations, δ⟨i,j⟩ enforces i and j to be near- est neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The expectation value of the Hamiltonian (6) in a given state |Ψ⟩ is ⟨Ψ| ˆH|Ψ⟩ = Ä ⟨Ψ| ˆU † S ä Ä ˆUS ˆH ˆU † S ä Ä ˆUS |Ψ⟩ ä .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (29) Suppose ˆUS |Ψ⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', |Ψ⟩ transformed by the Schrieffer- Wolff transformation) lies entirely in the ice manifold, then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (19), we get ⟨Ψ| ˆH|Ψ⟩ = Ä ⟨Ψ| ˆU † S ä ˆHeff Ä ˆUS |Ψ⟩ ä .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (30) For the ground state, |Ψg⟩ of the full Hamiltonian ˆH, ˆUS |Ψg⟩ lies entirely in the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, we pick an ansatz wavefunction for ˆUS |Ψ⟩ that also lies entirely in the ice manifold and compute the expectation value of ˆHeff in our ansatz state to get the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Before describ- ing our ansatz states in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A 3, we first consider the limit Ω = 0 in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Classical ground state of the long-range Hamiltonian Here, we will find the ground state selected by long- range interactions in the limit Ω = 0 where there are no quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Hamiltonian is ˆHcl = ˆH0 + ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We find the ground state by going to the Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the pyrochlore lattice is an FCC lattice with a four-site basis, we use the notation ˆSz r,µ for spins where r is an FCC lattice vector and µ ∈ {0, 1, 2, 3} labels the sites within the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The spin ˆSz r,µ is physically located at r+eµ/2 where eµ are the vectors joining a diamond A site to a neighboring diamond B site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(b) for the precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') As we are considering the classical limit in this section, we drop hats on quantities which would otherwise be operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Fourier transform of Sz r,µ is Sz r,µ = 1 √Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' � k eik·rSz k,µ, (31) where Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' is the number of FCC unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Substituting this in Hcl, we get Hcl = � µ,ν,k Vµν,kSz k,µSz −k,ν, (32) where k is a vector in the Brillouin zone of the FCC lattice and Vµν,k is the Fourier transform of the van der Waals potential: Vµν,k = V 2 � r eik·r Å a |r + (eµ − eν)/2| ã6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (33) Diagonalizing the matrix Vµν,k for each k gives Hcl = � µ,k εk,µ|S ′z k,µ|2, (34) where S ′z k,µ is related to Sz k,ν through a multiplication by a unitary matrix Uµν,k which diagonalizes Vµν,k: S ′z k,µ = � ν Uµν,kSz k,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Recall that Sz r,µ is either +1/2 or −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This imposes the following constraint: � k,µ |S ′z k,µ|2 = � k,µ |Sz k,µ|2 = � r,µ �Sz r,µ �2 = Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='. (35) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (34), the energy can be interpreted as a weighted sum of εk,µ with the corresponding weights be- ing |S ′z k,µ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Because of the constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (35), the energy is minimized by having the full weight on the smallest εk,µ and no weight on the rest of the εk,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This holds provided that such a configuration of S ′z k,µ in the momentum space corresponds to some configuration in the real space where Sz r,µ are ±1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Calculating the Fourier transform of the long-range po- tential, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (33), and its eigenvalues εk,µ, we find that the Figure 7: An ice ferromagnet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is an ice rule obeying state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', n = 2 on every tetrahedron) with k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' All the up-pointing tetrahedra are copies of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The same is true for the down-pointing tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' There are six (4C2) such states, and together they make up the ground subspace of ˆHcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' minimum of εk,µ occurs for k = 0 and is triply degener- ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In particular, Vµν,k=0 = Öv1 v2 v2 v2 v2 v1 v2 v2 v2 v2 v1 v2 v2 v2 v2 v1 è , (36) where v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='113V and v2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='12V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Its eigenvalues are ε0,0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V and ε0,1 = ε0,2 = ε0,3 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='004V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The unitary that diagonalizes the above matrix also relates S ′z 0,µ to Sz 0,ν as à S ′z 0,0 S ′z 0,1 S ′z 0,2 S ′z 0,3 í = 1 2 à 1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 í à Sz 0,0 Sz 0,1 Sz 0,2 Sz 0,3 í .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (37) Since ε0,1, ε0,2 and ε0,3 are the minimum eigenvalues, the energy is minimized by having all the weight distributed between S ′z 0,1, S ′z 0,2 and S ′z 0,3 and no weight on the remain- ing S ′z k,µ, that is, S ′z k̸=0,µ = 0 and S ′z 0,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' There indeed exist states satisfying these two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The first con- dition, S ′z k̸=0,µ = 0, implies that the ground state is a k = 0 state, while the second condition, S ′z 0,0 = 0, im- plies that the ground state satisfies the ice rule (so that the total spin, which is S ′z 0,0 is 0), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' There are six such states, and we refer to them as the “ice fer- romagnet” or “ice FM” states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' One of these is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ansatz wavefunctions for the ordered state and for the quantum spin liquid We now assume that, as one increases Ω starting from Ω = 0, the ground state remains adiabatically connected to the ice ferromagnet derived in the previous section till 10 the point where it undergoes the putative phase transi- tion to the QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, our ansatz for the ordered state is |Ψord⟩ = ˆU † S |ΨIFM⟩ , (38) where |ΨIFM⟩, a product state in the ˆSz basis, is the k = 0 ice ferromagnet defined Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This configu- ration is given by Sz r,µ = 1 2εµ (independent of r), where (ε0, ε1, ε2, ε3) ≡ (1, 1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We note that there are six such choices for εµ that satisfy the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We pick one such choice, but our calculations are not sensitive to which one we pick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' |ΨIFM⟩ lives entirely in the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Left-multiplication by ˆU † S takes it back to the original Hilbert space with ice rule violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our ansatz wave function for the spin liquid state is |ΨQSL⟩ = ˆU † S |ΨRK⟩ , (39) where |ΨRK⟩ is a uniform superposition of all dimer cov- erings [65] of the diamond lattice (with n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' |ΨRK⟩ lives in the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Like before, we left-multiply it by ˆU † S to take it back to the original Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The justification for our choice is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' |ΨRK⟩ is the ground state of the dimer model at the RK point [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (10)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When the RK potential is zero, |ΨRK⟩ has an energy expectation value of −4Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='Jringnflip, where nflip is the average fraction of flippable hexagons in the RK wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We find numerically that nflip = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='1757 (also calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, the energy of |ΨRK⟩ is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='7028JringNu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' which is not too far from the ground state energy of the dimer model (12) found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [51] to be −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='756JringNu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='. Even though |ΨRK⟩ has slightly higher energy, it has the advantage of being sim- pler to sample by classical Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This explains our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For comparison, we will also calculate the energy of a different ordered state |Ψ′ ord⟩ = ˆU † S |ΨIAFM⟩ that we call an ice antiferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Here |ΨIAFM⟩ is an ice rule obeying state with ordering wave vector k = π (b1 + b2), where b1, b2 and b3 are primitive reciprocal lattice vec- tors of the FCC lattice satisfying ai ·bj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This state is known elsewhere in literature as the 2π(001) state (this nomenclature uses an enlarged cubic unit cell of the FCC lattice) [76, 78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Numerical results—energy expectation values and phase diagram We now describe our computation of the expectation value of ˆHeff [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (22)] in |ΨRK⟩, |ΨIFM⟩ and in |ΨIAFM⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While the expectation value in |ΨIFM⟩ and |ΨIAFM⟩ can be computed straightforwardly, the expec- tation value in |ΨRK⟩ requires classical Monte Carlo sam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We use a system with 8×8×8 unit cells (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', con- taining 2048 pyrochlore sites) with periodic boundary conditions in the a1, a2, and a3 directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We restrict our sampling to sectors in which the total electric flux Figure 8: ⟨ ˆHeff⟩ in |ΨRK⟩, |ΨIFM⟩, and |ΨIAFM⟩ calcu- lated by inserting the values in Table I in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' piercing through any 2D torus cross-section (as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48]) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our sampling is done us- ing loop moves as described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48, 78, 79] – in each Monte Carlo run, we perform 512 × 500, 000 loop moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We calculate ¯nflip, HLR, W (2) LR and L(2) LR after every 512 loop moves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', we take 500,000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We cal- culate M (2) LR, W (3) LR, and W (4) LR after every 512 × 10, 000 loop moves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' we take 50 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We repeat this procedure for 9 independent runs in order to cal- culate the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our results are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' With these values at hand, we calculate the ex- pectation value of ˆHeff using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (22) in |ΨRK⟩, |ΨIFM⟩, and |ΨIAFM⟩, and the result is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As we turn on Ω, the transition point ΩC can be determined within our approximation as the Ω for which the energy of the ice ferromagnet becomes higher than that of the RK wavefunction, as calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We find ΩC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='43927(1)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (40) There is an important question on whether our use of perturbation theory is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First, we argue that treating ˆHLR perturbatively is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ¶ ˆHLR © , ¶ ˆW (2) LR, ˆL(2) LR, ˆ M (2) LR © , ¶ ˆW (3) LR © , and ¶ ˆW (4) LR © are sets of op- erators that are first, second, third, and fourth order re- spectively in ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As we can see from Table I, the ex- pectation values of these operators in |Ψ⟩RK drops by an order of magnitude each time one goes one order higher in ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Next, is perturbation theory in ˆHΩ justified, given that our calculated ΩC is outside the Ω ≪ V regime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We observe that the leading contribution to Jring that we dropped, 33833 2592 (ΩC)8 V 7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='018V [74], is smaller than the one we kept, 63 16 (ΩC)6 V 5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='028V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If we had kept higher order contributions to Jring, it would only decrease the energy of the QSL relative to the ice ferromagnet and ice antiferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Further, the energy of the QSL that we present is a conservative estimate since we used the RK wavefunction which has higher energy than the true ground state of Hamiltonian (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This gives us hope that 11 Operator |ΨRK⟩ |ΨIFM⟩ |ΨIAFM⟩ ˆR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='70288(4)Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0 0 ˆHLR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='6037(1) × 10−2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='4002 × 10−2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='8722 × 10−2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆ W (2) LR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='11778(1) × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='01642 × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='4994 × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆL(2) LR −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='7467(3) × 10−4Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='0829 × 10−4Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='5662 × 10−4Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆ M (2) LR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='96(3) × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='073 × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='66 × 10−3Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆ W (3) LR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='25(4) × 10−5Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='00665 × 10−5Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='81 × 10−5Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆ W (4) LR −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='57(2) × 10−6Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='0309 × 10−6Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='35 × 10−6Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Table I: The expectation values of the operators in the left column in ansatz wavefunctions |ΨRK⟩, |ΨIFM⟩ and |ΨIAFM⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The operator ˆR is defined as ˆR = � � �� � � �� + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the RK wavefunction, ⟨ΨRK| ˆR|ΨRK⟩ = 4¯nflipNu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='. To calculate expectation values in |ΨRK⟩, we have used classical Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' our result obtained using perturbation theory is qualita- tively correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Within our approximation, for Ω < ΩC, the ground state is an ice ferromagnet, an ordered state satisfying the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For Ω > ΩC but also close to ΩC, the ground state is in the QSL phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', the deconfined phase of a U(1) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From the point of view of the QSL, the ordered ice ferromagnet state is obtained when monopole excitations of the spin liquid proliferate, and the monopole-antimonopole string operator, to be defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV B, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (53), acquires an expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As a consequence of this, the fractional “electric charges”, or spinons, get confined [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The monopole creation operator (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV B and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48]) is diagonal in the ˆSz basis, and acts in the sector that obeys the ice rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is thus plausible that the confined phase is in- deed the ice ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While our calculation provides microscopic intuition for this transition, we emphasize that, to prove the existence of, locate and characterize this transition accurately, one needs to do a more careful quantum Monte Carlo calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Large Ω—Higgs transition From the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (6), it is clear that, in the limit Ω ≫ V , the ground state is a transverse-field- polarized (TFP) state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', a product state of (|g⟩ − |r⟩)i at each site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, as Ω is increased away from ΩC, the system should eventually go through a phase transition from the putative QSL phase into the TFP phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From the point of view of the QSL, this is a Higgs transition because the operator ˆSx that acquires expectation value in the TFP phase creates a pair of “electric”-charge ex- citations in the spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The perturbation theory in Ω/V that we performed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A relies on the ability to go to a basis where the Hilbert space decouples into ice rule obeying and ice rule disobeying sectors separated by an energy gap of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' But the ground state in the Ω ≫ V limit (TFP) straddles both of these sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So we do not expect perturbation theory in Ω/V to capture the phase Figure 9: Approximate ground state phase diagram of the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The ground state for Ω = 0 was calculated exactly to be an ice ferromagnet (ice FM) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We assume that, as Ω is increased, no phase transition occurs to a different ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The transition point from the ice ferromagnet (confined phase) to the QSL (deconfined phase) at ΩC ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='44V is obtained by comparing energies of ansatz wavefunc- tions in the effective Hamiltonian obtained using pertur- bation theory in ˆHΩ and ˆHLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For the Higgs transition to the TFP phase, we make an approximation by drop- ping ˆHLR, in which case ΩH was calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [62] to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='55(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' transition into the TFP phase that contains the Ω → ∞ ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hence, we will present an indirect reason- ing below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the Ω ≪ V limit, ˆHLR was important, since it was the dominant term splitting the degeneracy in the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, in the vicinity of the putative Higgs transition, ˆHLR may not be as impor- tant since the largest term in ˆHLR has magnitude V/27, and as justified above using Table I, the effect of ˆHLR is indeed perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, we drop ˆHLR as a zeroth- order approximation for calculating the Higgs transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The resulting Hamiltonian is the transverse field Ising model on the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [62] and [74] studied this model and found the transition point ΩH to be at ΩH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='55(5)V and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='6V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This leads us to expect that, in the window 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='44 < Ω < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='55, the ground state may be a QSL, leading us to sketch the phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Within our approxima- tion, ΩC < ΩH and there is a window where the QSL 12 is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, the introduction of ˆHLR may result in a lowering of the energy of the TFP state relative to the QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Calculating this effect and verifying that this does not bring down ΩH far enough to destroy the QSL phase requires a more careful calculation which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the remainder of this section, we provide some intu- ition for the Higgs transition by performing a gauge mean field theory (gMFT) calculation introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Gauge mean field theory—Higgs transition The main idea of this approach is to first recast the microscopic Hamiltonian as an exact gauge theory by introducing ancillary degrees of freedom followed by a mean-field decoupling of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This theory involves bosonic charges hopping in the presence of a fluctuating gauge field whose mean-field value is chosen self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If this mean-field gauge-field configura- tion is such that the hopping amplitudes of the bosonic charges is 0, then the theory is in a confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If not, the theory is in the deconfined phase as long as the bosons do not condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If the bosonic charges condense, then the theory is in a Higgs phase, which is adiabatically connected to the TFP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Concretely, the construction is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For r ∈ A, where A is a sublattice of the diamond lattice, ˆS+ r→r+eµ = ˆΦ† rˆs+ r→r+eµ ˆΦr+eµ, (41) where ˆS+ r→r+eµ ≡ ˆS+ r+eµ/2 = ˆS+ r,µ (and similarly ˆs+ r→r+eµ ≡ ˆs+ r+eµ/2 = ˆs+ r,µ) lives on a bond of the dia- mond lattice connecting sites r and r + eµ (recall that centers of the bonds of the diamond lattice are sites of the pyrochlore lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆsz is also a spin-1/2 operator and has eigenvalues ±1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Here, ˆΦ† r serves as a raising oper- ator for ˆQ r ≡ ηr(ˆn r − 2), where ηr = 1 for r ∈ A and ηr = −1 for r ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For convenience, we drop the sym- bol from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆQr and ˆΦ† r satisfy the commutation relation: î ˆQr, ˆΦ† r ó = ˆΦ† r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Note that ˆΦr is not a canonical boson but a rotor satisfying ˆΦ† r ˆΦr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (42) To recover the original spin Hilbert space, one imposes the constraint that the total gauge charge at r is ˆQr = ηr � µ ˆsz r+ηeµ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (43) Rewriting the Hamiltonian (6) in terms of the fictitious variables, ˆQr, ˆΦr and ˆsr,µ we get ˆH =V 2 � r∈A,B ˆQ2 r − Ω 2 � (r∈A),µ ÄˆΦ† rˆs+ r→r+eµ ˆΦr+eµ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ä + 1 2 � r,r′∈A � µ,ν Vµν(r − r′)ˆsz rµˆsz r′,ν, (44) where Vµν(r−r′) = V Ä a r−r′+eµ/2−eν/2 ä6 whenever (r, µ) and (r′, ν) are distinct and are not nearest-neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Vµν(r − r′) is 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [80], we perform the zeroth-order mean- field decoupling: ˆΦ† ˆΦˆs → ˆΦ† ˆΦ ⟨ˆs⟩ + ¨ˆΦ† ˆΦ ∂ ˆs − ¨ˆΦ† ˆΦ ∂ ⟨ˆs⟩ and ˆsˆs → ˆs ⟨ˆs⟩ + ⟨ˆs⟩ˆs − ⟨ˆs⟩ ⟨ˆs⟩ (where ˆs could either be ˆs+, ˆs−, or ˆsz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Upon doing so, the Hamiltonian de- couples into a Hamiltonian of bosons hopping on the diamond lattice and a Hamiltonian of spins in an ex- ternal field, which itself is set self-consistently by the Green’s function of the bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Before solving the re- sulting theory, one needs to enforce the constraints (42) and (43) using Lagrange multipliers λr and vr, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Within the mean-field theory, it is assumed that these Lagrange multipliers take a spatially homogeneous value at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We then find the minimum value of ΩMF H such that, for any Ω > ΩMF H , it is pos- sible to self-consistently choose λ only by macroscopi- cally occupying a boson mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This ΩMF H marks the lo- cation of the Bose-Einstein-condensation transition (or Higgs transition within the mean-field theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We find ΩMF H ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='7V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Appendix A, we present more details of this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' An artifact of this technique is that, although we include long-range interactions in our cal- culation, they do not play any role at the saddle point near the Higgs transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, the final steps and result of our calculation are identical to the ones carried out in [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Appendix A, we also point out a major limitation of this technique in the small-Ω limit that may not have been appreciated in previous literature on gauge mean field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Comments on dynamical state preparation So far, we have focused on the nature of the ground state of Hamiltonian (6) as a function of Ω/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' How- ever, what is often experimentally relevant is the nature of the state prepared by a ramping of parameters in a finite amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This issue was studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81] for a Z2 gauge theory in the context of the experiment in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the main ideas of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81] are gen- eral enough, here we will present an adaptation of the conclusions of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81] to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The excitations of a U(1) QSL are gapless “photons”, magnetic monopoles, and “electric charges” (spinons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The transition of a QSL to an ice ferromagnet is driven by the condensation of monopoles, while the transition to the TFP phase is driven by the condensation of spinons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The gapless “photons” are not directly involved in these transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Also, a state with “photon” modes excited on top of a QSL state is still in the deconfined phase of the U(1) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This allows us to ignore “pho- tons” in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the confined phase, ice fer- romagnet has an extensive number of monopoles, we use the difference per unit cell between the energies of 13 Figure 10: A qualitative sketch of the energy scales (per unit cell) in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For Ω > ΩC, the ground state is a U(1) QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ice ferromagnet is the ordered state obtained when monopoles proliferate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', the ice ferromagnet has an extensive number of monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We therefore use the energy difference per unit cell between the QSL and the ice ferromagnet at Ω = 0, obtained in Table I, as a proxy for the monopole energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This scale ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='03V is much smaller than the spinon energy scale (“electric charge”), which is ∼ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' the QSL and ice ferromagnet states as a proxy for the monopole energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' At Ω = 0, this difference is �¨ ˆHLR ∂ QSL − ¨ ˆHLR ∂ IFM � /Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='03V (see Table I), which is much smaller than the spinon energy scale (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 10 for a sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Suppose one starts with an initial state (for a small ϵ ∼ Ω/V ) ��Ψ(t=0) � = ⊗i (|g⟩i + ϵ |r⟩i) , (45) which is the ground state in the limit of large negative δ/V and small Ω/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II, the clas- sical ground state lies in the ice manifold when δ ∈ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Now suppose that δ is ramped up from its initial large negative value to a value in this range such that the ramp is adiabatic with respect to the spinon gap V , but is sudden with respect to the monopole scale ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='03V , while keeping Ω/V ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using arguments in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81], this protocol will not prepare the ground state, which, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 9, is an ice ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Instead, it will (approximately) project out violations of the ice rule (due to adiabaticity with respect to the spinon scale) from the initial state ��Ψ(t=0) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The resulting final state is |Ψfinal⟩ ≈ ˆP {⊗i (|g⟩i + ϵ |r⟩i)} = |ΨRK⟩ , (46) where ˆP is the projector onto the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The pro- jected wavefunction is an equal-weight superposition of all coverings, which is simply the RK wavefunction and which lies in the QSL phase [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' There is one catch to the above argument—the spinon gap closes during the above ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So it is impossible to be sudden with respect to the monopole scale and yet be strictly adiabatic with respect to the spinon gap throughout the ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For a short du- ration (while the ramp is going through the spinon gap closing), adiabaticity with respect to the spinon gap will be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' By the Kibble-Zurek mechanism, the result- ing state is composed of finite-size puddles of QSL-like re- gions with a nonzero density of spinons interspersed [81– 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, in summary, there are two different ways in which one can prepare a U(1) QSL-like state in ex- periment and study a confinement-deconfinement transi- tion1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ω/V ≪ 1: Perform a ramp of δ starting from a large negative value and ending in the range (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V ) for a fixed Ω/V ≪ 1 such that the ramp is adiabatic with respect to V (spinon gap) but sudden with respect to the monopole scale (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='03V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Even though the ground state is not a QSL for these parameters, this procedure would create puddles of QSL-like regions by the argument in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' To see a deconfinement- confinement transition, the ramp of δ should be slowed down and, once it is adiabatic with respect to the monopole gap, an ordered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' confined state will be prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Adiabatic: Perform a ramp of δ starting from a large negative value and ending in the range (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='46V ) and a ramp in Ω starting from Ω/V ≪ 1 and ending in a final value Ωf, such that both ramps are adiabatic with respect to the monopole scale always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The two ramps can be per- formed simultaneously, or such that the ramp in δ precedes the ramp in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This would approximately create the ground state of Hamiltonian (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As the final value Ωf goes through ΩC (ΩH), the nature of the final state prepared this way goes through a confinement-deconfinement (Higgs) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Once a state is prepared by either of the above schemes, one needs to devise measurements that can tell whether the state is in the confined phase or in the deconfined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We address this in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' DIAGNOSIS OF THE QUANTUM SPIN LIQUID Access to wavefunction snapshots in the ˆSz basis, com- bined with access to unitary evolution, allows one to use the Rydberg-atom platform to measure non-local observ- ables, a feature generally unavailable in traditional con- densed matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this section, we describe some 1 We note that the confinement-deconfinement transition of U(1) gauge theory in 3+1D is strictly speaking, a ground state tran- sition [71, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, in this paper, when we use the phrase confinement-deconfinement transition, we mean signatures of this transition in a finite-size state prepared in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 14 Figure 11: Notation for the plaquette correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' P and P ′ are two hexagonal plaquettes of the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' r, r′, r + uµ, and r′ + uν are the sites of the dual diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' uµ and uν are vectors perpendicular to P and P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' measurable correlators which can be used to observe the signatures of a quantum spin liquid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this section, we assume that the detuning is chosen such that ρ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Plaquette-plaquette correlators The plaquette operators are off-diagonal in the ˆSz ba- sis and map one dimer covering to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' They are important in distinguishing a coherent quantum super- position from a classical admixture of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We define two plaquette operators ˆXP and ˆYP for a hexagonal pla- quette P of the pyrochlore lattice as ˆXP = Ä ˆS+ 1 ˆS− 2 ˆS+ 3 ˆS− 4 ˆS+ 5 ˆS− 6 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ä , ˆYP = −i ˆS+ 1 ˆS− 2 ˆS+ 3 ˆS− 4 ˆS+ 5 ˆS− 6 + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', (47) where 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6 denote the sites around a plaquette P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Either of the two correlators, ⟨ ˆXP ˆXP ′⟩ and ⟨ ˆYP ˆYP ′⟩, of the plaquette operators on two plaquettes P and P ′ can distinguish a QSL phase from other phases including a classical spin ice (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We compare the two cor- relators and provide protocols to measure them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We as- sume throughout that the two plaquettes P and P ′ do not have any sites in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using the mapping between the spins and the effective U(1) gauge theory from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (13), we see that the opera- tors ˆXP and ˆYP are equal to (twice) the cosine and the sine of the magnetic field, respectively: ˆXP = 2 cos Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 ä = 2 cos Ĉbr,µ ä , ˆYP = 2 sin Ĉθ1 − ˆθ2 + ˆθ3 − ˆθ4 + ˆθ5 − ˆθ6 ä = 2 sin Ĉbr,µ ä , (48) where r belongs to the dual diamond lattice [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(c)], and µ ∈ {0, 1, 2, 3} labels the direction of mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆbr,µ is along uµ, which are vectors joining an A site of the dual diamond lattice to its neighboring B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' These vectors are perpendicular to the plaquettes of the pyrochlore lattice, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The effective theory in the deconfined phase (QSL) is Maxwell electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the 3 + 1D continuum Maxwell electromagnetism, the correlators of the Cartesian components of the mag- netic field ˆbr,i for i ∈ {x, y, z} can be expressed as Gaus- sian integrals which evaluate to [48] ⟨ˆb0,iˆbR,j⟩0 ∝ 1 R4 Å 2RiRj R2 − δij ã ≡ CB ij(R), (49) where ⟨·⟩0 is the expectation value with respect to the Maxwell action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The correlators of the magnetic field operators ˆbr,µ for µ ∈ {0, 1, 2, 3} on the pyrochlore plaquettes in the decon- fined phase are obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (49) by taking compo- nents of the Cartesian magnetic field along the vectors uµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The result is ⟨ ˆXP ˆXP ′⟩ − 4 ∝ 1 R8 � �� k,l (uµ)k(uν)l Å 2RlRk R2 − δk,l ã� � 2 , ⟨ ˆYP ˆYP ′⟩ ∝ 1 R4 � �� k,l (uµ)k(uν)l Å 2RlRk R2 − δk,l ã� � , (50) where the summation is over k, l ∈ {x, y, z}, R = r − r′, and R is assumed to be large compared to the monopole correlation length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The factors inside the square brack- ets are geometric factors, which depend on the direction of the vectors uµ, uν, and R, but are independent of the distance R between the two plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [48] also sep- arately studied the correlators precisely at the RK point (which sits at the phase boundary between deconfined and confined phases) where the effective field theory dif- fers from the regular Maxwell theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the RK wave- function, while the behavior of the plaquette correlators differs from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (50), it is still a power law with a slower decay [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We note that, if the experimentally prepared state is close to an RK wavefunction (see discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III C), then this distinction will be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Outside of the deconfined phase (QSL) of the com- pact U(1) gauge theory, Maxwell electromagnetism is no longer the effective theory, and the behavior of the corre- lators is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The ice ferromagnet state is an ordered state with the spins pointing in the z-direction, thus the two plaquette correlators are expected to decay exponen- tially with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The TFP phase has ⟨ ˆS+⟩ ̸= 0, and hence the plaquette X correlator approaches a nonzero constant at large R, while the plaquette Y correlator is 0 (or decays exponentially or faster in R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Lastly, in a classical spin liquid, which consists of an incoherent mixture of expo- nentially many dimer coverings, the plaquette correlators decay at least exponentially (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since the plaquette correlators involve off-diagonal op- erators, they cannot be read out directly from the snap- shots of a Rydberg-atom array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, we show that they can be measured by evolving the system under a modified Hamiltonian for a specific time duration fol- lowed by measurement of a diagonal operator [26, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We describe the protocols to measure both plaquette X and plaquette Y correlators in the sections below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 15 Correlator Confined (Ice FM) Deconfined (QSL) Higgs (TFP) Classical Spin Ice ⟨ ˆ XP ˆ XP ′⟩ − 4 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay 1/R8 Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay ⟨ ˆYP ˆYP ′⟩ Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay 1/R4 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay ¨ ˆ M† ˆ M(r1 → r2) ∂ Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' decay Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' decay Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay χE C Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='a Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay χM C Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='a Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' or faster decay ⟨ ˆSz ri ˆSz r′j⟩ Nonzero const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1/R4 Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' decay 1/R4 a Distinguishing this non-zero constant from zero for χE C in the confined phase (Ice FM) and for χM C in the Higgs phase (TFP) may be practically challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Table II: Behavior of various correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆXP and ˆYP are plaquette operators defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆ M† ˆ M(r1 → r2) is a monopole string operator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' χE C and χM C are BFFM order parameters defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (60) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (64), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this table, we have omitted the form factors multiplying 1/R4 and 1/R8 that are provided in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (50) and (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Plaquette X correlator For a state |Ψ0⟩ completely within the ice manifold, the expectation value of ˆXP ˆXP ′ is the same as that of a product of ˆSx operators on the 12 sites of P and P ′, that is, ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩ = ⟨Ψ0| 12 � i=1 2 ˆSx i |Ψ0⟩, (51) where the sites i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6 are on the plaquette P and sites i = 7, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 12 are on the plaquette P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This can be seen by writing 2 ˆSx i = ˆS+ i + ˆS− i and noticing that the only terms that preserve the ice rule are ring exchanges over P and P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When the remaining terms act on a state in the ice manifold, they either take the state outside of the ice manifold or annihilate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus the expectation value of these remaining operators in |Ψ0⟩ is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For ex- ample, ˆS+ 1 ˆS+ 2 ˆS+ 3 ˆS− 4 ˆS− 5 ˆS− 6 ˆS+ 7 ˆS− 8 ˆS+ 9 ˆS− 10 ˆS+ 11 ˆS− 12 acting on a state in the ice manifold would either annihilate this state or give a state that violates the ice rule on four of the tetrahedra surrounding P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The protocol to measure the plaquette X correlator is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We abruptly change the phase and the amplitude of the Rabi frequency, so that the new Hamiltonian is ˆHY ≈ ΩY � i ˆSy i with ΩY ≫ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (Achieving ΩY ≫ V may require working with atom spacings that are sufficiently large and/or with Rydberg principal quantum numbers that are sufficiently low, but not low enough to make Rydberg lifetime a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') It is assumed that this change of the Hamiltonian is done sufficiently rapidly so that the sudden approximation is valid and the state of the system does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We evolve the system under ˆHY for a time tY = π/(2ΩY ), which amounts to a π/2 pulse about the y-axis, and then measure all atoms in the {|g⟩ , |r⟩} basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thanks to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (51), this allows us to compute the plaquette X correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (51) assumes that the state |Ψ0⟩ is in the ice man- ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, the ground state |Ψg⟩ of the system is not completely in the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The error introduced because of assuming |Ψg⟩ to be in the ice manifold is of sixth order in Ω/V , as we show in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The intuitive reason for this is that six factors of ˆHΩ are re- quired to give a state that can have nonzero overlap with �12 i=1 2 ˆSx i |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We note that, for Ω/V ≫ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' in the TFP phase), the experimentally measured quantity ⟨Ψg| �12 i=1 2 ˆSx i |Ψg⟩ is not the same as the plaquette X correlator ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, but their behaviors are nevertheless the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', both approach a nonzero constant as the dis- tance between the plaquettes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The plaquette X correlator behaves as ⟨ ˆXP ˆXP ′⟩ − 4 ∝ 1/R8 (the angular dependence is not shown here) in the QSL phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Although it is a power law, the decay is very rapid, and it might be practically difficult to distinguish it from an exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This issue is less promi- nent in the plaquette Y correlator, and we now provide a protocol to measure it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Plaquette Y correlator This protocol relies on the fact that, for a state |Ψ0⟩ within the ice manifold, we have an identity similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (51): ⟨Ψ0| ˆYP ˆYP ′|Ψ0⟩ = ⟨Ψ0| 6 � i=1 (2 ˆSx 2i−1)(2 ˆSy 2i)|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (52) Recall that sites 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6 are on P, while sites 7, 8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 12 are on P ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This protocol is the same as the protocol for measuring the plaquette X correlator, except that now the π/2 pulses on sites 2i for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6 are about the x-axis on the Bloch sphere while the π/2 pulses on sites 2i − 1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' , 6 are around the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' After applying these π/2 pulses, �12 i=1 2 ˆSz i is measured by taking a snapshot of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Note that, for any set of non-overlapping plaquettes, we can simultaneously measure Y correlators between all pairs of plaquettes in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This protocol requires control over individual sites and has an error of order (Ω/V )6 (the estimation of the er- 16 ror is similar to the one for the plaquette X correlator done in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The advantage of the plaquette Y correlator over the plaquette X correlator is that the for- mer decays as 1/R4, which is slower than the decay of the plaquette X correlator and can be easier to observe experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Monopole-monopole correlator In the deconfined phase, monopoles are gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, the expectation value of an (equal-time) op- erator that creates a string with a monopole and an- timonopole at its endpoints should decay exponentially with the length of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, in the confined phase, monopoles are condensed, and hence the expectation value should approach a nonzero con- stant as the length of the string increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the contin- uum, the following operator inserts a string that creates a monopole at r1 and an antimonopole at r2 [48]: ˆ M† ˆ M(r1 → r2) ∼ ei � d3r′A(r′)·ˆe(r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (53) Here A(r′) is a classical (non single-valued) vector poten- tial such that the flux φΣ of B = ∇×A through a closed surface Σ is φΣ ≡ � Σ B · dS = 2πqQΣ, (54) where QΣ = 1 when Σ encloses r1 and not r2, QΣ = −1 when Σ encloses r2 and not r1, and QΣ = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' q is an integer and denotes the “charge” of the monopole string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For simplicity, we will set q = 1 in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We clarify that B and φΣ are classical numbers and are different from ˆb and ˆΦΣ which are operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆb ≡ ∇׈a, for gauge-field (operator) ˆa, and ˆΦΣ is defined as ˆΦΣ ≡ � Σ ˆb · dS = 2π ˆm, (55) where ˆm takes integer eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The form of the monopole string operator is chosen so that it increases the flux through Σ by 2πQΣ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', îˆΦΣ, ˆ M† ˆ M(r1 → r2) ó = 2πQΣ ˆ M† ˆ M(r1 → r2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (56) We now adapt this operator to the Rydberg setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Consider the diamond lattice formed by the centers of tetrahedra of the pyrochlore lattice, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Unlike the continuum, it is now important to specify that the endpoints of the monopole string r1 and r2 belong to the dual diamond lattice [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(c)], whose sites are centers of “polyhedra” made of four puckered-hexagonal “plaquettes” of the diamond lattice2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Let 2 In terms of the original pyrochlore lattice, the vertices of the dual diamond lattice are centers of the truncated tetrahedra [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 12(b)] which fill the voids between the tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Figure 12: (a) The “polyhedron” formed by four puck- ered hexagons of the diamond lattice is shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The centers of these “polyhedra” form the dual diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (b) The center of the “polyhedron” in (a) is also the center of a truncated tetrahedron (shown in red) of the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' x ≡ r + eµ/2 be a site on the pyrochlore lattice, where r is an A-site of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ax ≡ Ar,r+eµ is the discrete version of A integrated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1(b) shows the vectors eµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' along the line pointing from the center of an A tetrahedron (centred at r) to the B tetrahedron (centred at r+eµ) such that the two tetrahedra touch at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Ax is required to satisfy the discrete version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (54), and hence depends on r1, r2, the “magnetic field” configuration B and the gauge choice for Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For the pyrochlore lattice, we have ˆ M† ˆ M(r1 → r2) = ei � x∈pyrochlore Ax(ˆnx−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (57) This operator is purely diagonal in the ˆnx basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', in the ˆSz-basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So, experimentally, one can calculate this phase for each snapshot and average over shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Theoretically, one expects ��� ¨ ˆ M† ˆ M(r1 → r2) ∂��� ∼ ® e−|r2−r1|/λ, deconfined phase, constant, confined phase, (58) where λ is a correlation length that depends on the monopole gap and the “photon” velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 13, we provide an example of one configuration of the classi- cal numbers Ax that defines a monopole string operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Below, we comment on the freedom in choosing Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Choice of A The classical numbers Ax should of course obey the constraint that the flux of ∇×A through a closed surface Σ is 2πQΣ, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, one still has a freedom in the choice of A in the following two respects: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Freedom in the arrangement of the field lines of ∇ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For example, they can be confined to a thin tube connecting r1 and r2, or be spread out according to Coulomb’s law, or be something in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Different such arrangements, due to their different energy costs, would differ in sub-leading corrections to the exponentially decaying behavior, 17 Figure 13: An example of the monopole string operator ˆ M† ˆ M(r1 → r2) for which we provide Ax explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In our example, the string carries 2π flux through a tube with a width of 7 puckered hexagons of the diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (a) A schematic of the tube running along the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The diamond lattice (whose vertices are centers of tetrahedra of the pyrochlore lattice) can be seen as ABC stacking of layers of “honeycomb” lattices made of chair-like puckered hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The tube consists of three types of layers shown in yellow, orange, and cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Each layer is made of 7 puckered hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' To convey a sketch, we depict such a layer by a big hexagon with some thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (b) A side view of the stack showing three of its layers, which then need to be repeated in the z direction to get the entire string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For bonds x with arrows, the value of Ax is written next to the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For bonds x without arrows, Ax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (c) Top view of three of the layers of the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It can be seen from all three sub-figures (a)-(c) that the flux through any closed surface Σ that completely encloses an integer number of layers, such that the bottom layer is included but not the top, is 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, if Σ partially encloses a layer, then ΦΣ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This difficulty in defining arbitrary integer multiples of 2π flux through a volume enclosed by a finite number of plaquettes has been observed before [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore in our construction, r1 and r2 have to be seen as being smeared across 7 points of the dual diamond lattice below the bottom layer and above the top layer respectively, in order to be consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' but the leading behavior would be unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 13, we provide a choice of A, such that the monopole string is localized to a thin tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For a fixed choice of field lines, we still have a gauge choice for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Consider a gauge transforma- tion Ar,r+eµ → Ar,r+eµ + λr+eµ − λr, where λr is 18 an r-dependent real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It results in ˆ M† ˆ M(r1 → r2) → ˆ M† ˆ M(r1 → r2)e−i � r λrηr(ˆn r−2), (59) where ηr = 1 for r ∈ A and ηr = −1 for r ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the Ω/V ≪ 1 limit, we have ˆn r = 2, so the ex- pectation value is invariant under the gauge trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Away from this limit, a gauge transfor- mation on Ar,r+eµ generically results in a physical transformation on the monopole string operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, as long as the external field h = 0 [h is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (5)], by particle-hole symmetry, we have �ˆn r � = 2, and �(ˆn r − 2)2� is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hence we do not expect the gauge transformation on Ar,r+eµ to qualitatively change the behavior of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' BFFM order parameter It is known that the confined and deconfined phases of a gauge theory without matter fields can be distinguished by the scaling of the Wilson loops WL = ¨ ei � L Aµdxµ∂ , where Aµ is the gauge field and L is a closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the deconfined phase, the Wilson loop follows the perimeter law, WL ∝ e−Perimeter of L, while in the confined phase, it follows the area law, WL ∝ e−Area of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, in the presence of matter fields (which are generically al- ways present), the Wilson loop follows the perimeter law in both phases [86, 87], and it cannot be used to distin- guish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Bricmont-Fr¨olich-Fredenhagen-Marcu (BFFM) order parameter is useful in such cases and has a different behavior in the two phases [26, 37, 88–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The BFFM order parameter, denoted here by χE C , is defined as χE C = ���ei � C ˆarr′ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='��� »���ei � L ˆarr′ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='���, (60) where C is an open curve and L is the closed loop formed by combining C with its mirror image about a plane that intersects C only at its end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This order parameter detects long-range order in the “electric charge”-creation string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the Higgs phase, “electric charges” are con- densed, and hence χE C approaches a nonzero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the deconfined phase, the numerator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (60) (cal- culated on an open curve) decays to zero faster than the denominator (calculated on a closed loop, giving the Wil- son loop), as the length of C is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, in the deconfined phase, χE C goes to 0 as the length of C is in- creased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the confined phase, it was argued in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [89] that while both the numerator and the denominator go to zero as the length of C is increased, the limit of their ratio approaches a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However distinguishing this constant from zero in finite systems for finite length of C may be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Below we explain how to measure χE C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using the mapping from spin operators to gauge fields, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (13) and (15), we see that ei � C ˆarr′ ≃ ˆS+ 1 ˆS− 2 ˆS+ 3 · · · , (61) where the product of ˆS+ and ˆS− operators is over the sites on the curve C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The denominator in χE C has a similar expression in terms of spin operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From the point of view of measurement, it is more convenient to consider another quantity, which has the same behavior as χE C in the three phases, defined as: ˜χE C ≡ ��� ¨� i∈C ˆSx i ∂��� …��� ¨� i∈L ˆSx i ∂��� , (62) In the transverse-field-polarized (Higgs) phase, ˜χE C ap- proaches a nonzero constant, just like χE C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Now, we argue that even in the QSL and confined phases, ˜χE C and χE C have the same behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For a state |Ψ⟩ that dominantly lies in the ice manifold, with corrections from outside the ice manifold being of order Ω/V (such as the ground state |Ψg⟩), we have ⟨Ψ| ˆS+ 1 ˆS− 2 ˆS+ 3 · · · + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='|Ψ⟩ = ⟨Ψ|(2 ˆSx 1 )(2 ˆSx 2 )(2 ˆSx 3 ) · · · |Ψ⟩ + Θ �(Ω/V )L� , (63) where L is the number of sites on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The correction is of order (Ω/V )L by an argument similar to the one used to show that the error is sixth order in the protocol to measure the plaquette X correlator (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus, for small Ω/V , χE C and ˜χE C are equal up to order (Ω/V )L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The numerator and the denominator of ˜χE C can be mea- sured by applying π/2 pulses about the y-axis and mea- suring, from the snapshots, products of ˆSz along C and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This procedure is similar to the protocol to measure the plaquette X correlator, described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The operator ei � C ˆarr′ creates two opposite “electric charges” at the endpoints of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So a magnetic analogue of χE C can also be defined, where the numerator is the expectation value of the operator that creates a monopole and an antimonopole at the endpoints of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Such an order parameter, χM C , detects long-range order in the monopole string operator and is given by χM C = � ˆ M† ˆ M � r1 C−→ r2 �� …� ˆ M† ˆ M � r1 L −→ r1 ��, (64) where ˆ M† ˆ M(r1 C−→ r2) inserts a monopole-antimonopole string along C and was defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In this section, we use the notation where the path of the monopole- antimonopole string is explicitly written in the argument of ˆ M† ˆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since this operator is diagonal in the ˆSz basis, it can be measured straightforwardly from the snapshots of the Rydberg-atom array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 19 In the confined phase, monopoles are condensed, so χM C should be a nonzero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the deconfined phase, by the argument of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [89], the numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (64) decays to zero faster than the denominator as the length of C increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore, in the deconfined phase, χM C goes to zero as the length of C increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the Higgs phase, even though there is no long-range order in the monopole string and both the numerator and denomina- tor go to zero, by the argument in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [89], the ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' χM C ) approaches a nonzero constant as the length of C increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' But distinguishing this non-zero constant from zero in finite-size numerics and experiment may be challenging (similar to the situation for χE C in the con- fined phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The behavior of the BFFM order parameters in various phases is summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Before proceeding, we note that our protocols to mea- sure the plaquette correlators and the BFFM order pa- rameter χE C work in the limit Ω/V ≪ 1, which is outside the window in which the ground state of Hamiltonian (6) is a QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, we explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III C that it is possible to dynamically prepare finite puddles of QSL re- gions even in the Ω/V ≪ 1 limit when the ground state is not a QSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our protocols can then be applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Two-point ˆSz correlator Consider two spins ˆSz r,µ and ˆSz r′,ν located on the sites r + eµ/2 and r′ + eν/2, where r and r′ are the centers of two up-pointing tetrahedra and µ, ν ∈ {0, 1, 2, 3} label the sites of the tetrahedra (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From the map- ping of spins to gauge theory, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (13) and (15), it can be seen that the two-point correlator of these two spins ⟨ ˆSz r,µ ˆSz r′,ν⟩ is the same as the two-point correlator of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The effective theory in the deconfined phase is the Maxwell electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the 3 + 1D continuum Maxwell electromagnetism, the correlator of the Carte- sian components of the electric field ˆer,i for i ∈ {x, y, z} can be expressed as Gaussian integral which evaluate to [48] ⟨ˆe0,iˆeR,j⟩0 ∝ 1 R4 Å 2RiRj R2 − δij ã , (65) where ⟨·⟩0 denotes expectation value with respect to the Maxwell action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The correlator of the electric field operators ˆer,µ for µ ∈ {0, 1, 2, 3} along the links of the diamond lattice are obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (65) by taking components of the Cartesian electric field along the vectors eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The result is ⟨ ˆSz r,µ ˆSz r′,ν⟩ = � k,l∈{x,y,z} (eµ)k(eν)l⟨ˆer,kˆer′,l⟩0, (66) In the confined phase (ice ferromagnet), which is pri- marily diagonal in the ˆSz basis, this correlator should Figure 14: Notation for the two-point ˆSz correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' r and r′ are the positions of the centers of the tetrahe- dra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' eµ are the vectors joining the center of an up- pointing tetrahedron to the centers of its neighboring down-pointing tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' approach a constant for large R = |r − r′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' On the other hand, in the TFP phase, which is primarily a product state in the ˆSx basis, this correlator decays at least ex- ponentially with R (see Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since ˆSz r,µ ˆSz r′,ν is a diagonal operator, its correlator can be measured exper- imentally by capturing snapshots of the Rydberg-atom array and averaging over them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' DISCUSSION In this work, we have presented a proposal to prepare and detect the deconfined phase of the U(1) gauge the- ory in 3+1 dimensions on a Rydberg atom simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We first showed that laser-driven neutral atoms trapped in a pyrochlore lattice using optical tweezer arrays naturally realise a U(1) quantum spin liquid as the ground state when the laser detuning lies in a specified window and the interactions between Rydberg atoms are restricted to nearest-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We then studied the effect of van der Waals interactions beyond nearest-neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In the classical limit obtained by dropping the Rabi frequency term, we showed that long-range interactions break the degeneracy to select an ice ferromagnet as the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We then studied the competition between the long-ranged interactions that prefer an ordered state and quantum fluctuations that prefer a QSL state, by cal- culating the energies in ansatz wavefunctions using per- turbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We found that, for Rabi frequencies greater than ΩC ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='44V , the ground state is a QSL within our approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When Ω is increased fur- ther, we argued that the QSL goes into a transverse- field-polarized state via a Higgs transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While we have focused on the ground state, we also commented on the effect of dynamical state preparation in deciding the nature of the prepared state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We then provided ex- perimental protocols for measuring the plaquette correla- tors, Bricmont-Fr¨olich-Fredenhagen-Marcu order param- eters, the monopole-monopole correlator, and the “elec- tric field” correlator that can distinguish a QSL phase from ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our ground state phase diagram is the result of an approximate calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While it is possible that the 20 Figure 15: A lattice made of corner-sharing tetrahedra different from the pyrochlore lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The lattice consists of ABAB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' stacking of the blue (A) and the orange (B) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A configuration satisfying n = 2 is shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' true phase diagram differs from what we found, we note that there are other knobs one can tune to get a de- sired phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Dressed states created from multi- ple Rydberg and possibly ground levels can be used to customize the interactions away from the isotropic 1/r6 form we considered in this paper [94–99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Designing a dressing scheme compatible with the symmetries of the pyrochlore lattice and exploring the resulting phase di- agrams is an interesting direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We also note that our proposal requires two Rydberg excita- tions per tetrahedron, meaning that it lies outside of the Rydberg-blockade regime and is therefore sensitive to im- perfections and thermal fluctuations in nearest-neighbor spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It will therefore be useful to extend our pro- posal to the blockade regime of one excitation per tetra- hedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' While previous numerical work on dimer models have required a nonzero RK potential (6-body term) to achieve this, it will be worthwhile to study if one can engineer long-range Rydberg interactions that stabilize a spin-liquid in the blockade regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' One can also look for other lattices that could real- ize a U(1) QSL ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' One such possibility is a lattice of corner-sharing tetrahedra where all up-pointing tetrahedra (and separately all down-pointing tetrahedra) form a hexagonal close-packed lattice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' If only nearest-neighbor interactions are considered be- tween atoms positioned on the sites of this lattice, then, by perturbation theory in Ω/V for a particular range of detunings, one gets ring exchange terms similar to the ones obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' II A, and the system maps onto a dimer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is not known if this dimer model is in the QSL phase when the RK potential is zero and long range van der Waals interactions are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Another open problem is to construct lattices where a dimer model can be realized within the blockade regime without the RK potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Next, we note that, formally, a distinction between the confined and deconfined phases exists only in the thermo- dynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Experimentally, there are two finiteness effects that can be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First, a realistic three dimensional Rydberg array will likely have a relatively small linear dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Some of the correlators presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV require asymptotic behavior in distance to dis- tinguish different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Second, as found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [81] and mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III C, a finite-time state prepara- tion scheme would generically prepare puddles of spin- liquid regions as opposed to an entire spin liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is therefore necessary to quantitatively study how the be- havior of the correlators is modified under these condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We also note that, to translate field-theory observables into microscopic variables, we relied on the perturbative limit of small Ω/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However in the phase diagram that we found, the region where the spin liquid is a ground state does not satisfy Ω/V ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Understanding how the field-theory operators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' plaquette, monopole, and electric-field operators) get renormalized away from the perturbative limit is important both from fundamental and practical standpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our work is a proposal to prepare a gapless U(1) spin liquid using unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' An interesting research di- rection would be to come up with schemes that also use projective measurements to expedite the state prepara- tion along the lines of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [44, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' One can also ex- plore how other exotic phases of matter such as fractons and 3+1D topological order can potentially be realized on a Rydberg simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Nikita Astrakhantsev, Peter Lunts, Nathan Schine, Alexander Schuckert, Dayal Singh, and Ashvin Vishwanath for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' were supported by NSF DMR-2037158, US-ARO Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='W911NF1310172, and Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' were supported in part by AFOSR, NSF QLCI (award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' OMA-2120757), DoE QSA, DoE ASCR Ac- celerated Research in Quantum Computing program (award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' DE-SC0020312), the DoE ASCR Quantum Testbed Pathfinder program (award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' DE-SC0019040), NSF PFCQC program, ARO MURI, AFOSR MURI, and DARPA SAVaNT ADVENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Anderson, Resonating valence bonds: A new kind of insulator?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Bulletin 8, 153 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Gorshkov, Asymmetric blockade and multi- qubit gates via dipole-dipole interactions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 127, 120501 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [100] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Lavasani, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Luo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Vijay, Monitored quan- tum dynamics and the kitaev spin liquid, arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='02877 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Appendix A: Gauge mean field theory In this appendix, we first provide details of the gauge mean field theory calculation sketched in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III B 1, with a focus on capturing the Higgs transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Then, we attempt to use the same technique in the small-Ω limit to obtain the confinement-deconfinement transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We find that, in this limit, the technique is fraught with a serious limitation stemming from neglecting gauge fluc- tuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (44) of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III B 1 and performing the mean-field decoupling, we get ˆHMF = ˆHΦ + ˆHs + ˆHc, where ˆHΦ =V 2 � r∈A,B ˆQ2 r − Ω 2 � (r∈A),µ ÄˆΦ† r ˆΦr+eµ �ˆs+ r,µ � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ä , ˆHs = − Ω 2 � (r∈A),µ ĨˆΦ† r ˆΦr+eµ ∂ ˆs+ r,µ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ä + � (r∈A),µ ˆsz r,µ � (r′∈A),ν �Vµν(r − r′) �ˆsz r′,ν �� , ˆHc =Ω 2 � (r∈A),µ ĨˆΦ† r ˆΦr+eµ ∂ �ˆs+ r,µ � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ä − 1 2 � (r∈A),µ ˆsz r,µ � (r′∈A),ν �Vµν(r − r′) �ˆsz r′,ν �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A1) ˆHc is a constant, and Vµν(r − r′) was de- fined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆHs above is of the form − � (r∈A),µ �hx r,µˆsx r,µ + hz r,µˆsz r,µ �, where hx r,µ = Ω ¨ˆΦ† r ˆΦr+eµ ∂ , hz r,µ = − � (r′∈A),ν �Vµν(r − r′) �ˆsz r′,ν �� , (A2) and ¨ˆΦ† r ˆΦr+eµ ∂ is calculated in the ground state of ˆHΦ, which in turn depends on ⟨ˆs+⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (We have implicitly as- sumed here that ¨ˆΦ† r ˆΦr+eµ ∂ is real, which we will show can be assumed self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') This implies that, in the ground state, �ˆsi r,µ � = hi r,µ 2|hr,µ| for i = x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A3) Our goal is to self-consistently minimize the ground-state energy of the mean-field Hamiltonian subject to the con- straints in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (42) and (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We showed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A 2 that the ordered ground state at Ω = 0 has momentum 24 k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Also, the TFP state in the large-Ω limit is a k = 0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So we start with a mean-field ansatz with full translation symmetry (similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [80]): �s+ r,µ � = 1 2 cos θ, �sz r,µ � = 1 2εµ sin θ, (A4) where εµ = 1, 1, −1, −1 for µ = 0, 1, 2, 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' To solve the matter sector, it is convenient to deal with the Lagrangian instead of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The imaginary- time Lagrangian for the matter sector is L = 1 2V � r∈A,B |(∂τ − ivr)Φr|2 − Ω cos θ 4 � (r∈A),µ (Φ∗ rΦr+eµeiar,µ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') − i � r∈A,B � ηrvr �� µ sz r+ηreµ/2 � + ˜λr(|Φr|2 − 1) � , (A5) where the Lagrange multiplier ˜λr (which gets integrated over) enforces the constraint |Φr|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The Lagrange multiplier vr enforces the constraint (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' To zeroth or- der, we ignore the gauge fluctuation ar,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The matter Lagrangian alone, despite being quadratic in the rotor variables, is nevertheless interacting because a quadratic term in rotor operators is nonlinear in terms of canonical bosons (in other words, it is a cosine term in the phase of the rotor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') In order to make progress, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [80] as- sumes that, at the saddle point, ˜λr takes on a spatially uniform and purely imaginary value iλ, and also implic- itly assumes that vr is 0 at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Here, we will follow suit while acknowledging that these approxi- mations are uncontrolled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Making these simplifications, we obtain L = 1 2V � r |∂τΦr|2 − Ω cos θ 4 � (r∈A),µ �Φ∗ rΦr+eµ + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='� + λ � r (|Φr|2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A6) The constraints now simplify to �Φ† rΦr � = 1, (A7) hx = Ω �Φ† rΦr+eµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A8) Now, we have a quadratic Lagrangian, which we solve by Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Our Fourier transformation convention is (for α ∈ {A, B}) Φr,α(τ) = T � ωn � k∈BZ Φk,α(ωn)ei(k·r−ωnτ), (A9) where T is the temperature, ωn are Matsubara frequen- cies and we eventually take the limit T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A6) becomes L = T � k,ωn �Φ∗ k,A(ωn) Φ∗ k,B(ωn)� G−1 k (ωn) ÅΦk,A(ωn) Φk,B(ωn) ã , (A10) where G−1 k (ωn) = Ç ω2 n 2V + λ − Ω cos θ 4 fk − Ω cos θ 4 f ∗ k ω2 n 2V + λ å .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A11) Here, fk = 1 + e−ik1 + e−ik2 + e−ik3, (A12) where k ≡ k1b1 + k2b2 + k3b3, and b1, b2 and b3 are reciprocal lattice vectors of the FCC lattice satisfying ai · bj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Upon inverting the above matrix, we find that the eigenvalues of Gk(ωn) are 2V ω2n+(ω± k (λ,θ)) 2 , where the dis- persion of the two bosonic bands is ω± k (λ, θ) = 2V Å λ ± Ω cos θ 4 |fk| ã .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A13) As long as the spinon dispersion is gapped, spinons will not condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' From the dispersion above, we see that the dispersion becomes gapless when λ = Ω cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, as we will see below, for fixed θ and Ω, λ is determined by the constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Therefore the condition λ = Ω cos θ is met for a specific Ω = ΩMF H , which we will calculate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Before that, will go through a few intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First, the matrix form of Gk(ωn) is (assuming Ω > 0) Gk(ωn) = V Ñ 1 ω2n+(ω+ k )2 + 1 ω2n+(ω− k )2 gk � 1 ω2n+(ω+ k )2 − 1 ω2n+(ω− k )2 � g∗ k � 1 ω2 n+(ω+ k )2 − 1 ω2 n+(ω− k )2 � 1 ω2 n+(ω+ k )2 + 1 ω2 n+(ω− k )2 é , (A14) where gk = ® − fk |fk| when 0 ≤ θ < π/2, 0 when θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A15) With the Green’s function in hand, we are now ready 25 to impose the constraints, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First, we calculate equal-time correlation functions of Φ (by performing the Matsubara sum on the Green’s function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using these, the constraints in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A8) become, respectively, F1(λ, θ) ≡ V 2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' � k Ç 1 ��ω+ k �� + 1 ��ω− k �� å = 1, (A16) ΩF2(λ, θ) ≡ Ω V 2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' � k gk Ç 1 ��ω− k �� − 1 ��ω+ k �� å = hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A17) Next, by imposing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A3) with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A2), we get hz = −B sin θ 2 , where B = sin θ 2 � (r′∈A),ν V0,ν(−r′)εν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A18) For a given θ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A16) determines λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We see that there are three self-consistent solutions for θ: θ = � � � � � 0, π/2, cos−1 Ä 2ΩF2(λ,θ) B ä .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A19) Within gMFT (gauge mean field theory), these three so- lutions correspond to a QSL, a “Coulomb ferromagnet” (spin liquid with nonzero ice ferromagnetic order param- eter), and an ice ferromagnet, respectively [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For a fixed parameter Ω, the true solution depends on which of the three solutions above has lower energy with respect to the mean-field Hamiltonian (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Suppose that, for large enough Ω, one is in the QSL phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', θ = 0 and ¨ˆΦr ∂ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Now, the bosons will condense when their dispersion becomes gapless, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', λ = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using constraint (A16), we find that this transition point is ΩH MF ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='7V , as also found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For Ω > ΩH MF, the ground state is in the TFP phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Having identified the Higgs transition point, we now attempt to identify the confinement-deconfinement tran- sition for low Ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=', find Ω at which θ = 0 becomes the lowest-energy saddle-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A1), we get the following expression for the mean-field energy: EMF = K − Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Å 2ΩF2(λ, θ) cos θ + B 2 sin2 θ ã , (A20) where K is the total kinetic energy of the bosons and can be calculated to be K = 1 2 � k �ω+ k + ω− k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A21) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 16, we plot the energy EMF for θ = 0 (QSL) and θ = π/2 (ice ferromagnet), and find a transition at Ω ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='13V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (The third solution for θ becomes the lowest- energy solution only in a minuscule window around Ω ≈ Figure 16: The energy per unit cell (in units of V ) of saddle points θ = 0 (QSL) and θ = π/2 (ice ferromagnet) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (A20) up to an overall additive constant that is the same for θ = 0 and θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' We also plot − Ω2 4V 2 arising from trivial spin-flip pairs: this plot almost overlaps with the energy of the θ = 0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='13V , so we ignore it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=') However, we will now argue that this result is misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In gMFT, the energy reduction in the QSL phase with respect to the ordered phase (ice ferromagnet) arises from the minimization of kinetic energy of the bosonic charges ˆΦr that are allowed to hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' When θ = 0, the hopping coefficient is maximized, while, for θ = π/2, the hopping coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, microscopically, this hopping corresponds to a single spin-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' A pair of spin-flips at the same site leads to a constant reduction of energy coming from second order perturbation theory, given by −Ω2Nu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content='/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' It is constant in the sense that this reduc- tion is obtained for any state including the QSL and the ice ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The mean-field calculation, however, unfairly assigns this reduction to the QSL but not to the ordered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In fact, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' 16, we have also plotted −Ω2/(4V ) (the factor of 1/4 can perhaps be attributed to using spin-1/2 and classical spins at the same time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' As can be seen, this plot almost completely overlaps with the energy of the QSL calculated within gMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' So it is clear that, within gMFT, the difference between the ener- gies of the QSL and the confined phase is quadratic in Ω to leading order even though we know from perturbation theory that the leading order term should be proportional to Ω6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Hence, gMFT cannot be used in the vicinity of the confinement-deconfinement transition unless gauge- fluctuations are properly taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Appendix B: Corrections in the measurement protocol of the plaquette X correlator In the protocol described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV A 1, we equated the expectation value of �12 i=1(2 ˆSx i ) with ˆXP ˆXP ′ in the ground state (more generally, the state prepared in ex- 26 periment) assuming that the state is supported entirely on the ice manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Corrections arise because this is satisfied only approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' However, we show in this appendix that these corrections are of sixth order in Ω/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Let |Ψg⟩ be the ground state of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' This im- plies that |Ψ0⟩ = ˆUS |Ψg⟩ is in the ice manifold, where ˆUS is the unitary operator that implements the Schrieffer- Wolff transformation (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' III A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First we show that the first order correction in Ω/V is zero in the limit Ω ≪ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The quantity measured in an experiment implementing the protocol of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV A 1, assuming that the experiment prepares the ground state, is Cexp X = ⟨Ψg|ei ˆ HYtY 12 � i=1 (2 ˆSz i )e−i ˆ HYtY|Ψg⟩ ≈ ⟨Ψ0| ˆUS ˆA ˆU † S|Ψ0⟩, where, ˆA = �12 i=1 2 ˆSx i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Now, ˆUS ˆA ˆU † S = ˆA + î ˆS, ˆA ó + 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' î ˆS, î ˆS, ˆA óó + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' î ˆS, î ˆS, î ˆS, ˆA óóó + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (B1) Since |Ψ0⟩ is in the ice manifold, the expectation value of the first term above is the required plaquette X cor- relator, ⟨Ψ0| ˆA|Ψ0⟩ = ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' IV A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The expectation value of the remaining terms is the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' ˆS can be written as �∞ i=1 ˆSi where ˆSi is of order (Ω/V )i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' First, we will show that the first order error, E1 = ⟨Ψ0| î ˆS1, ˆA ó |Ψ0⟩, is zero using the ex- pressions for ˆS up to first order in Ω/V from Appendix B of [40]: ˆS1 = ˆP ˆHΩ ˆD − ˆD ˆHΩ ˆP, where ˆD = − ˆ1 − ˆP ˆH0 − E0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (B2) Here, ˆP is the projector into the ice manifold, E0 is the energy of the ice manifold, and ˆH0 and ˆHΩ are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Plugging in the expression for ˆS1 in E1 gives E1 = ⟨ ˆP ˆHΩ ˆD ˆA − ˆA ˆP ˆHΩ ˆD − ˆD ˆHΩ ˆP ˆA + ˆA ˆD ˆHΩ ˆP⟩0 = ⟨Ψ0| ˆHΩ ˆD ˆA|Ψ0⟩ + ⟨Ψ0| ˆA ˆD ˆHΩ|Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (B3) We have used ˆD |Ψ0⟩ = 0 and ˆP |Ψ0⟩ = |Ψ0⟩ above, and the expectation values are taken in |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Since ˆA flips the 12 spins on P and P ′, while ˆHΩ flips only one spin, it is not possible to go back to the ice manifold after applying ˆHΩ ˆD ˆA on |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' Thus ⟨ ˆHΩ ˆD ˆA⟩0 = 0, implying E1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The corrections at order i involves expectation value in |Ψ0⟩ of operators that consist of i factors of ˆHΩ and one factor of ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' For example, the second-order error is E2 = ⟨Ψ0| î ˆS2, ˆA ó + 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' î ˆS1, î ˆS1, ˆA óó |Ψ0⟩ , (B4) where ˆS2 =P ˆHΩ ˆD ˆHΩ ˆD − ˆD ˆHΩ ˆD ˆHΩ ˆP − ˆP ˆHΩ ˆP ˆHΩ ˆD2 + ˆD2 ˆHΩ ˆP ˆHΩ ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (B5) It can be seen that, at order i = 1, 2, 3, 4, and 5, the error will be zero because ˆA would flip 12 spins, while i factors of ˆHΩ would flip only i spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' At sixth order, however, the error can be nonzero if |Ψ0⟩ has support on a configuration where one of the two plaquettes, say P ′, is flippable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In such a situation, ˆA will map P ′ to the complementary flippable configuration and will also flip the spins on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' The six factors of ˆHΩ can bring P back to the original configuration, giving a state within the ice manifold which could have nonzero overlap with |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' In conclusion, we find that the experimentally mea- sured correlator, Cexp X , is the same as the theoretically needed correlator, ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩, with corrections of order (Ω/V )6: Cexp X = ⟨Ψ0| ˆXP ˆXP ′|Ψ0⟩ + Θ �(Ω/V )6� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} +page_content=' (B6)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE3T4oBgHgl3EQfrArn/content/2301.04657v1.pdf'} diff --git a/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf b/ANE1T4oBgHgl3EQfowXh/content/2301.03325v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a0729031f9893b5fd6cfef688338a16567a23c3e --- /dev/null +++ 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The standard framework takes a query +plan and materializes the intermediate views, which incurs high +polynomial costs in both space and time, with the join operator be- +ing the culprit. In this paper, we propose a new change propagation +framework without joins, thus naturally avoiding this polynomial +blowup. Meanwhile, we show that the new framework still supports +constant-delay enumeration of both the deltas and the full query +results, the same as in the standard framework. Furthermore, we +provide a quantitative analysis of its update cost, which not only +recovers many recent theoretical results on the problem, but also +yields an effective approach to optimizing the query plan. The new +framework is also easy to be integrated into an existing stream- +ing database system. Experimental results show that our system +prototype, implemented using Flink DataStream API, significantly +outperforms other systems in terms of space, time, and latency. +1 +INTRODUCTION +We study the problem of query evaluation under updates, a.k.a. +incremental view maintenance. Given a query 𝑄, a database 𝐷, and +a sequence of updates, where each update is either the insertion or +deletion of a tuple, the goal is to maintain the query results 𝑄p𝐷q +continuously. More precisely, there are two modes to return the +updated𝑄p𝐷q to the user (an end user or an upper-level application): +full enumeration and delta enumeration. The former is pull-based, +i.e., the system returns 𝑄p𝐷q passively upon request of the user; +while in the latter case, we push the delta Δ𝑄p𝐷,𝑡q, i.e., the change +to 𝑄p𝐷q caused by the insertion/deletion of 𝑡, to the user after each +update 𝑡. These two modes are applicable to different scenarios. +Full enumeration cannot be done too frequently if 𝑄p𝐷q is large, +and it may miss some ephemeral events in between two requests. +Delta enumeration offers real-time responses with low latency, but +it requires the user to have the ability to “consume” the deltas in +a timely fashion. It can be considered as a stream-in-stream-out +operator, where the input is a stream of updates to the base tables, +while the output is a stream of updates to the query result (i.e., +a stream of deltas). If the user wishes to always have a complete +and accurate 𝑄p𝐷q, it has to maintain 𝑄p𝐷q and update it with the +deltas as they are received. If approximation is acceptable, some +more efficient streaming algorithms can be used instead. +Change propagation. Change propagation [12, 28, 35] is a widely +used framework in database systems for solving this problem. It +can be instantiated with any query plan, which is a tree where the +leaves are the base relations and each internal node is a relational +operator. At each internal node, it maintains the results of the sub- +query corresponding to the subtree at this internal node, which +is often called a materialized view. Figure 1(a) shows a particular +𝑉4 “ 𝑉2 1 𝑉3 +𝑉1 “ 𝑅2 1 𝑅3 +𝑉3 “ 𝜋𝑥4𝑅4 +𝑅4 +𝑉2 “ 𝑅1 1 𝑉1 +𝑅2 +𝑅3 +𝑅1 +(a) Old plan +𝑉2 “ 𝑅1 1 𝑅2 1 𝑅3 1 𝑉1 +𝑅1 +𝑅2 +𝑅3 +𝑉1 “ 𝜋𝑥4𝑅4 +(b) Another old plan +X : 𝑉𝑠pr𝑥3sq +𝜋 : 𝑉𝑝p𝑅2q +𝜋 : 𝑉𝑝p𝑅3q +˙ : 𝑉𝑠p𝑅2q +˙ : 𝑉𝑠p𝑅3q +𝜋 : 𝑉𝑝p𝑅1q +𝑅2 +𝑅3 +𝜋 : 𝑉𝑝p𝑅4q +𝑉𝑠p𝑅1q “ 𝑅1 +𝑉𝑠p𝑅4q “ 𝑅4 +𝑅1 +𝑅2 +𝑅3 +𝑅4 +r𝑥3s +(c) Our new plan +Figure 1: For 𝑄 “ 𝜋𝑥1,𝑥2,𝑥3,𝑥4𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q 1 𝑅3p𝑥3,𝑥4q +1 𝑅4p𝑥4,𝑥5q, 1(a) and 1(b) are two plans under the standard +change propagation framework and 1(c) is our new plan. +query plan for the query 4-Hop query from benchmark [31] +𝑄 :“ 𝜋𝑥1,𝑥2,𝑥3,𝑥4𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q 1 𝑅3p𝑥3,𝑥4q 1 𝑅4p𝑥4,𝑥5q. +Under the standard change propagation framework, we maintain +four materialized views𝑉1,𝑉2,𝑉3,𝑉4 “ 𝑄 (if only delta enumeration +is needed, then 𝑉4 need not be maintained). When a tuple 𝑡 is +inserted or deleted in a relation, say 𝑅1, it follows the leaf-to-root +path to propagate the deltas to the root. More precisely, it first +computes Δ𝑉2 “ Δ𝑅1 1 𝑉1 “ 𝑡 1 𝑉1, then computes Δ𝑄 “ Δ𝑉4 “ +Δ𝑉2 1 𝑉3. Note that with the help of the materialized views, it +avoids re-computing some of the sub-queries during updates. +However, the penalty is space: both𝑉1 and𝑉2 can have quadratic +size in the worst case [5]. To avoid space blowup, one can use a +different query plan, say, the one shown in Figure 1(b). This query +plan does not have any materialized views (except 𝑉1 “ 𝜋𝑥4𝑅4, +which has at most linear size), but it has to compute a multi-way +join, e.g., 𝑅1 1 𝑅2 1 𝑅3 1 𝑡 upon each update in 𝑅4, which could +take quadratic time. Making things worse, this quadratic blowup +exacerbates for queries involving more relations [5]. +Prior work has designed advanced techniques to address this +space or time blowup. The Dynamic Yannakakis algorithm [21–23] +has linear space and linear update time while supporting constant- +delay enumeration for free-connex queries1; the update time fur- +ther reduces to 𝑂p1q amortized2 for q-hierarchical queries. Concur- +rently, Berkholz et al. [8] designed a different algorithm for the +q-hierarchical case with the same space/time guarantees. However, +these algorithms have not been integrated into any full-fledged +database or data warehouse products, possibly due to the complica- +tions of the techniques and the use of non-standard operations not +routinely found in existing database systems. +Change propagation without joins. The main contribution of +this paper is to achieve (and improve for certain classes of queries +1All technical terms in the introduction are formally defined in Section 3. +2All update time bounds are amortized in this paper. +arXiv:2301.04003v1 [cs.DB] 10 Jan 2023 + +and/or update sequences) the results above, but still under the +standard change propagation framework. Our observation is that +the only relational operator that may cause a super-linear blowup +is join. Thus, if the query plan has no joins, then both space and +update time will be at most linear. To avoid joins, our high-level +strategy is to replace each join in the query plan by a semi-join +(or an intersection) plus a projection. However, not every query +plan is amenable to this replacement strategy. The key technical +contribution of this paper, therefore, is the construction of such a +query plan for every free-connex conjunctive query. For example, +such a join-free query plan for the earlier query is shown in Figure +1(c), which will be elaborated in Section 4. +Since our query plan has no joins, linear space and linear up- +date time follow straightforwardly. Still, two technical challenges +remain: (1) how to support constant-delay enumeration, and (2) +how to achieve an update time better than linear. (1) is trivial under +a traditional query plan where the root corresponds to the query +results 𝑄p𝐷q. Since our query plan is join-free, no node in the plan +corresponds to 𝑄p𝐷q. Instead, our query plan can be considered +as a compact, linear-size representation of a polynomially sized +𝑄p𝐷q. By borrowing ideas from the static case [6], we show how to +enumerate 𝑄p𝐷q with constant delay, by appropriately traversing +this compact representation. Supporting constant-delay enumera- +tion of the delta Δ𝑄p𝐷,𝑡q, on the other hand, is quite different from +the static case, and we need new techniques which exploit some +important properties of our query plan. +To address issue (2), Wang and Yi [37] introduced the notion of +enclosureness 𝜆 of an update sequence, which captures the hardness +of the update sequence. It is linear in the worst case, but is often a +constant in many common cases, such as any first-in-first-out (FIFO) +update sequence. They also designed an algorithm with update cost +𝑂p𝜆q for foreign-key acyclic queries. Such queries are relatively easy +to handle since their result size is at most linear, so they are immune +to the polynomial blowup problem caused by non-key joins, such +as free-connex queries. Indeed, we show (c.f. Theorem 6.2) that +there is a simple free-connex query for which it is impossible to +achieve 𝑂p|𝐷|1{2´𝜀q update time even over FIFO update sequences, +which implies that the previous definition of 𝜆 is not achievable +for free-connex queries. Nevertheless, we show that, after a simple +relaxation of the definition, 𝜆 is still an appropriate measure of the +update complexity; in particular, we show that change propagation +under our query plan achieves 𝑂p𝜆q update time for every free- +connex query under the new definition. To further illustrate the +usefulness of our new definition of 𝜆, we show that for certain +queries (such as q-hierarchical queries) and/or update sequences +(such as FIFO or insertion-only), 𝜆 is indeed a small constant. For +general queries, 𝜆 also provides guidance on what would constitute +a good query plan for change propagation. +Our results. Specifically, this paper achieves the following results: +(1) We show how to construct a change propagation query plan +without joins for any free-connex conjunctive query, such that +the space needed by the query plan is linear and the update time +is 𝑂p𝜆q, for an appropriately defined notion of enclosureness 𝜆 +of the update sequence. +(2) We show how to support constant-delay enumeration of both +full query results and each delta in our query plan. +(3) We show that 𝜆 is a constant for certain classes of conjunctive +queries (such as q-hierarchical queries) and/or special update +sequences (such as FIFO or insertion-only). These results not +only recover the prior known result of [8, 21] on q-hierarchical +queries, but also extend it to cover many other cases commonly +encountered in practice. +(4) We show how our framework can handle various extensions +such as selections, aggregations, and non-free-connex queries. +(5) We demonstrate the practicality of our new framework by im- +plementing it on top of Flink and comparing it with state-of- +the-art view maintenance and SQL-over-stream systems. +2 +RELATED WORK +Our new change propagation framework is inspired by several lines +of research. In the static case, the classical Yannakakis algorithm +[38] has runtime 𝑂p|𝐷| ` |𝑄p𝐷q|q for every free-connex query. It +consists of two stages. The first stage uses a series of semi-joins +to remove all the dangling tuples in 𝑂p|𝐷|q time, and the second +stage performs pairwise joins to compute 𝑄p𝐷q in 𝑂p|𝑄p𝐷q|q time. +The Dynamic Yannakakis algorithm [21] extends the algorithm +to the dynamic case, but it deviates from the change propagation +framework, making it harder to integrate into existing database +systems. Our algorithm can also be viewed as a dynamic version +of the Yannakakis algorithm, but it strictly follows the standard +change propagation framework while achieving a better runtime. +The Dynamic Yannakakis algorithm has an update cost of 𝑂p|𝐷|q +for free-connex queries, while our algorithm achieves 𝑂p𝜆q up- +date time, where 𝜆 is the enclosureness of the update sequence. We +have 𝜆 ď |𝐷| for all update sequences, while the former is usually +much smaller on real-world update sequences. Furthermore, Dy- +namic Yannakakis achieves𝑂p1q update time only for q-hierarchical +queries, while our algorithm also achieves 𝑂p1q update time for +non-q-hierarchical queries if the update sequences enjoy some spe- +cial properties, such as first-in-first-out or insertion-only (formally +defined in Section 6.1). The gap between Dynamic Yannakakis and +our algorithm can be as large as Θp|𝐷|q on some non-q-hierarchical +queries (see Example 6.12). +Bagan et al. [6] observe that, in the static case, the second stage +of the Yannakakis algorithm can be enhanced to support constant- +delay enumeration. We adapt their ideas to support enumeration +in the dynamic case for our query plan. However, as there is no +notion of delta in the static case, we need some new ideas to support +delta enumeration with constant delay, which non-trivially relies +on some nice features of our query plan. +Kara et al. [27] show that it is possible to increase the enumera- +tion delay in exchange for faster update time, on hierarchical (but +non-q-hierarchical) queries. We have not considered this trade-off, +as we believe the constant delay is important, and our update cost +𝜆 is low enough for most queries and update sequences already. +Furthermore, their trade-off only applies to full enumeration, not +delta enumeration. Nevertheless, for cases where 𝜆 is high, it would +be an interesting direction to explore such a trade-off. +In the standard change propagation framework, a single update +to a base relation may incur many changes in the intermediate +views. Higher-Order Incremental View Maintenance (HIVM) [4] +has been proposed to remedy this problem. It takes the changes to a +2 + +view as another query (delta query) and maintains this delta query +recursively. HIVM improves upon IVM for many complex queries +in practice, and it can also extend to accelerate several machine +learning tasks [33, 34], but there is no theoretical guarantee on its +update time. Furthermore, HIVM still uses super-linear space. +The problem is also related to stream joins. In particular, a cash- +register stream corresponds to an insertion-only update sequence, +while a turnstile stream is an update sequence with arbitrary inser- +tions and deletions. The sliding-window stream model is a special +case of a FIFO update sequence. Most stream processing systems +like Flink [9] and Trill [11] use standard change propagation for +multi-way stream joins, which we will compare against in Section +8. Some specialized systems are designed for two-way stream joins +[14, 17, 25, 30, 36], but they do not extend to multi-way joins. +3 +PRELIMINARIES +3.1 +Problem Definition +Conjunctive queries. We focus on conjunctive queries (CQ) of the +following form: +𝑄 :“ 𝜋y p𝑅1p𝑒1q 1 𝑅2p𝑒2q 1 ¨ ¨ ¨ 1 𝑅𝑛p𝑒𝑛qq, +(1) +where each 𝑅𝑖 is a relation with a set of attributes/variables 𝑒𝑖, +𝑖 “ 1, . . . ,𝑛. Each tuple 𝑡 P 𝑅𝑖 assigns a value to each attribute in +𝑒𝑖. For any 𝑥 P 𝑒𝑖, 𝑡r𝑥s “ 𝜋𝑥𝑡 denotes the value of 𝑡 on attribute 𝑥. +Similarly, for a subset of attributes 𝑒 Ď 𝑒𝑖, 𝑡r𝑒s “ 𝜋𝑒𝑡 denotes the +tuple formed by the values of 𝑡 on the attributes in 𝑒. +Let V “ 𝑒1 Y ¨ ¨ ¨ Y 𝑒𝑛 be the set of all attributes in the query. +We call y Ď V the output attributes, while ¯y “ V ´ y are the non- +output attributes, also known as the existential variables. If y “ V, +such a query is known as a full join query; otherwise, it is said to +be join-project query. For simplicity, we assume that each 𝑅𝑖 in 𝑄 +is distinct, i.e., the query does not have self-joins. Nevertheless, +self-joins can be taken care of easily: Suppose a relation 𝑅 appears +twice in the query (with different attribute renamings). Then we +consider them as two identical copies of 𝑅, and for any update to 𝑅, +we apply the update to both copies of 𝑅. +Given a database 𝐷, we write 𝑄p𝐷q for the query results of 𝑄 +on 𝐷. We use 𝑄p𝐷 ˙ 𝑡q to denote the query results that depend on +a given tuple 𝑡, and call 𝑄p𝐷 ˙ 𝑡q the query results witnessed by 𝑡. +Such a witness query will be frequently used in this paper. Given a +query 𝑄 in the form of (1) and a tuple 𝑡 P 𝑅𝑖, it is clear that +𝑄p𝐷 ˙ 𝑡q “ 𝜋y p𝑅1 1 ¨ ¨ ¨ 1 𝑅𝑖´1 1 t𝑡u 1 𝑅𝑖`1 1 ¨ ¨ ¨ 1 𝑅𝑛q . +Note that for a full join CQ, we have 𝑄p𝐷 ˙ 𝑡q “ 𝑄p𝐷 ` 𝑡q ˙ 𝑡; for +join-project queries, 𝑡 itself may not appear in 𝑄p𝐷 ˙ 𝑡q due to the +projection on y. When analyzing the costs of algorithms, we adopt +the notion of data complexity, i.e., the size of the query 𝑄 is taken +as a constant while |𝐷| is an asymptotic parameter. +Semi-joins. The semi-join 𝑅𝑖p𝑥𝑖q ˙ 𝑅𝑗p𝑥𝑗q is defined as +𝑅𝑖p𝑥𝑖q ˙ 𝑅𝑗p𝑥𝑗q “ t𝑡|𝑡 P 𝜋𝑥𝑖𝑅𝑖 1 𝑅𝑗u. +Updates and Deltas. An update to a database 𝐷 is either the inser- +tion or deletion of a tuple 𝑡 in some relation 𝑅𝑖 of 𝐷. In this paper, +we adopt set semantics. We denote 𝐷 ` 𝑡 as the database after +inserting 𝑡 and 𝐷 ´ 𝑡 as the database after deleting 𝑡. In particular, +this means that if 𝑅𝑖 already contains 𝑡, then inserting 𝑡 into 𝑅𝑖 will +not change 𝑅𝑖; if 𝑅𝑖 does not contain 𝑡, deleting 𝑡 from 𝑅𝑖 has no +effect, either. We ignore these non-effective updates. +The delta of an update to 𝑄 is defined as +Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷 ` 𝑡q ´ 𝑄p𝐷q +in case of the insertion of 𝑡 and +Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷q ´ 𝑄p𝐷 ´ 𝑡q +in the case of deletion. For a full join query, Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷 ˙ 𝑡q. +For join-project queries, Δ𝑄p𝐷,𝑡q Ď 𝑄p𝐷 ˙ 𝑡q. In particular, it is +possible to have Δ𝑄p𝐷,𝑡q “ H even if 𝑄p𝐷 ˙ 𝑡q ‰ H. +We target constant delay [6] for both full and delta enumeration, +i.e., the time between the start of the enumeration process to the first +tuple in 𝑄p𝐷q (or Δ𝑄p𝐷,𝑡q), the time between any consecutive pair +of tuples, and the time between the last tuple and the termination +of the enumeration process should all be bounded by a constant. +3.2 +Classification of CQs +Acyclic queries. The acyclicity of a CQ 𝑄 is defined by the acyclic- +ity of the hypergraph pV, t𝑒1, . . . ,𝑒𝑛uq. More precisely, 𝑄 is acyclic +if there exists a join tree T, whose nodes are the relations in 𝑄 such +that, for each attribute 𝑥 P V, all nodes of T containing 𝑥 form a +connected component of T. For example, Figures 2(a) and 2(b) are +two possible join trees for the query 𝑄1 :“ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q. +We will often not distinguish between a node in T and the relation +it represents, or its set of attributes. +In this paper, we use an equivalent definition based on generalized +relations [13, 21]. Different from previous definition of generalized +relation, it now can be a proper subset of any 𝑒𝑖. We can show that +the following is an equivalent definition of acyclic queries3: +Definition 3.1 (Acyclic queries). A CQ 𝑄 is acyclic if there exists +a rooted join tree T satisfying the following properties: +(1) each input relation in𝑄 corresponds to a unique node in T, each +leaf of T corresponds to an input relation, and each internal +node in T corresponds to either an input relation in 𝑄 or one +of its generalized relations (some generalized relations may not +appear in T); +(2) for each attribute 𝑥, all nodes of T containing 𝑥 form a con- +nected component of T; +(3) a node corresponding to a generalized relation must appear +above any node corresponding to an input relation; and +(4) if 𝑒 is the parent of 𝑒1 in T and 𝑒 is a generalized relation, 𝑒 Ď 𝑒1. +An example is given in Figure 2(c). In a generalized join tree T, +we use 𝑟 to denote the root, and T𝑒 for the subtree rooted at node 𝑒, +C𝑒 for the set of children of node 𝑒 and 𝑝p𝑒q for the parent of node +𝑒. Let keyp𝑒q “ 𝑒 X 𝑝p𝑒q be the join key between node 𝑒 and 𝑝p𝑒q. +The height of T is defined as the maximum number of relations on +any leaf-to-root path, without counting generalized relations. +Free-connex queries. A CQ 𝑄 is free-connex if the hypergraphs +pV, t𝑒1, . . . ,𝑒𝑛uq and pV, t𝑒1, . . . ,𝑒𝑛, yuq are both acyclic. By defi- +nition, any free-connex query must by acyclic, and an acyclic full +join query must be free-connex. For our development, we need the +following equivalent definition of free-connex queries: +3Proof of equivalence is given in the Appendix A. +3 + +𝑅2p𝑥2,𝑥3q +𝑅1p𝑥1,𝑥2q +(a) T1 +𝑅1p𝑥1,𝑥2q +𝑅2p𝑥2,𝑥3q +(b) T2 +𝑅2p𝑥2,𝑥3q +𝑅1p𝑥1,𝑥2q +r𝑥2s +(c) T3 +Figure 2: Three (generalized) join trees for 𝑄1 “ 𝑅1p𝑥1,𝑥2q 1 +𝑅2p𝑥2,𝑥3q. In 2(c), node r𝑥2s is a generalized relation with one +attribute 𝑥2. The height of T1, T2 is 2 and that of T3 is 1. +Definition 3.2 (Free-connex queries). A CQ 𝑄 is free-connex if it +has a generalized join tree T, such that 𝑟 Ď y, and for every 𝑥1 P y +and every 𝑥2 P V ´ y, topp𝑥2q is not an ancestor of topp𝑥1q in T, +where topp𝑥q is the highest node in T that contains 𝑥. Such a T is +called a free-connex join tree. +For example, for the query 𝑄1 +1 :“ 𝜋𝑥2𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q, all +three join trees in Figure 2 are valid free-connex join trees. If the +output attribute is 𝑥1, then only Figure 2(a) is a valid free-connex +join tree (so it does not have a height-1 free-connex join tree). If the +output attributes are p𝑥1,𝑥3q, then the query is not free-connex. +Q-hierarchical queries. A more restricted subclass of free-connex +queries is q-hierarchical query. Let E𝑥 denote the set of relations +containing attribute 𝑥. +Definition 3.3 (Q-hierarchical queries). A CQ 𝑄 is q-hierarchical if +(1) for every pair of attributes 𝑥1,𝑥2, either E𝑥1 Ď E𝑥2 or E𝑥2 Ď E𝑥1 +or E𝑥1 X E𝑥2 “ H; and (2) for every pair of attributes 𝑥1,𝑥2, if +𝑥1 P y and E𝑥1 Ĺ E𝑥2, then 𝑥2 P y. +These classifications capture the hardness of evaluation or enu- +meration for a CQ. Firstly, a full join query can be evaluated in linear +time in terms of input and output size if and only if it is acyclic; +for join-project CQs, this complete class extends to free-connex +queries. Furthermore, free-connex and q-hierarchical CQs have +played important roles in query enumeration. [6] showed that in +static settings, constant-delay enumeration after a linear-time pre- +processing step is possible for a CQ if and only if it is free-connex. +Berkholz et al. [8] showed that in dynamic settings, constant-delay +enumeration is possible for a CQ from a data structure that can be +updated in constant time if and only if it is q-hierarchical. +4 +CHANGE PROPAGATION WITHOUT JOINS +4.1 +A New Query Plan +Given a free-connex query𝑄, our new query plan is guided by a free- +connex (generalized) join tree T of 𝑄. We illustrate the construction +using the query in Figure 1 with the join tree highlighted in red +(note that the join tree is not unique). A normal query plan following +this join tree would compute a series of joins p𝑅1 1 𝑅2q 1 p𝑅3 1 +𝜋𝑥4𝑅4q. In our new query plan, we replace each join with a semi-join +followed by a projection. More precisely, we maintain two views +for each node 𝑒 P T, a semi-join view 𝑉𝑠p𝑅𝑒q and a projection view +𝑉𝑝p𝑅𝑒q, defined recursively as follows. +Every non-root node 𝑒 P T has a projection view +𝑉𝑝p𝑅𝑒q :“ 𝜋keyp𝑒q𝑉𝑠p𝑅𝑒q. +(2) +Noted that the root node does not have a projection view. +To define the semi-join view 𝑉𝑠p𝑅𝑒q, we distinguish three cases. +Algorithm 1: PlanGenerationp𝑄,𝑇q +Input +:A generalized join tree 𝑇 for query 𝑄; +Output:A new query plan 𝑇 for 𝑄; +1 foreach node 𝑒 in a postorder traversal of 𝑇 do +2 +Replace node 𝑒 with 𝑉𝑠p𝑅𝑒q in 𝑇; +3 +if 𝑒 is not the root of 𝑇 then +4 +Add 𝑉𝑝p𝑅𝑒q between 𝑉𝑠p𝑅𝑒q and the parent of 𝑒; +5 Return 𝑇; +(i) If 𝑒 is a leaf, 𝑅𝑒 is an input relation, and 𝑉𝑠p𝑅𝑒q :“ 𝑅p𝑒q. +(ii) If 𝑒 is an internal node and 𝑅𝑒 is an input relation, then +𝑉𝑠p𝑅𝑒q :“ 𝑅𝑒 ˙ 𝑉𝑝p𝑅𝑒1q ˙ ¨ ¨ ¨ ˙ 𝑉𝑝p𝑅𝑒𝑘 q, +(3) +where C𝑒 “ t𝑒1, . . . ,𝑒𝑘u are the children of 𝑒. +(iii) If 𝑒 is an internal node that corresponds to a generalized virtual +relation 𝑅𝑒, since all the 𝑉𝑝p𝑅𝑒𝑖 q’s have the same attributes +keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 “ 𝑒 for every 𝑖 (by the last property in +Definition 3.1), (3) simplifies to an intersection: +𝑉𝑠p𝑅𝑒q :“ 𝑉𝑝p𝑅𝑒1q X ¨ ¨ ¨ X 𝑉𝑝p𝑅𝑒𝑘 q. +(4) +Our query plan simply connects these views together using the +formulae above. Algorithm 1 takes as input a generalized join tree, +and outputs a new query plan under our framework. +Figure 1(c) shows the new query plan for the example query. +Note that 𝑅2 and 𝑅4 fall into case (ii), while the root node r𝑥3s is +under case (iii). +As neither projection nor semi-join (including intersection as a +special case) enlarges the input relations, the following is straight- +forward: +Lemma 4.1. All views in our query plan have size 𝑂p|𝐷|q. +Example 4.2. Figure 3(a) shows the initial index built for the +query in Figure 1. For𝑅1 and𝑅4, both semi-join and projection views +are defined as themselves. 𝑉𝑠p𝑅2q contains tuples in 𝑅2 that can +join with 𝑉𝑝p𝑅1q, which include p2, 2q and p2, 4q. 𝑉𝑠p𝑅3q is defined +similarly including 4 tuples from 𝑅3. For the generalized node r𝑥3s, +we define the virtual relation 𝑅pr𝑥3sq “ 𝑉𝑝p𝑅2q Y 𝑉𝑝p𝑅3q. Only +tuple p4q belongs to 𝑅pr𝑥3sq, since every other tuple in 𝑅pr𝑥3sq fails +to join with 𝑉𝑝p𝑅2q and 𝑉𝑝p𝑅3q: their counters need to be 2. +4.2 +Change propagation +Change propagation using our new query plan can be done using +standard (actually, even simpler for certain operators) propagation +formulae [12]. For completeness, we briefly describe them below, +which are also needed to understand the algorithms in Section 5. +S-Update When there is an update to 𝑉𝑠p𝑅𝑒q for some 𝑒, we use +an S-Update to update 𝑉𝑝p𝑅𝑒q by formula (2). This can be done +in 𝑂p1q time by derivation counting [12], a standard technique to +propagate changes through a projection. Specifically, we associate a +counter countr𝑡1s for each tuple 𝑡1 P 𝑉𝑝p𝑅𝑒q that stores the number +of tuples 𝑡 P 𝑉𝑠p𝑅𝑒q such that 𝑡rkeyp𝑒qs “ 𝑡1. The detailed process, +which needs to distinguish between an insertion and a deletion, is +given in Algorithm 2. Note that for the algorithm to run in 𝑂p1q +time, we need a hash index on 𝑉𝑝p𝑅𝑒q. +P-Update Let 𝑒𝑖 be a child of 𝑒. When there is an update to some +𝑉𝑝p𝑅𝑒𝑖 q, we use a P-Update to update 𝑉𝑠p𝑅𝑒q by formula (3) in the +4 + +Algorithm 2: S-Updatep𝑒,𝑡q +Input +:An update 𝑡 from 𝑉𝑠p𝑅𝑒q; +Output:Updated 𝑉𝑝p𝑅𝑒q; +1 𝑡1 Ð 𝑡rkeyp𝑒qs; +2 if 𝑡 is an insertion into 𝑉𝑠p𝑅𝑒q then +3 +if 𝑡1 P 𝑉𝑝p𝑅𝑒q then countr𝑡1s Ð countr𝑡1s ` 1; +4 +else +5 +𝑉𝑝p𝑅𝑒q Ð 𝑉𝑝p𝑅𝑒q Y t𝑡1u, countr𝑡1s Ð 1, +P-Update(𝑝p𝑒q,𝑡1); +6 else +7 +if countr𝑡1s “ 1 then +8 +𝑉𝑝p𝑅𝑒q Ð 𝑉𝑝p𝑅𝑒q ´ t𝑡1u, P-Update(𝑝p𝑒q,𝑡1); +9 +else countr𝑡1s Ð countr𝑡1s ´ 1; +Algorithm 3: P-Update(𝑒,𝑡) +Input +:An update 𝑡 from 𝑉𝑝p𝑅𝑒𝑖 q for some 𝑒𝑖 P C𝑒; +Output:Updated 𝑉𝑠p𝑅𝑒q; +1 if 𝑡 is an insertion into 𝑉𝑝p𝑅𝑒𝑖 q then +2 +foreach 𝑡1 P 𝑅𝑒 with 𝑡1rkeyp𝑒𝑖qs “ 𝑡 do +3 +countr𝑡1s Ð countr𝑡1s ` 1; +4 +if countr𝑡1s “ |C𝑒| then +5 +𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q Y t𝑡1u, S-Updatep𝑒,𝑡1q; +6 else +7 +foreach 𝑡1 P 𝑅𝑒 with 𝑡1rkeyp𝑒𝑖qs “ 𝑡 do +8 +countr𝑡1s Ð countr𝑡1s ´ 1; +9 +if countr𝑡1s “ |𝐶𝑒| ´ 1 then +10 +𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q ´ t𝑡1u, S-Updatep𝑒,𝑡1q; +case where 𝑒 is an input relation or (4) in case 𝑒 is a generalized +relation. We consider the former case first; the latter case is similar. +The standard change propagation formula for a semi-join [19] +rewrites it as a join followed by a projection, e.g., 𝑅𝑒 ˙ 𝑅𝑒𝑖 :“ +𝜋𝑒p𝑅𝑒 1 𝑅𝑒𝑖 q. This defeats the whole purpose of avoiding joins. +However, observe that in our query plan, 𝑅𝑒𝑖 has already been +projected onto keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 Ď 𝑒 before the semi-join, thus this +allows a very simple and efficient way to maintain the whole multi- +way semi-join (3) as one operator, which can also be considered +as a “horizontal” version of derivation counting. More precisely, +we maintain a counter countr𝑡1s for every tuple 𝑡1 in 𝑅𝑒, storing +the number of child nodes 𝑒𝑖 P C𝑒 such that 𝑡1rkeyp𝑒𝑖qs P 𝑉𝑝p𝑅𝑒𝑖 q. +A tuple 𝑡1 appears in 𝑉𝑠p𝑅𝑒q if and only if countr𝑡1s “ |C𝑒|. The +algorithm is then immediate, as shown in Algorithm 3. We also +need a hash index (that needs to support 𝑒 X 𝑒𝑖 as the key for each +𝑒𝑖 P C𝑒) on 𝑅𝑒 so that each counter change can be done in 𝑂p1q +time. However, unlike the S-Update, a P-Update may take more +than constant time since multiple tuples may change their counters. +In fact, this is the only place where the update time blows up during +change propagation in our query plan. +R-Update. The last case is when there is an update in an input +relation 𝑅𝑒, we also need to update 𝑉𝑠p𝑅𝑒q by formula (3). We call +this an R-Update. The detailed procedure, given in Algorithm 4, +Algorithm 4: R-Update(𝑒,𝑡) +Input +:An update 𝑡 from an input relation 𝑅𝑒; +Output:Updated 𝑉𝑠p𝑅𝑒q; +1 if 𝑡 is an insertion into 𝑅𝑒 then +2 +countr𝑡s Ð 0; +3 +foreach 𝑒𝑖 P C𝑒 do +4 +if 𝑡rkeyp𝑒𝑖qs P 𝑉𝑝p𝑒𝑖q then +countr𝑡s Ð countr𝑡s ` 1; +5 +if countr𝑡s “ |C𝑒| then +6 +𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q Y t𝑡u, S-Updatep𝑒,𝑡q; +7 else +8 +if countr𝑡s “ |C𝑒| then +9 +𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q ´ t𝑡u, S-Updatep𝑒,𝑡q; +simply maintains the counters in 𝑅𝑒, and then 𝑉𝑠p𝑅𝑒q, in a straight- +forward manner. It is obvious that an R-Update takes 𝑂p1q time +(also using the hash index on 𝑉𝑝p𝑅𝑒𝑖 q). +Example 4.3. Figure 3(b) shows the index after inserting p1, 1q +into 𝑅1. This new tuple first triggers an insertion to 𝑉𝑝p𝑅1q, which +further increments counters of the three tuples in 𝑉𝑠p𝑅2q with +𝑥2 “ 1, which are then brought into 𝑉𝑠p𝑅2q. From here, the propa- +gation diverges into three paths. Tuple p1, 2q P 𝑅2 increments the +counter of p1q P 𝑉𝑝p𝑅2q but this propagation path stops here. Tuple +p1, 1q P 𝑉𝑠p𝑅2q first inserts a new tuple p1q to 𝑉𝑝p𝑅2q, which then +further increments the counter of tuple p1q in the root, bringing it to +𝑉𝑠pr𝑥3sq. Tuple p1, 4q P 𝑅2 increments the counter of p4q P 𝑉𝑝p𝑅2q +and the propagation stops. +Figure 3(c) shows the index after deleting p1, 1q from𝑅4. This dele- +tion first decrements the counter of tuple p1q P 𝑉𝑝p𝑅4q, removing it +from 𝑉𝑝p𝑅4q, and further decrements the counter of p1, 1q P 𝑉𝑠p𝑅3q, +removing it from𝑉𝑠p𝑅3q as well. Finally, the counter of p1q P 𝑉𝑝p𝑅3q +decreases from 2 to 1, and the propagation stops here. +Lemma 4.4. All projection and semi-join views in our query plan +can be updated in 𝑂p|𝐷|q time. +5 +ENUMERATION +5.1 +Full Result Enumeration +We first consider how to perform constant-delay enumeration of +𝑄p𝐷q from our query plan. We need the following lemma: +Lemma 5.1. For any node 𝑒, 𝑉𝑠p𝑅𝑒q “ 𝜋𝑒p1𝑒1PT𝑒 𝑅𝑒1q. +Proof of Lemma 5.1. We prove it by the induction on the height +of generalized join tree T. First, it holds for any leaf node 𝑒, since +𝑉𝑠p𝑅𝑒q “ 𝑅𝑒. We next consider an arbitrary internal node 𝑒. Let +𝐶𝑒 “ t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u be the set of children of node 𝑒. By hypothesis, +we assume this lemma holds for every 𝑒𝑖 P 𝐶𝑒, i.e. +𝑉𝑠p𝑅𝑒𝑖 q “ 𝜋𝑒𝑖 p1𝑒1PT𝑒𝑖 𝑅𝑒1q +5 + +Algorithm 5: FullEnumpT,𝑒,𝑡q +Input: A free-connex generalized join tree T, a node 𝑒 P T +and a tuple 𝑡 P 𝑅𝑒; +Output: Query results over T𝑒 that can be joined with 𝑡; +1 if 𝑒 ´ y ‰ H then +2 +if 𝑒 X y ´ 𝑝p𝑒q “ H then Yield xy; +3 +else Yield 𝜋yX𝑒p𝑅𝑒 ˙ 𝑡q; +4 else +5 +Let C𝑒 “ t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u; +6 +foreach 𝑡1 P FullEnumpT,𝑒1,𝑡rkeyp𝑒1qsq do +7 +foreach 𝑡2 P FullEnumpT,𝑒2,𝑡rkeyp𝑒2qs do +8 +¨ ¨ ¨ +9 +foreach 𝑡𝑘 P FullEnumpT,𝑒𝑘,𝑡rkeyp𝑒𝑘qsq do +10 +Yield 𝑡 1 𝑡1 1 𝑡2 1 ¨ ¨ ¨ 1 𝑡𝑘; +Then, we can rewrite 𝑉𝑠p𝑅𝑒q as follows: +ô 𝑅𝑒 ˙ 𝑉𝑝p𝑅𝑒1q ˙ ¨ ¨ ¨ ˙ 𝑉𝑝p𝑅𝑒𝑘 q +ô 𝑅𝑒 1 𝑉𝑝p𝑅𝑒1q 1 ¨ ¨ ¨ 1 𝑉𝑝p𝑅𝑒𝑘 q +ô 𝑅𝑒 1 +´ +𝜋keyp𝑒1q𝑉𝑠p𝑅𝑒1q +¯ +1 ¨ ¨ ¨ 1 +´ +𝜋keyp𝑒𝑘q𝑉𝑠p𝑅𝑒𝑘 q +¯ +ô 𝑅𝑒 1 +´ +𝜋keyp𝑒1qp1𝑒1PT𝑒1 𝑅𝑒1q +¯ +1 ¨ ¨ ¨ 1 +´ +𝜋keyp𝑒𝑘qp1𝑒1PT𝑒𝑛 𝑅𝑒1q +¯ +ô 𝜋𝑒p𝑅 1𝑒1PT𝑒 𝑅𝑒1qq +where the first equation follows the definition of semi-join views, +the second equation follows the fact that keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 Ď 𝑒, the +third equation follows the definition of projection views, the fourth +equation follows the hypothesis, and the last equation follows the +facts that T𝑒 “ t𝑒u Y T𝑒1 Y T𝑒2 Y ¨ ¨ ¨ Y T𝑒𝑘 and keyp𝑒𝑖q is exactly +the set of join attributes shared by 𝑅𝑒 and p1𝑒1PT𝑒𝑖 𝑅𝑒1q. +□ +In plain language, the semi-join view of node 𝑒 is essentially the +projection of the join results of relations in the subtree rooted at 𝑒, +to attributes in 𝑒. An immediate corollary is +Corollary 5.2. 𝑉𝑠p𝑅𝑟q “ 𝜋𝑟𝑄p𝐷q. +This means that the semi-join view at the root 𝑟 (recall that 𝑟 +does not have a projection view) contains precisely all the query +results projected onto 𝑟. Using the notion of a witness query, this +leads to the following useful fact for full enumeration, where Ţ +denotes disjoint union: +Lemma 5.3. 𝑄p𝐷q “ Ţ +𝑡P𝑉𝑠p𝑅𝑟 q 𝑄p𝐷 ˙ 𝑡q. +Lemma 5.1, Corollary 5.2, and Lemma 5.3 allow us to use essen- +tially the same algorithm from Bagan et al. [6] to achieve constant- +delay enumeration of 𝑄p𝐷q; see Algorithm 5, which takes as input +a node 𝑒 P T and a tuple 𝑡 P 𝑅𝑒, and yields the query results over +T𝑒 that can be joined with 𝑡. To enumerate 𝑄p𝐷q, we simply invoke +FullEnumpT,𝑟,𝑡q for every tuple 𝑡 P 𝑉𝑠p𝑅𝑟q. +Lemma 5.4. Algorithm 5 enumerates 𝑄p𝐷q with 𝑂p1q delay. +Proof of Lemma 5.4. We prove it by induction on the height of +T. The algorithm stops if the root contains no output attributes. For +ease of expression, assume any node 𝑒 with 𝑒 X y “ H is removed +from T. We first establish a based case, in which T contains only +one node. The algorithm returns 𝜋yX𝑒𝑉𝑠p𝑅𝑒q in 𝑂p|𝜋yX𝑒𝑉𝑠p𝑅𝑒q|q +time, since all tuples in 𝜋yX𝑒𝑉𝑠p𝑅𝑒q can be enumerated in 𝑂p1q +delay. Hence, this base case can be handled with 𝑂p1q delay. +In general, we have the hypothesis holds on all child nodes 𝑒𝑖 of +𝑒 (line 6-9): Algorithm 5 can enumerate all join results that agree +with values 𝜋keyp𝑒𝑖q𝑡1 over attributes keyp𝑒𝑖q in the subtree T𝑒𝑖 . Let +𝑡𝑖 be a join result returned from T𝑒𝑖 . From the properties of join tree, +line 10 will return a valid join result. Emitting every combination +of join results over all subtrees of 𝑒1 just takes 𝑂p1q time. +□ +5.2 +Delta Enumeration +Delta enumeration is straightforward in a standard query plan, as +the root node corresponds to 𝑄p𝐷q, so all changes propagated to the +root are precisely Δ𝑄p𝐷,𝑡q. However, it becomes tricky in our new +query plan, as no node corresponds to 𝑄p𝐷q, which is necessarily +the case if a linear-size representation of 𝑄p𝐷q is desired. In our +query plan, one cannot just inspect the root, because not every +change propagates to the root, and many propagations stop mid- +way, which is actually the main reason why our query plan is +not only space-efficient but also time-efficient. Recall that the full +enumeration algorithm relies on Lemma 5.3. Then the key question +is, can we have an analogy of Lemma 5.3 for the delta Δ𝑄p𝐷,𝑡q? In +other words, can we identify a set of witness tuples 𝑡1 for 𝑡 such that +the delta Δ𝑄p𝐷,𝑡q is the disjoint union of 𝑄p𝐷 ˙ 𝑡1q? Fortunately, +the answer is yes, but the answer is not as simple as Lemma 5.3. +Let’s first consider the insertion case. When we insert 𝑡 into some +𝑅𝑒, the propagation follows the path from 𝑒 to 𝑟, by (possibly) ap- +plying an R-Update first, then an S-Update, P-Update, S-Update, +P-Update, .... Recall that both S-update and R-update only propa- +gate a single change upward (see line 8, 12 in Algorithm 2 and Algo- +rithm 4), but P-update may propagate multiple changes upward (see +line 6, 12 in Algorithm 2. Hence, there could be multiple propagation +paths starting from 𝑡. To be more precise, we denote the nodes lying +on the path from 𝑒 to 𝑟 as 𝑒0 “ 𝑒,𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘 “ 𝑟. Every propaga- +tion path inserts a tuple into each of the views on the path, and we +denote the inserted tuples on such a path as p𝑡,𝑡𝑠 +0,𝑡𝑝 +0 ,𝑡𝑠 +1,𝑡𝑝 +1 , ¨ ¨ ¨ , q, +where 𝑡𝑠 +𝑖 P 𝑉𝑠p𝑅𝑖q and 𝑡𝑝 +𝑖 P 𝑉𝑝p𝑅𝑖q for 𝑖 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u. +Now, we distinguish three cases of a propagation path with +respect to its ending tuple: (1) 𝑡; (2) 𝑡𝑝 +𝑗 for some 𝑗 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u; +(3) 𝑡𝑠 +𝑖 for some 𝑖 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u. +Case (1) happens when the first update is an R-Update and does +not propagate any further change. This means that in Algorithm 4, +there exists some child node 𝑒1 of 𝑒 such that 𝑡rkeyp𝑒1qs R 𝑉𝑝p𝑅𝑒1q, +i.e.,𝑡 cannot join with T𝑒1. In this case,𝑡 will not produce any change +to 𝑄p𝐷q, thus can be ignored. +Case (2) happens when P-Update(𝑒𝑗,𝑡𝑝 +𝑗 ) does not propagate any +further change. Putting it into Algorithm 3, this means that either +there exists no tuple 𝑡1 P 𝑅𝑝p𝑒𝑗 q that can join with 𝑡𝑝 +𝑗 , or if such +a tuple exists, but it cannot join with any query result over T 1𝑒 +for some child node 𝑒1 of 𝑝p𝑒𝑗q, since its counter is smaller than +|C𝑝p𝑒𝑗 q|. In either case, this propagation path will not cause any +change to 𝑄p𝐷q, thus can also be ignored. +Case (3) happens when S-update(𝑒𝑖,𝑡𝑠 +𝑖 ) does not propagate any +further change. Putting it into Algorithm 2, this means that either +we have reached the root, or there exists some other tuple 𝑡1 P +𝑉𝑝p𝑅𝑖q such that 𝑡1 ‰ 𝑡𝑠 +𝑖 and 𝑡𝑠 +𝑖 rkeyp𝑒𝑖qs “ 𝑡1rkeyp𝑒𝑖qs. This is the +6 + +Algorithm 6: DeltaEnum(T,𝑡) +Input: A free-connex generalized join tree T; an updated +tuple 𝑡. +Output: Delta results induced by 𝑡. +1 Let 𝑒0,𝑒1, ¨ ¨ ¨ ,𝑒𝑘 “ 𝑟 be the nodes on 𝑡’s propagation path; +2 foreach witness tuple 𝑡1 of 𝑡 do +3 +Let 𝑒𝑖 be the node such that 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒𝑖,𝑡q; +4 +𝑆 Ð 𝑡1 1 𝑉𝑙p𝑅𝑒𝑖`1q 1 ¨ ¨ ¨ 1 𝑉𝑙p𝑅𝑒𝑘 q; +5 +foreach 𝑞 P 𝑆 do +6 +𝑆𝑖 ÐFullEnumpT𝑒𝑖,𝑒𝑖,𝑞r𝑒𝑖sq; +7 +𝑆𝑗 ÐFullEnumpT𝑒𝑗 ´ T𝑒𝑗´1,𝑒𝑗,𝑞r𝑒𝑗sq, 𝑗 P r𝑖 ` 1,𝑘s; +8 +Yield 𝑆𝑖 1 𝑆𝑖`1 1 ¨ ¨ ¨ 1 𝑆𝑘; +only case where changes to 𝑄p𝐷q can possibly happen. We will give +a more detailed characterization of this case later. +Live views. To support constant-delay delta enumeration, we +maintain a live view for each node 𝑒 such that 𝑒 X y ‰ H: +𝑉𝑙p𝑅𝑒q :“ 𝜋𝑒𝑄p𝐷q, +which are the “live” tuples (i.e., appearing in the query results) +projected onto 𝑒. Note that 𝑉𝑙p𝑅𝑒q Ď 𝜋y𝑉𝑠p𝑅𝑒q, which means for +𝑒 Ď y, it can be implemented by simply adding an extra bit in +𝑉𝑠p𝑅𝑒q, indicating if the corresponding tuple is in 𝑉𝑙p𝑅𝑒q. +For the root 𝑟, there is no need to maintain 𝑉𝑙p𝑅𝑟q separately +since𝑉𝑙p𝑅𝑟q “ 𝑉𝑠p𝑅𝑟q by Corollary 5.2. For the leaf nodes, their live +views need not be maintained, either, since they will not be needed +by delta enumeration. The other live views can be maintained by +the following observation: +Lemma 5.5. For any non-root node 𝑒 such that 𝑒 Xy ‰ H and any +tuple 𝑡 P 𝜋y𝑉𝑠p𝑅𝑒q, 𝑡 P 𝑉𝑙p𝑅𝑒q if and only if 𝑡 1 𝑉𝑙p𝑅𝑝p𝑒qq ‰ H. +Based on the Lemma 5.5, the maintenance of 𝑉𝑙p𝑅𝑒q can pig- +gyback on the delta enumeration: After enumerating a result +𝑡1 P Δ𝑄p𝐷,𝑡q, we update the live views in a top-down fashion. +For every non-root 𝑒 such that 𝑒 Xy ‰ H, if the update is insertion, +then we always add 𝑡1r𝑒s to𝑉𝑙p𝑅𝑒q; if the update is deletion, then we +delete 𝑡1r𝑒s from 𝑉𝑙p𝑅𝑒q if 𝑡1r𝑒s cannot join with 𝑉𝑙p𝑅𝑝p𝑒qq, which +can be done in 𝑂p1q time with a hash index on 𝑉𝑙p𝑅𝑝p𝑒qq (which is +physically the same hash index on 𝑉𝑠p𝑅𝑝p𝑒qq for 𝑒 Ď y). This only +adds another constant to the delay of delta enumeration. +Witness tuples. We now are ready to give a more precise char- +acterization of the ending tuples falling into Case (3) that actually +cause changes to 𝑄p𝐷q, called witness tuples: +Definition 5.6 (Witness tuple). Suppose 𝑡 is inserted into or +deleted from 𝐷. A tuple 𝑡1 is a witness of 𝑡 if +𝑡1 P Δ𝑉𝑠p𝑅𝑟,𝑡q, or +(5) +𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q ˙ 𝑉𝑙p𝑅𝑝p𝑒qq +(6) +for some non-root 𝑒 such that 𝑒 X y ‰ H. +Here Δ𝑉𝑠p𝑅𝑒,𝑡q denotes the tuples to be inserted into (or deleted +from) 𝑉𝑠p𝑅𝑒q due to 𝑡 and 𝑉𝑙p𝑅𝑝p𝑒qq is the live view before the +update. We give some intuition behind Definition 5.6. First, (5) is +the counterpart of Corollary 5.2 for delta enumeration and such a 𝑡1 +is guaranteed to generate changes to 𝑄p𝐷q. (6) is specific for delta +enumeration, addressing the situation mentioned earlier, where +the propagation stops mid-way yet still causes changes to 𝑄p𝐷q. +Note that in this case, the attributes of 𝑡1 are 𝑒 X y. Then (6) implies +that 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q and 𝑡1rkeyp𝑒qs P 𝜋keyp𝑒q𝑉𝑙p𝑅𝑝p𝑒qq. Since +𝑡1rkeyp𝑒qs P 𝜋keyp𝑒q𝑉𝑙p𝑅𝑝p𝑒qq, it must have 𝑡1rkeyp𝑒qs P 𝑉𝑝p𝑅𝑒q, i.e. +𝑡1rkeyp𝑒qs R Δ𝑉𝑝p𝑅𝑒,𝑡q, which means that the propagation stops at +node 𝑒 under case (3). In addition, each witness tuple 𝑡1 should (i) +contribute to the delta over T𝑒 induced by 𝑡, and (ii) join with tuples +from the remaining relations in T ´ T𝑒. For (i), it suffices to require +𝑡1 P Δ +` +𝜋y𝑉𝑠p𝑅𝑒q +˘ +“ 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q, since Δ +` +𝜋𝑒Xyp1𝑒1PT𝑒 𝑅𝑒1q +˘ +“ +Δ +` +𝜋y𝑉𝑠p𝑅𝑒q +˘ +. For (ii), it suffices to require 𝑡1 ˙𝑉𝑙p𝑅𝑝p𝑒qq ‰ H, and +this is exactly the reason we introduced 𝑉𝑙p𝑅𝑒q in the first place. +Lemma 5.7. Δ𝑄p𝐷,𝑡q “ Ţ +𝑡1:a witness of 𝑡 𝑄p𝐷 ˙ 𝑡1q. +We are now ready to state the counterpart of Lemma 5.3 for delta +enumeration, in Lemma 5.7. Unlike Lemma 5.3, the proof of Lemma +5.7 is nontrivial, and the details are given in Appendix B. +The algorithm. To perform delta enumeration using Lemma 5.7, +we still need to address two issues: (1) how to find all witness tuples +𝑡1, and (2) how to enumerate 𝑄p𝐷 ˙ 𝑡1q with constant delay. +To find all the witness tuples, we consider the two cases in +Definition 5.6: (5) can be computed easily after updating 𝑉𝑠p𝑅𝑟q; +for (6), just an extra check with 𝑉𝑙p𝑅𝑝p𝑒qq is needed, which can be +done in 𝑂p1q time using the hash index on 𝑉𝑙p𝑅𝑝p𝑒qq. These steps +only increase the update cost by a constant factor. +It remains to describe how to enumerate 𝑄p𝐷 ˙ 𝑡1q for each +witness 𝑡1. As before, let 𝑒0,𝑒1, . . . ,𝑒𝑘 “ 𝑟 be the nodes on the +propagation path, and suppose we are given a witness tuple 𝑡1 P +𝜋yΔ𝑉𝑠p𝑅𝑒𝑖,𝑡q for some 𝑖. We first enumerate the query results par- +ticipated by 𝑡1 together with relations on the path from 𝑒𝑖`1 to the +root 𝑟, denoted as 𝑆. This can be done by joining 𝑡 with the live +views associated with these nodes. For each such result 𝑞 P 𝑆, we +enumerate the query results that participated by 𝑞. This enumer- +ation is done by partitioning the whole generalized join tree into +disjoint subtrees T𝑒𝑖, T𝑒𝑖`1 ´T𝑒𝑖, ¨ ¨ ¨ , T𝑒𝑘 ´T𝑒𝑘´1, and invoking Ful- +lEnum for each subtree separately. Finally, we join these subtrees +together. The detailed process is given in Algorithm 6. Note that, +as written, the algorithm does not achieve constant-delay enumer- +ation. However, this can be easily fixed. First, the join in line 4 can +be enumerated with constant delay using (a variant of) FullEnum +starting from 𝑡1. Then we interleave the two enumeration processes: +After enumerating each 𝑞 P 𝑆, we immediately call line 6–8. Finally, +line 6–8 can be rewritten into nested loops so as to enumerate the +join 𝑆𝑖 1 ¨ ¨ ¨ 1 𝑆𝑘 with constant delay. In fact, this join is more like +a cross product (common attributes must have the same value, the +same as those in 𝑞), and a total of Π𝑘 +𝑗“𝑖|𝑆𝑗| results will be yielded. +Example 5.8. In figure 3(a), there are two query results p1, 2, 4, 4q +and p2, 2, 4, 4q. In figure 3(b), when the propagation stops, we have +‚ Tuple p1, 2q P 𝑅2 is not a witness as it cannot join with any tuple +in 𝑉𝑙pr𝑥3sq, thus no delta is produced; +‚ Tuple p1q P r𝑥3s is a witness, which triggers delta enumeration. +For a witness in the root, DeltaEnum simply degenerates to +FullEnumpT𝑟,𝑟, p1qq, which outputs tp1, 1, 1, 1q, p1, 1, 1, 2qu. +‚ Tuple p1, 4q P 𝑅2 is a witness, which triggers delta enumeration. +DeltaEnum finds 𝑆 “ p1, 4q 1 𝑉𝑙pr𝑥3sq “ tp1, 4qu. For p1, 4q P 𝑆, +7 + +r𝑥3s +𝑉𝑠pr𝑥3sq +𝑥3 +𝑐r𝑡s +1 +1 +2 +1 +3 +1 +4 +2 +𝑅2 +𝑉𝑝p𝑅2q +𝑥3 +𝑐r𝑡s +4 +1 +2 +1 +Ô +𝑉𝑠p𝑅2q +𝑥2 +𝑥3 +𝑐r𝑡s +1 +2 +0 +2 +2 +1 +4 +3 +0 +1 +1 +0 +2 +4 +1 +1 +4 +0 +𝑅3 +𝑉𝑝p𝑥3q +𝑥3 +𝑐r𝑡s +1 +2 +3 +1 +4 +1 +Ô +𝑉𝑠p𝑅3q +𝑥3 +𝑥4 +𝑐r𝑡s +1 +1 +1 +2 +5 +0 +3 +3 +1 +1 +2 +1 +4 +4 +1 +𝑅1 +𝑉𝑝p𝑅1q +𝑥2 +𝑐r𝑡s +2 +2 +3 +1 +Ò +𝑉𝑠p𝑅1q +𝑥1 +𝑥2 +1 +2 +2 +2 +3 +3 +𝑅4 +𝑉𝑝p𝑅4q +𝑥4 +𝑐r𝑡s +1 +1 +2 +1 +3 +1 +4 +1 +Ò +𝑉𝑠p𝑅4q +𝑥4 +𝑥5 +1 +1 +2 +2 +3 +3 +4 +4 +(a) Initialization. +r𝑥3s +𝑉𝑠pr𝑥3sq +𝑥3 +𝑐r𝑡s +1* +2 +2 +1 +3 +1 +4 +2 +𝑅2 +𝑉𝑝p𝑅2q +𝑥3 +𝑐r𝑡s +4 +2 +2 +2 +1 +1 +Ô +𝑉𝑠p𝑅2q +𝑥2 +𝑥3 +𝑐r𝑡s +1 +2 +1 +2 +2 +1 +4 +3 +0 +1 +1 +1 +2 +4 +1 +1* +4* +1 +𝑅3 +𝑉𝑝p𝑥3q +𝑥3 +𝑐r𝑡s +1 +2 +3 +1 +4 +1 +Ô +𝑉𝑠p𝑅3q +𝑥3 +𝑥4 +𝑐r𝑡s +1 +1 +1 +2 +5 +0 +3 +3 +1 +1 +2 +1 +4 +4 +1 +𝑅1 +𝑉𝑝p𝑅1q +𝑥2 +𝑐r𝑡s +2 +2 +3 +1 +1 +1 +Ò +𝑉𝑠p𝑅1q +𝑥1 +𝑥2 +1 +2 +2 +2 +3 +3 +1 +1 +𝑅4 +𝑉𝑝p𝑅4q +𝑥4 +𝑐r𝑡s +1 +1 +2 +1 +3 +1 +4 +1 +Ò +𝑉𝑠p𝑅4q +𝑥4 +𝑥5 +1 +1 +2 +2 +3 +3 +4 +4 +(b) After the insertion of p1, 1q into 𝑅1. +r𝑥3s +𝑉𝑠pr𝑥3sq +𝑥3 +𝑐r𝑡s +1 +2 +2 +1 +3 +1 +4 +2 +𝑅2 +𝑉𝑝p𝑅2q +𝑥3 +𝑐r𝑡s +4 +2 +2 +2 +1 +1 +Ô +𝑉𝑠p𝑅2q +𝑥2 +𝑥3 +𝑐r𝑡s +1 +2 +1 +2 +2 +1 +4 +3 +0 +1 +1 +1 +2 +4 +1 +1 +4 +1 +𝑅3 +𝑉𝑝p𝑥3q +𝑥3 +𝑐r𝑡s +1 +1 +3 +1 +4 +1 +Ô +𝑉𝑠p𝑅3q +𝑥3 +𝑥4 +𝑐r𝑡s +1* +1* +0 +2 +5 +0 +3 +3 +1 +1 +2 +1 +4 +4 +1 +𝑅1 +𝑉𝑝p𝑅1q +𝑥2 +𝑐r𝑡s +2 +2 +3 +1 +1 +1 +Ò +𝑉𝑠p𝑅1q +𝑥1 +𝑥2 +1 +2 +2 +2 +3 +3 +1 +1 +𝑅4 +𝑉𝑝p𝑅4q +𝑥4 +𝑐r𝑡s + +1 + +1 +2 +1 +3 +1 +4 +1 +Ò +𝑉𝑠p𝑅4q +𝑥4 +𝑥5 +�1 +�1 +2 +2 +3 +3 +4 +4 +(c) After the deletion of p1, 1q from 𝑅4. +r𝑥3s +𝑉𝑠pr𝑥3sq +𝑥3 +𝑐r𝑡s +1 +2 +2 +1 +3 +1 +4 +2 +𝑅2 +𝑉𝑝p𝑅2q +𝑥3 +𝑐r𝑡s +4 +2 +2 +2 +1 +1 +Ô +𝑉𝑠p𝑅2q +𝑥2 +𝑥3 +𝑐r𝑡s +1 +2 +1 +2 +2 +1 +4 +3 +0 +1 +1 +1 +2 +4 +1 +1 +4 +1 +𝑅3 +𝑉𝑝p𝑥3q +𝑥3 +𝑐r𝑡s +1 +1 +3 +1 +4 +1 +Ô +𝑉𝑠p𝑅3q +𝑥3 +𝑥4 +𝑐r𝑡s +1 +1 +0 +2 +5 +0 +3 +3 +1 +1 +2 +1 +4 +4 +1 +𝑅1 +𝑉𝑝p𝑅1q +𝑥2 +𝑐r𝑡s +2 +2 +3 +1 +1 +2 +Ò +𝑉𝑠p𝑅1q +𝑥1 +𝑥2 +1 +2 +2 +2 +3 +3 +1 +1 +4* +1* +𝑅4 +𝑉𝑝p𝑅4q +𝑥4 +𝑐r𝑡s +2 +1 +3 +1 +4 +1 +Ò +𝑉𝑠p𝑅4q +𝑥4 +𝑥5 +2 +2 +3 +3 +4 +4 +(d) After the insertion of p4, 1q into 𝑅1. +Figure 3: A running instance for query in Figure 1 using the plan in Figure 1(c). Tuples in white are in 𝑉𝑠p𝑅q, in grey are in +𝑅z𝑉𝑠p𝑅q, in cyan are in 𝑉𝑙p𝑅q (live views for leaf nodes are not needed, but we still show them for clarity), with star symbols are +the witness tuples. Changes in each step are marked in red. +it invokes FullEnumpT𝑟 ´ T𝑅2,𝑟, p4qq with tp4, 4qu returned and +FullEnumpT𝑅2, 𝑅2, p1, 4qq with tp1, 1, 4qu returned. Joining them +yields the delta tp1, 1, 4, 4qu. +Finally, as each new result is enumerated, we update the live views. +In figure 3(c), tuple p1, 1q P Δ𝑉𝑠p𝑅3q is a witness. DeltaEnum +first finds 𝑆 “ tp1, 1qu. For p1, 1q P 𝑆, it invokes FullEnumpT𝑟 ´ +T𝑅3,𝑟, p1qq with tp1, 1, 1qu returned, and FullEnumpT𝑅3, 𝑅3, p1, 1qq +with tp1, 1, 4qu returned (delta enumeration upon a deletion is done +before the tuple deletion so as to find the delta). Joining them yields +the delta tp1, 1, 1, 1qu. Finally, we update live views with the delta. +Lemma 5.9. Algorithm 6 enumerates Δ𝑄p𝐷,𝑡q with constant delay. +We have now closed the loop: while enumerating Δ𝑄p𝐷,𝑡q, we +update the live views as described earlier, which are needed for +enumerating the next delta. +6 +UPDATE COST ANALYSIS +We have shown that the enumeration delay of both full query +results and deltas is a constant, and this holds for the query plan +defined by any free-connex join tree as in Section 4.1. On the other +hand, the update cost differs for different query plans and can be +as large as 𝑂p|𝐷|q in the worst case. This is caused by P-Update, +which may trigger an S-Update to every tuple in its parent node. +However, such a worst-case behavior only happens on contrived +update sequences, and the actual update cost can be much better. +Characterizing the update cost will be important for constructing a +good query plan, as there can be many free-connex join trees for +a given free-connex query. As we will see, the height of the join +tree is an important parameter, and this is precisely the reason why +we make our framework applicable to any generalized join tree, +as the height of a generalized join tree can be lower than that of +any standard join tree. For example, the query in Figure 2 has a +generalized join tree of height 1 while the two standard join trees +have height 2; the query in Figure 1 has a generalized join tree of +height 2 while any standard join tree has height as least 3. +6.1 +Enclosureness +Update sequences and lifespans. Given an update sequence 𝑆𝐷, +the lifespan of tuple 𝑡 is an interval 𝐼p𝑡q “ r𝑡`,𝑡´s, where 𝑡` +denotes the timestamp when 𝑡 is inserted into 𝐷 and 𝑡´ denotes +the timestamp when 𝑡 is deleted from 𝐷. We set 𝑡` “ ´8 to +indicate that 𝑡 exists in the initial 𝐷 and 𝑡´ “ `8 indicates that +𝑡 still exists in 𝐷 after the update sequence. Note that if a tuple is +repeatedly inserted and deleted, it will be treated as multiple tuples, +which have the same values but disjoint lifespans. +Although our algorithms will be able to handle arbitrary update +sequences, their performance can be better if the update sequences +possess some nice properties. In particular, the following two re- +strictive classes of update sequences are of practical importance: +‚ First-in-first-out (FIFO). A update sequence 𝑆𝐷 is FIFO if for +any two tuples 𝑡1,𝑡2 P 𝑆𝐷, 𝑡` +1 +ă 𝑡` +2 implies 𝑡´ +1 +ă 𝑡´ +2 . FIFO +sequences are commonly used in practice, such as sliding-window +or tumbling-window models over streaming data. +‚ Insertion-only or deletion-only. A update sequence 𝑆𝐷 is +insertion-only (w.r.t. deletion-only) if for any tuple 𝑡 P 𝑆𝐷, 𝑡´ “ +`8 (w.r.t. 𝑡` “ ´8). The two cases are symmetric, so we will +only discuss the insertion-only case in this paper. +8 + +The notion of enclosureness was first introduced in [37] to give +an instance-specific characterization of the hardness of the update +sequence, which we briefly review next. +Definition 6.1 (Enclosureness). Given an update sequence 𝑆𝐷, the +enclosureness of a tuple 𝑡 P 𝑆𝐷 is +𝜆p𝑡q :“ +max +JĎ𝑆𝐷 +@𝑡1PJ,𝐼p𝑡1qĂ𝐼p𝑡q +@𝑡2,𝑡3PJ,𝐼p𝑡2qX𝐼p𝑡3q“H +|J|, +(7) +i.e., the largest number of disjoint lifespans in 𝑆𝐷 contained in +𝐼p𝑡q. Then the enclosureness of the update sequence is the average +enclosureness of all the tuples (but at least 1), i.e., +𝜆p𝑆𝐷q :“ max +˜ř +𝑡P𝑆𝐷 𝜆p𝑡q +|𝑆𝐷| +, 1 +¸ +. +(8) +We often omit 𝑆𝐷 and simply write 𝜆 :“ 𝜆p𝑆𝐷q for the enclosure- +ness of an update sequence. +Then, they give an algorithm that can update any foreign-key +acyclic query in 𝑂p𝜆q time for any 𝑆𝐷 while supporting 𝑂p1q-delay +enumeration. This is appealing, since while 𝜆 can be as large as +𝑂p|𝑆𝐷|q in the worst case, it is often a small constant for many com- +mon update sequences, including FIFO, FILO (first-in-last-out), and +insertion-/deletion-only sequences. The worst-case situation only +happens when there are many tuples with long lifespans joining +with many tuples with short lifespans, something that is uncommon +in practice (i.e., many big but ephemeral changes to the query). +However, their analysis crucially relies on the nice property of +foreign-key acyclic queries, that their result size is at most linear, +which is not the case for non-key joins. In fact, we show below +that the 𝑂p𝜆q update time is unachievable for free-connex queries, +which follows from the negative result that we prove below: +Theorem 6.2. Consider the query 𝑄 “ 𝑅1p𝑥1q 1 𝑅2p𝑥1,𝑥2q 1 +𝑅3p𝑥2,𝑥3q 1 𝑅4p𝑥3,𝑥4q 1 𝑅5p𝑥4q over a FIFO update sequence. If +there is an algorithm for 𝑄 with update time 𝑂p|𝐷|1{2´𝜖q while sup- +porting 𝑂p|𝐷|1´𝜖q-delay enumeration of full results for any constant +𝜖 ą 0, then the OuMv conjecture4 fails. +Note that this theorem separates the difficulty of (at least one +of) free-connex queries from foreign-key acyclic queries, for which +𝑂p1q update time is possible for FIFO sequences [37]. +6.2 +Join-tree-specific Enclosureness +Hope is not all lost despite the negative result above. First, Theorem +6.2 only holds for a particular free-connex query; other queries may +still be updated in 𝑂p1q time. Secondly, the definition of enclosure- +ness in [37] only considers the time dimension while ignoring the +structure dimension, i.e., which relation each update is applied to. +These observations motivate a more refined definition of enclosure- +ness that also depends on the join tree (which nodes the updates +are applied to). As we will see, a hard query like the one in Theorem +6.2 can still be solved efficiently, when information from both the +structural dimension and the time dimension is taken into account. +4The OuMv conjecture [20] is that the following problem cannot be solved in𝑂p𝑛3´𝜖q +time for any constant𝜖 ą 0: Given an𝑛ˆ𝑛 matrix 𝑀 and a sequence of𝑛-dimensional +vectors𝑢1, 𝑣1,𝑢2, 𝑣2, ¨ ¨ ¨ ,𝑢𝑛, 𝑣𝑛, compute𝑢𝑖𝑀𝑣𝑖 for each𝑖 over the Boolean semiring. +The algorithm must return 𝑢𝑖𝑀𝑣𝑖 before 𝑢𝑖`1, 𝑣𝑖`1 are revealed. +(1, 1) +· · · +(n, 1) +(1, 2) +· · · +Time +(n, 2) +(1, 1) +· · · +(1, n) +(2, 1) +· · · +(2, n) +R2 +n/1/1 +n/1/1 +n/1/1 +1/1/1 +1/1/1 +1/1/1 +1/1/1 +1/1/1 +1/1/1 +1/n/1 +1/n/1 +1/n/1 +R1 +Figure 4: As update sequence in Example 6.5. Each interval +is the lifespan of a tuple, and three numbers above each in- +terval are its enclosureness over T1, T2 and T3 in Figure 2. +Definition 6.3 (Effective lifespan). Given a free-connex query 𝑄, +a free-connex generalized join tree T of 𝑄, a database 𝐷, and an +update sequence 𝑆𝐷, the two effective lifespans of a tuple 𝑡1 P 𝑅𝑒 +with lifespan 𝐼p𝑡1q “ r𝑡` +1 ,𝑡´ +1 s are +p𝐼p𝑡1q “ +« +𝑡` +1 , min +˜ +𝑡´ +1 , +min +𝑡2P𝑅𝑒1:𝑒1PT𝑒´t𝑒u,𝑡´ +2 ą𝑡` +1 +𝑡´ +2 +¸ff +; +q𝐼p𝑡1q “ +« +max +˜ +𝑡` +1 , +max +𝑡2P𝑅𝑒1:𝑒1PT𝑒´t𝑒u,𝑡` +2 ă𝑡´ +1 +𝑡` +2 +¸ +,𝑡´ +1 +ff +. +In plain language, p𝐼p𝑡1q is obtained from 𝐼p𝑡1q by moving forward +its ending time to the first deletion of a tuple from any descendent +of 𝑒, while to obtain q𝐼p𝑡1q, we move its starting to the last insertion +from any descendent of 𝑒. +We can now define the join-tree-specific enclosureness of a tuple: +Definition 6.4. Given a free-connex query 𝑄, a free-connex gen- +eralized join tree T of 𝑄, a database 𝐷, and an update sequence 𝑆𝐷, +for a node 𝑒 P T and a tuple 𝑡 P 𝑅𝑒, its enclosureness is +𝜆Tp𝑡q “ +max +@𝑡1PJ,D𝑒1PT𝑒´t𝑒u,𝑡1P𝑅𝑒1 +@𝑡1PJ,9𝐼p𝑡1qĎ𝐼p𝑡q +@𝑡2,𝑡3PJ,9𝐼p𝑡2qX9𝐼p𝑡3q“H +|J|, +(9) +where each 9𝐼 is either p𝐼 or q𝐼, i.e., the largest number of disjoint +effective lifespans of tuples in the descendants of 𝑒, which are +contained in the lifespan of 𝑡. Then the enclosureness of the update +sequence is still the average: +𝜆Tp𝑆𝐷q :“ max +˜ř +𝑡P𝑆𝐷 𝜆Tp𝑡q +|𝑆𝐷| +, 1 +¸ +. +We often write 𝜆T :“ 𝜆Tp𝑆𝐷q for the enclosureness of an update +sequence with respect to T. +Example 6.5. Consider 𝑄 :“ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q in Figure 2 +with T1, T2, T3. For the update sequence in Figure 4, 𝜆T1 “ 𝜆T2 “ 𝑛 +and 𝜆T3 “ 1. In fact, 𝜆T3 “ 1 for any update sequence. +The main analytical result of this paper is the following theorem, +whose proof is quite technical given in Appendix C: +Theorem 6.6. For any free-connex query 𝑄, the update cost of the +query plan in Section 4 induced by any given free-connex generalized +join tree T of 𝑄 is 𝑂p𝜆Tq under any update sequence. +9 + +This result is complemented with a matching lower bound, for +at least one particular query: 𝑄 “ 𝜋𝑥1p𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2qq, which +has one join tree as shown in Figure 2(a) (one could add a general- +ized relation r𝑥1s at the top, but it does not change the enclosure- +ness). Thus, for this query, 𝜆T does not really depend on T. +Theorem 6.7. [37] Suppose there is an algorithm for the query +𝑄 “ 𝜋𝑥1p𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2qq with update time 𝑂p𝜆1´𝜖q while +supporting𝑂p𝜆1´𝜖q-delay enumeration of full results for any constant +𝜖 ą 0, then the OMv conjecture5 fails. +6.3 +Implications of Enclosureness +We present some implications of our join-tree-specific enclosure- +ness and Theorem 6.6, exhibiting an interesting trade-off between +the hardness of update sequences and the complexity of queries. +Arbitrary update sequences. For arbitrary update sequences, +prior work [8, 21] has shown how to achieve 𝑂p1q update time +while supporting 𝑂p1q-delay enumeration for any q-hierarchical +query. It turns out that this is an easy consequence of Theorem 6.6, +plus the following structural property of q-hierarchical queries, as +well as the simple fact that 𝜆T “ 1 if the height of T is 1: +Lemma 6.8. Every q-hierarchical query has a free-connex general- +ized join tree of height 1. +For arbitrary update sequences, q-hierarchical queries are pre- +cisely the class of queries for which 𝑂p1q update time is possible +[8]. Thus, for queries outside this class, we must restrict the update +sequence in order to achieve 𝑂p1q update time. We consider the +following two classes of update sequences. +FIFO sequences. The update time is shown to be 𝑂p1q for foreign- +key acyclic joins over FIFO sequences [37], but nothing is known +for non-key joins (except for q-hierarchical queries which do not +rely on FIFO). We present the first extension in this direction: +Lemma 6.9. For any free-connex query 𝑄 with a free-connex gener- +alized join tree T of height at most 2, 𝜆T “ 1 for any FIFO sequence. +Note that the height limit of 2 is the best one can hope for, since +the query in Theorem 6.2 has a join tree of height 3 and the theorem +shows that it cannot be updated in 𝑂p1q time over FIFO sequences. +Although the height-2 limitation restricts the class of queries, this +already includes some fairly complex queries, such as the one in +Figure 1; more examples can be found in Section 8. +Insertion-only sequences. As we restrict the update sequence +further, we can handle more queries in 𝑂p1q time. For simplicity, +the following result only considers insertion-only sequences, but +the same result holds for deletion-only or FILO sequences as well. +Lemma 6.10. For any free-connex query 𝑄 and any join tree T, +𝜆T “ 1 for any insertion-only update sequence. +Combining Theorem 6.6 and Lemma 6.10, the following theorem +is straightforward. +Theorem 6.11. For a free-connex query 𝑄, there is an index that +can be updated in 𝑂p1q amortized time under any insertion-only +update sequence, while supporting 𝑂p1q-delay enumeration. +5The OMv conjecture is similar to the OuMv conjecture, except that the algorithm +needs to compute 𝑀𝑣𝑖 for every 𝑣𝑖. +Note that Lemma 6.10 incorporates the static result [6] as a +special case. Given a static database 𝐷, we can simply insert every +tuple from 𝐷 into our query plan. By Lemma 6.10, this builds a data +structure in 𝑂p|𝐷|q time that supports 𝑂p1q-delay enumeration of +𝑄p𝐷q. Also, the dichotomy result of [6] states that 𝑂p|𝐷|q-time +preprocessing and 𝑂p1q-delay enumeration are possible only for +free-connex queries, thus Lemma 6.10 cannot be extended to beyond +free-connex queries, either, even over insertion-only sequences. +Example 6.12. Consider an insertion-only update sequence for +the query in Figure 1: (1) tuples p𝑖, 𝑗q P r𝑛s ˆ r𝑛s are inserted into +𝑅2, 𝑅3 and 𝑅4 initially; (2) tuples p𝑖, 𝑗q P r𝑛s ˆ r𝑛s are inserted into +𝑅1 later. Standard change propagation or HIVM needs to materi- +alize Δp𝑅1 1 𝑅2 1 𝑅3q, hence incurs 𝑂p𝑛3q cost; the Dynamic +Yannakakis algorithm [21] needs to scan all tuples p𝑖1, 𝑗1q P 𝑅2 for +𝑖1 “ 𝑗, once p𝑖, 𝑗q is inserted into 𝑅1, hence incurs Θp𝑛q cost; and +our framework only incurs 𝑂p1q cost. +Query plan optimization. If the given query and/or the update +sequence do not fall into any of the three cases above where 𝑂p1q +update time can be guaranteed, our enclosureness analysis still +yields an effective heuristic for choosing a good T, which in turn +determines the query plan. First, it is clear that T with a smaller +height is always preferred. Furthermore, Definition 6.4 suggests +that we should put nodes with more updates higher in T, as a tuple +in a node might increase the enclosureness of tuples in its ancestors. +Thus, in our implementation, we construct all join trees and use +the one that minimizes +ÿ +𝑒PT +𝑑p𝑒q𝑁p𝑒q, where 𝑑p𝑒q is the depth of 𝑒 +in T (not counting generalized relations and itself) and 𝑁p𝑒q is the +number of updates to 𝑒. If 𝑁p𝑒q is unavailable, we can estimate it +by observing (and buffering) the first few updates. +7 +EXTENSIONS TO GENERAL QUERIES +7.1 +General CQs +Acyclic but non-free-connex queries. Consider such a query +𝜋𝑥1,𝑥3𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q. We simply add 𝑥2 as an output at- +tribute to turn it into a free-connex query, and then do a projection +over 𝑥1,𝑥3 during enumeration. Note that enumeration may con- +tain duplicates. Thus, if a DISTINCT keyword is declared explicitly, +duplicates need to be removed, hence making the delay more than +constant, but this is inevitable due to the lower bound [6]. +Cyclic queries. Cyclic queries can also be easily handled in our +framework by resorting to Generalized Hypertree Decomposition +(GHD) [18]. More specifically, by grouping several relations into a +bag, an arbitrary CQ can be converted into a free-connex one. For +example, Figure 5(a) shows a GHD for the “dumbbell” query with 3 +bags. We can use standard change propagation within each bag, and +apply our framework across the bags. This results in the query plan +in Figure 5(b), which has 𝑂p𝑁 2q space and 𝑂p𝑁q update time while +supporting constant-delay enumeration. On the other hand, the +standard change propagation framework would use a query plan +like the one in Figure 5(c), which has 𝑂p𝑁 3q space and update time. +Of course, all these are worst-case bounds; on realistic inputs, the +costs are lower, but our new query plan is still order-of-magnitude +better than the old plan, as shown in Section 8. +10 + +𝑥1 +𝑥2 +𝑥3 +𝑥4 +𝑥5 +𝑥6 +𝑅1 +𝑅2 +𝑅3 +𝑅4 +𝑅5 +𝑅6 +𝑅7 +(a) Hypergraph and a GHD +˙ : 𝑉𝑠p𝑅7q +𝜋 : 𝑉𝑝p𝐵1q +𝑅7 +𝜋 : 𝑉𝑝p𝐵2q +𝑉𝑠p𝐵1q : 𝑅3 1 𝑉1 +𝑉𝑠p𝐵2q : 𝑅6 1 𝑉2 +𝑅3 +𝑉1 : 𝑅2 1 𝑅1 +𝑉2 : 𝑅4 1 𝑅5 +𝑅6 +𝑅2 +𝑅1 +𝑅4 +𝑅5 +𝐵1 +𝐵2 +(b) Query plan under new change propagation +𝑉6 “ 𝑉4 1 𝑉5 +𝑉4 “ 𝑉2 1 𝑅7 +𝑉5 “ 𝑉3 1 𝑅6 +𝑉2 “ 𝑉1 1 𝑅3 +𝑅7 +𝑉3 “ 𝑅4 1 𝑅5 +𝑅6 +𝑅3 +𝑉1 “ 𝑅2 1 𝑅3 +𝑅5 +𝑅4 +𝑅1 +𝑅2 +(c) Query plan under standard change propagation +Figure 5: 5(a) is the hypergraph of the “dumbbell” query 𝑄 “ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥1,𝑥3q 1 𝑅3p𝑥2,𝑥3q 1 𝑅4p𝑥5,𝑥6q 1 𝑅5p𝑥4,𝑥5q 1 +𝑅6p𝑥4,𝑥6q 1 𝑅7p𝑥3,𝑥4q, with GHD illustrated in red circle. 5(c),5(b) are query plans under the standard, new change propaga- +tion framework respectively. In 5(c), 𝐵1, 𝐵2 are treated as two basic relations, on which projection and semi-join views are +constructed. +If one is interested in further improving the theoretical bounds, +the algorithm for maintaining the query results inside each bag +can be replaced by a better algorithm. For example, Kara et al. [26] +present an algorithm for maintaining the triangle join. Replacing +with the new algorithm can improve the space usage from 𝑂p𝑁 2q +to 𝑂p𝑁 +? +𝑁q. On the other side, although the algorithm [26] can +improve the update cost for each bag to 𝑂p +? +𝑁q, the “dumbbell” +query still suffers from 𝑂p𝑁q update cost. This is indeed unavoid- +able as a single tuple update can change as large as 𝑂p𝑁q results +materialized for one bag, which further propagates to the overall +framework. Hence, the update cost for this GHD-based change +propagation framework is determined by updates not only inside +each bag but also across bags. +Beyond the triangle join, not many results are known. This is +still an actively researched problem; any improvement here will +also improve general CQs when plugged into our framework. +Theorem 7.1. Given a CQ 𝑄 with a free-connex GHD of width6 +𝑤, there is an index of 𝑂p𝑁 𝑤q size that can be updated in Ωp𝑁 𝑤q +time while supporting 𝑂p1q-delay enumeration. +Proof. Maintaining any bag of relations requires Ωp𝑁 𝑤q time, +and it needs 𝑂p𝑁 𝑤q space to store all query results in the bag. +After maintaining each bag of relations, the algorithms proposed +in Section 4 can use to maintain between each bag, which takes +𝑂p|𝐷|q time for maintenance. Noted that current database size |𝐷| +is bounded by the largest bag size, which will be 𝑁 𝑤, makes the +total maintenance time to Ωp𝑁 𝑤q and space cost to 𝑂p𝑁 𝑤q. +□ +The following lemma can be easily derived from the above theo- +rem. +Lemma 7.2. For the “dumbbell” query, there is an index of 𝑂p𝑁 1.5q +size that can be updated in 𝑂p𝑁q time per tuple update, while sup- +porting 𝑂p1q-delay enumeration. +6The definition of width depends on the algorithm used for maintaining query results +inside each bag. If adopting the standard change propagation framework, the width is +defined as the maximum width over all bags, where the width of a bag is the optimal +integral edge covering number of the corresponding subquery derived for this bag. +7.2 +Selection, union, and set difference +The query plan in Section 4.1 works for CQs with joins and projec- +tions, but it can be equipped with other operators easily. +‚ If there is a selection 𝜎𝜙 on an input relation 𝑅𝑒 where 𝜙 is a +predicate on 𝑒, then for an update with tuple 𝑡 P 𝑅𝑒, we simply +check if 𝜙p𝑡q is true, and discard this update if not. This only +adds 𝑂p1q time to the update cost. +‚ For the union of CQs 𝑄 “ 𝑄1 Y ¨ ¨ ¨ Y 𝑄𝑘, we just maintain +each 𝑄𝑖 separately. Full enumeration can be supported with 𝑂p1q +delay using the technique in [10]. We note that [10] assumes that +the data structure on each 𝑄𝑖p𝐷q can check if 𝑡 P 𝑄p𝐷𝑖q in 𝑂p1q +time for any given 𝑡, which is indeed supported by our query +plan. For delta enumeration, we can use the same technique to +enumerate Δ𝑄1p𝐷q Y ¨ ¨ ¨ Y Δ𝑄𝑘p𝐷q. However, this is not the +same as Δp𝑄1 Y ¨ ¨ ¨ Y 𝑄𝑘q, and we need to check, say, if some +new result 𝑡 P Δp𝑄1q already exists in 𝑄2p𝐷q. Thus, while the +technique of [10] is still correct for delta enumeration, the delay +is not bounded by a constant. How to support 𝑂p1q-delay delta +enumeration for UCQs remains an interesting open problem. +‚ For a query like 𝑄 “ 𝑄1 ´ 𝑄2, we can as above maintain 𝑄1 and +𝑄2 separately. For enumeration, we enumerate every 𝑡 P 𝑄1p𝐷q +and check if 𝑡 P 𝑄2p𝐷q, although this does not guarantee con- +stant delay. In fact, even in the static case, it is an open question +whether 𝑄1p𝐷q ´ 𝑄2p𝐷q can be enumerated in 𝑂p1q delay after +linear-time preprocessing where 𝑄1 and 𝑄2 are both free-connex. +7.3 +Aggregations +Standard relational algebra can be extended to support aggregations, +and we adopt the following formalism [2, 24]. Let p𝑆, ‘, bq be a +commutative ring7. Every tuple 𝑡 P 𝑅𝑒 has an annotation 𝑣p𝑡q P 𝑆. +For a full CQ 𝑄 in the form of p1q, the annotation for any join +result 𝑡 P 𝑄p𝐷q is defined as 𝑣p𝑡q :“ b +𝑒P𝑄 𝑣p𝑡r𝑒sq. For a non-full +query 𝜋y𝑄, the projection becomes GROUP BY y, and the annotation +for each result 𝑡 P 𝜋y𝑄p𝐷q (i.e., the aggregate of each group) is +𝑣p𝑡q :“ +‘ +𝑡1P𝑄p𝐷q:𝜋y𝑡1“𝑡 +𝑣p𝑡1q. +7In the static case, p𝑆, ‘, bq is only required to be a semi-ring, but we need additive +inverses to support deletions. +11 + +Our new change propagation framework can support aggrega- +tions easily. For any relation 𝑒, let 𝑣p𝑡q be the annotation for 𝑡 P 𝑅𝑒, +𝑣𝑠p𝑡q be the annotation for 𝑡 P 𝑉𝑠p𝑅𝑒q and 𝑣𝑝p𝑡q be the annotation +for 𝑡 P 𝑉𝑝p𝑅𝑒q. Following the definitions of semi-join view 𝑉𝑠 and +projection view 𝑉𝑝 (see Section 4), the annotation of 𝑡 P 𝑉𝑠p𝑅𝑒q can +be written as +𝑣𝑠p𝑡q :“ +˜ +b +𝑒𝑖Pt𝑒1,¨¨¨,𝑒𝑘u:𝑒𝑖Xy“H +𝑣𝑝p𝜋𝑒𝑖𝑡q +¸ +b +" +𝑣p𝑡q +𝑒 X y “ H +1 +𝑒 X y ‰ H +(10) +where t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u are the children nodes of 𝑒. The annotation +of 𝑡 P 𝑉𝑝p𝑅𝑒q can be written as +𝑣𝑝p𝑡q :“ +‘ +𝑡1P𝑉𝑠p𝑅𝑒q:𝜋keyp𝑒q𝑡1“𝑡 +𝑣𝑠p𝑡1q. +(11) +Now, we can rewrite 𝑣p𝑡q for each query result 𝑡 P 𝑄 as: +𝑣p𝑡q :“ b +𝑒P𝑄 p𝑣p𝜋𝑒𝑡q b 𝑣𝑠p𝜋𝑒𝑡qq . +(12) +We store these annotations alongside their counters in the query +plan. Then, the change propagation and enumeration procedures +should be modified according to the formulas above. More precisely, +whenever the counter of a tuple is updated, we also update its +annotation by (10) or (11). One technical difference is that, during +an S-Update, we need to keep propagating the change upwards +whenever the counter changes, not just when the counter changes +from 0 to 1 or vice versa as in Algorithm 2. Finally, after enumerating +a result, we compute its annotation by (12). +8 +EXPERIMENTS +8.1 +Setup +Prototype implementation. We have implemented our algo- +rithms and built a system prototype called CROWN (Change +pROpagation Without joiNs) on top of Flink DataStream API. All +of our algorithms are implemented as DataStream functions, which +take as input an update stream. Each tuple in the update stream +is associated with a flag indicating whether the update is an inser- +tion or deletion, as well as the name of the updated relation. After +processing an update, the DataStream function outputs the deltas +triggered by this update. Enumeration of full query results can be +invoked upon the user’s request. Implementing the prototype over +Flink allows us to inherit all the benefits of Flink, such as fault- +tolerance and the ability to work with a variety of data sources and +sinks. To dispatch tuples in a load-balanced fashion, we borrow a +similar idea from massively parallel algorithms, such as HyperCube +[3, 7, 37]. +We have evaluated our algorithms in both centralized and dis- +tributed settings. The centralized version runs on a single machine +with a single thread, where we disable certain Flink features such +as false tolerance, serialization, and dispatching. This is for a fair +comparison with other centralized systems (DBToaster and Trill) +that do not support these features. The distributed version has all +these features enabled. It runs over two machines, each equipped +with two Intel Xeon 2.1GHz processors with 48 cores and 416 GB +memory. The machine runs Linux, with Scala 2.11.12, dotnet 5.0.403, +Flink 1.13.5, and Spark 2.2.3. Each query is evaluated 10 times on +each engine and we report the average runtime. We set a 4-hour +time limit for each run. +CROWN +Flink +DBToaster +DBToaster +Trill +Spark +Distributed +✓ +✓ +✓ +Full +✓ +✓ +✓ +✓ +enumeration +Delta +✓ +✓ +enumeration +Updates +Arbitrary +FIFO +Arbitrary +Batch +Arbitrary +Internal +This +Standard +HIVM +HIVM +Standard +paper +change +change +propagation +propagation +Table 1: Comparison of different query processing engines. +Query processing engines compared. We compare CROWN +with (1) DBToaster [4], the best HIVM engine that supports multi- +way joins over arbitrary update streams in centralized settings; (2) +DBToaster Spark [32], which can support IVM with batch updates +in a distributed/parallel setting; (3) Trill [11], a continuous query +evaluation system over streaming data using the standard change +propagation framework; and (4) the native Flink SQL engine over +streaming data. +Table 1 summarizes various features of these systems. Note that +only CROWN supports both full enumeration and delta enumera- +tion. Flink can support insertion-only update streams or window +streams, but not arbitrary update streams. We run every experiment +twice: one for delta enumeration, and the other for full enumera- +tion. For full enumeration, we request the full query results after +processing every 10% of the update sequence. As Trill does not +support full enumeration, we ask Trill to report the entire delta +stream for full enumeration. +Queries and updates. We evaluate all systems over two classes +of queries. The first class contains graph pattern queries from the +benchmark by Nguyen et al. [31], over the SNAP dataset (Stanford +Network Analysis Project) [29]. Such a benchmark evaluates the +performance of each system for join queries over static data, and +we modify it to adapt to the dynamic scenario. We test all acyclic +queries from the benchmark, such as hop (path) queries, star queries +and comb queries. We also test the dumbbell query, which is a +variant of the lollipop query. The detailed query definition is given +in the Appendix D and one example of the 3-Hop query is given +below, where we use a filter over to control the output size. +SELECT G1.src as A, G2.src as B, G3.src as C, G3.dst as D +FROM G G1, G G2, G G3 +WHERE G1.dst = G2.src AND G2.dst = G3.src +AND FILTER OVER (G3.dst) +The second class includes more complex analytical queries over +the LDBC Social Network Benchmark (LDBC-SNB) [15], which +accesses the neighborhood of a given node in the graph with con- +tinuous updates. The following shows one example, which finds +the number of distinct messages associated with a particular tag +ID, while satisfying the filter conditions: +SELECT t_name, t_tagid, COUNT(DISTINCT m_messageid) +FROM tag, message, message_tag, knows +WHERE m_messageid = mt_messageid AND mt_tagid = t_tagid +AND m_creatorid = k_person2id AND m_c_replyof IS NULL +AND FILTER OVER (k_person1id) +GROUP BY t_name, t_tag_ids +12 + +𝐴 +𝐵 +𝐶 +𝐷 +𝐺1 +𝐺2 +𝐺3 +(a) 3-Hop Query +𝐴 +𝐵 +𝐶 +𝐷 +𝐺1 +𝐺2 +𝐺3 +(b) 2-Comb Query +𝐴 +𝐵 +𝐶 +𝐷 +𝐸 +𝐺1 +𝐺2 +𝐺4 +𝐺3 +(c) 4-Hop Query, SNB Q2 +𝐵 +𝐶 +𝐴 +𝐷 +𝐸 +𝐺1 +𝐺2 +𝐺4 +𝐺3 +(d) Star Query +𝑖𝑑2 +𝑖𝑑1 +𝑖𝑑3 +𝑀𝑖𝑑 +𝑇𝑖𝑑 +𝐾1 +𝐾2 +𝑀 +𝑀𝑇 +𝑇 +(e) SNB Q3 +𝑖𝑑2 +𝑖𝑑1 +𝑀𝑖𝑑 +𝑇𝑖𝑑 +𝑛𝑎𝑚𝑒 +𝐾 +𝑀 +𝑇 +𝑀𝑇 +(f) SNB Q4 +Figure 6: The relational hypergraphs of queries. The solid dots are output attributes for join-project and aggregation queries. +1e+00 +1e+01 +1e+02 +1e+03 +1e+04 +1e+05 +3-Hop +4-Hop +2-Comb +SNB Q1 +SNB Q2 +SNB Q3 +dumbbell +3-Hop +4-Hop +dumbbell +Star +SNB Q4 +Processing Time (Sec) + +CROWN +Flink +DBToaster CPP +DBToaster Spark +CROWN Delta +Trill +Aggregate Queries +Join-Project Queries +Full Join Queries +Figure 7: Processing times of CROWN, Flink, DBToaster, and Trill +1e-02 +1e-01 +1e+00 +0.3 +1 +3 +10 +Avg. Processing Time (ms) +Scale Factor +CROWN +Flink +DBToaster +Trill +(a) SNB Q1 +1e-02 +1e-01 +1e+00 +1e+01 +0.3 +1 +3 +10 +Avg. Processing Time (ms) +Scale Factor +CROWN +DBToaster +Trill +(b) SNB Q2 +1e-03 +1e-02 +1e-01 +1e+00 +1e+01 +0.3 +1 +3 +10 +Avg. Processing Time (ms) +Scale Factor +CROWN +DBToaster +Trill +(c) SNB Q4 +Figure 8: Average Processing Time v.s. Scale Factor +1e+00 +1e+01 +1e+02 +1e+03 +1e+04 +1 +2 +4 +10 +20 +Processing Time (Sec) +λ +Epinion +Google +Bitcoin +BerkStan +Figure 9: Runtime v.s. enclosureness 𝜆. +1e+01 +1e+02 +1e+03 +1e+04 +1 +2 +4 +8 +16 +32 +Processing Time (Sec) +Parallelism +CROWN 4-Hop +CROWN SNB Q3 +Flink 4-Hop +DBToaster SNB Q3 +Figure 10: Runtime v.s. parallelism 𝑝. +1e+01 +1e+02 +1e+03 +1e+04 +25% +50% +75% +Latency (ms) +Percentage of stream being processed +CROWN +Trill +Figure 11: Average latency. +Figure 6 shows the join hypergraphs of all queries. Except for +2-Comb, SNB Q3 and Q4, they have a height-2 free-connex general- +ized join tree. The star query (figure 6(d)) has a height-1 free-connex +generalized join tree, so it is q-hierarchical. The 4-Hop query (fig- +ure 6(c)) and SNB Q4 query (figure 6(f)) have the same hypergraph +structure but different output attributes, and the 4-Hop query has +a height-2 free-connex generalized join trees while SNB Q4 query +does not. +We create FIFO streams with a parameter 𝑤. For graph queries, +we assign a distinct integer 𝑡𝑒 to each edge 𝑒 in the graph, where +𝑒 has its lifespan r𝑡𝑒,𝑡𝑒 ` 𝑤s. For LDBC-SNB queries, each tuple 𝑡 +in the benchmark already has an insertion timestamp 𝑡`, and we +set its deletion time as 𝑤 days after its insertion, i.e., 𝑡´ “ 𝑡` ` 𝑤. +Note that the sliding window for graph queries is count-based, i.e., +the window always contains the same number of tuples. On the +other hand, the window for LDBC-SNB queries is time-based, so +the number of tuples in a window fluctuates over time. +8.2 +Experiment Results +Runtime. Figure 7 shows the total runtime of evaluating each +graph query over a mid-sized graph Epinions and each SNB query +in the centralized setting. The graph contains approximately 500K +edges and 76K vertices, as well as 3.7B 3-Hop paths and 378B 4-Hop +paths. On the other hand, we use the default scale factor of 1 for +all SNB queries. Under the scale factor, the total size of raw data is +1.5𝐺𝐵, and the largest relation contains 15 attributes. We set a filter +13 + +condition that only keeps 10% of the designated endpoints for all +queries. A missing bar in the figure indicates that the corresponding +system did not finish within the 4-hour limit or aborted with an +error (mostly out-of-memory errors and garbage collection timeout). +Only CROWN can finish all queries successfully. Trill only handles +a few graph queries. One possible explanation is that graph queries +tend to generate a large number of deltas. On the other hand, Flink +ran out of memory when evaluating SNB Q2, Q3, and Q4. For +those queries where the systems can finish, we see that CROWN +provides a speedup from 2x to 67x compared with Flink, 1.8x to +234x compared with DBToaster, and 2.7x to 523x compared with +Trill. Moreover, in handling join-project queries, CROWN requires +much less time than handling the corresponding full join queries, +while Flink requires more time. In addition, CROWN performs well +for both full and delta enumeration, and different modes of output +do not affect the overall performance of CROWN. +Enclosureness. To test the influences of enclosureness, we create +multiple update sequences with different 𝜆, over different graphs +from the SNAP dataset. We disable the output to see how the update +cost would change with different 𝜆. The experiment results are +shown in Figure 9. From the results, we can see the maintenance +cost of CROWN increases almost linear as 𝜆 increases. +Distributed processing. To compare CROWN with DBToaster +Spark and Flink in a distributed setting, we built a small cluster +with 32 task slots, and tested 4-Hop as well as SNB Q3 query, on +which DBToaster and Flink cannot finish in a centralized setting. +Figure 10 shows the results; missing data points or lines indicate the +system cannot finish within the time limit. Although we adopt the +HyperCube algorithm to dispatch all tuples, CROWN can still obtain +linear speedup with 𝑝 ă 16, where 𝑝 is the number of workers. +When more workers are available, the margin gain becomes smaller. +This is as expected, since (1) speedup becomes sublinear when +adding more workers implied by HyperCube; (2) the processing +time is already short, causing the system’s overhead to dominate +the entire runtime. For all finished data points, CROWN can provide +a speedup from 45x to 324x. +As Flink and DBToaster cannot finish all experiments with 128GB +memory, so we increase the memory usage for these two systems +to 500GB, where these two systems still only complete a tiny por- +tion of the experiments. On the other hand, CROWN can finish +all experiments with only 128GB of memory. If we further limit +the memory usage of CROWN to 16GB, i.e., 500MB per worker, +CROWN still works well without much change in its performance. +Latency. Finally, we tested the latency of delta enumeration, i.e., +the time between an update being received and its deltas being out- +putted. Figure 11 shows the result. The average latency of CROWN +is less than 90ms, while that of Trill is more than 6s. In addition, +the average latency is stable for CROWN, but it keeps growing for +Trill, making it infeasible to process streams for long periods. +Scalability. To test the scalability of different platforms, we change +the scale factor of the SNB benchmark and compare the average +update cost between different platforms. The experiment results +are shown in Figure 8. The results show that the average processing +time of CROWN is stable under different data sizes. In contrast, the +data size will affect the average processing time of other platforms, +suggesting CROWN has better scalability than the competitors. +1e+01 +1e+02 +1e+03 +1e+04 +0.1% +0.5% +1% +5% +10% +20% +50% 75% 1 +1e+05 +1e+06 +1e+07 +1e+08 +1e+09 +Processing Time (Sec) +Number of Results +Percentage of Input Size Compares with no Filter Condition +CROWN +Flink +DBToaster CPP +DBToaster Spark +Input +Output +Intermediate Join +(a) 3-Hop query +1e+01 +1e+02 +1e+03 +1e+04 +1e+05 +0.1% +0.5% +1% +5% +10% +20% +50% 75% 1 +1e+04 +1e+05 +1e+06 +1e+07 +1e+08 +Processing Time (Sec) +Number of Results +Percentage of Input Size Compares with no Filter Condition +(b) 4-Hop query +Figure 12: Runtime v.s. selectivity +Selectivity. Figure 12(a) shows the runtime when varying selectiv- +ity of join conditions. For standard change propagation and HIVM, +the maintenance cost depends not only on the input and output +size, but also on the size of intermediate views. For the 3-Hop +query 𝐺1p𝐴, 𝐵q 1 𝐺2p𝐵,𝐶q 1 𝐺3p𝐶, 𝐷q for 𝐺1 “ 𝐺2 “ 𝐺 and +𝐺3 “ Filterp𝐺q, the maintenance cost will be bounded by the size +of the view 𝐺1 1 𝐺2 even when 𝐺3 is empty. In the meantime, +the maintenance cost of CROWN only depends on the input and +output size. To better show such a property, we adjust the filter +condition in the 3-Hop query, which only changes |𝐺3| instead of +|𝐺1 1 𝐺2|. Trill is omitted here as it exceeded the 4-hour limit for +all data points except for the first one. When |𝐺3| ě 0.5%|𝐺|, the +output size exceeds the input size; and when |𝐺3| ě 20%|𝐺|, the +output size exceeds the intermediate join size |𝐺1 1 𝐺2|. From the +results, we can see the runtime of CROWN scales almost linearly as +|𝐺|`|𝑄|, which is as expected since the update sequence has 𝜆 “ 1. +On the other hand, the runtime of the DBToaster and Flink scales +proportionally to |𝐺1 1 𝐺2|`|𝑄|, which leads to poor performance +when |𝐺3| ď 20%|𝐺|. 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In Proc. +International Conference on Very Large Data Bases. 82–94. +A +MISSING PROOFS IN SECTION 3 +Lemma A.1 ([1, 16]). A query is 𝛼-acyclic if it has a join tree. +Lemma A.2. Given a query with a generalized join tree T, there +exists a join tree T 1 without generalized relation for the query, i.e., +the query is 𝛼-acyclic query. +Proof. It suffices to show that any generalized join tree T can +be transformed into a tree T 1 satisfying the following properties: (i) +there is a one-to-one mapping between input relations and nodes +in T 1; (ii) for each attribute 𝑥, all nodes of T 1 containing 𝑥 form a +connected component of T 1. If T does not contain any generalized +relations, then we are done, since (1)-(2) in Definition 3.1 already +implies (i)-(ii). +We start with the base case that T only contains one generalized +relation, say ˆ𝑅. Implied by (3) in Definition 3.1, ˆ𝑅 must be the root +of T. In this case, we can arbitrarily pick one of its child relations +to be the root, say 𝑅𝑐, and make all 𝑅𝑐’s siblings be the child node +of 𝑅𝑐. It is clear that the resulted tree T 1 is a join tree, since both (i) +and (ii) are preserved automatically. +In general, we choose a subtree T 2 that only contains one gen- +eralized relation which must also be the root of T 2, and perform a +local transformation on T 2 as we have done in the base case. Then, +there is no generalized relation in T 2. We repeat this procedure +until all generalized relations are removed. It can be easily checked +15 + +that (ii) is always preserved in the whole procedure. After all gen- +eralized relations are removed, the resulted tree must be a join tree, +as both (i) and (ii) are satisfied. +□ +B +MISSING PROOFS IN SECTION 5 +Proof of Lemma 5.5. We first prove the “only if" direction. For +any 𝑡 P 𝑉𝑙p𝑅𝑒q, there exists a 𝑡1 P 𝑄p𝐷q, such that 𝜋𝑒𝑡1 “ 𝑡. Mean- +while, it indicates that 𝑡2 “ 𝜋𝑝p𝑒q𝑡1 must satisfy 𝑡2 P 𝑉𝑙p𝑅𝑝p𝑒qq, +because 𝑡2 P 𝜋𝑝p𝑒q𝑄p𝐷q. Hence, 𝑡2 can join 𝑡, indicates 𝑡 1 +𝑉𝑙p𝑅𝑝p𝑒qq ‰ H. +For the “if" direction. Let 𝑡1 P 𝑉𝑙p𝑅𝑝p𝑒qq be a tuple that can join +with 𝑡. We divide the join tree T into two subtrees pT𝑒, TzT𝑒q and +divide the output attributes y into two sets py𝑒, yzy𝑒q accordingly. +Because 𝑡1 P 𝑉𝑙p𝑅𝑝p𝑒qq, 𝜋yzy𝑒𝑄p𝐷 ˙ 𝑡1q ‰ H and we let 𝑡2 be one +tuple from 𝜋yzy𝑒𝑄p𝐷˙𝑡1q. On the other side, since 𝑡 P 𝑉𝑠p𝑅𝑒q, there +also exists a tuple 𝑡𝑒 in 𝜋y𝑒 p1𝑒1PT𝑒 𝑅𝑒1q. 𝑡𝑒 can join with 𝑡2 as 𝑡 +can join with 𝑡1 and 𝜋𝑒𝑡𝑒 “ 𝑡, 𝜋𝑝p𝑒q𝑡2 “ 𝑡1. Hence, 𝑡𝑒 1 𝑡2 P 𝑄p𝐷q, +indicates that 𝑡 P 𝑉𝑙p𝑅𝑒q. +□ +Proof of Lemma 5.7. W.l.o.g, we assume that 𝑡 is inserted. The +case that 𝑡 is deleted follows the same argument. +Direction Ě. We show that each result in 𝑄p𝐷 ˙ 𝑡1q also appears +in Δ𝑄p𝐷,𝑡q, for every witness tuple 𝑡1 of 𝑡. Wlog, consider a query +result 𝑞 P 𝑄p𝐷 ˙𝑡1q for some witness tuple 𝑡1, where either 𝑡1 P 𝑅𝑒 +for 𝑒 Ď y or 𝑡1 P 𝜋y𝑅𝑒 for 𝑒 X y ´ 𝑝p𝑒q ‰ H. +First, 𝑄p𝐷 ˙ 𝑡1q Ď 𝑄p𝐷 ` 𝑡q since we have 𝑡1 P Δ +` +𝜋y𝑉𝑠p𝑅𝑒q +˘ +and 𝜋keyp𝑒q𝑡1 P 𝑉𝑝p𝑅𝑒q after the insertion of 𝑡. Hence, all results +witnessed by 𝑡1 appear in 𝑄p𝐷 ` 𝑡q after the insertion of 𝑡, i.e., +𝑞 P 𝑄p𝐷 ` 𝑡q. We next show 𝑞 R 𝑄p𝐷q. Now let’s go back to the +timestamp before the insertion of 𝑡. Implied by the definition of +witness tuple, 𝑡1 R 𝜋y𝑉𝑠p𝑅𝑒q then. We distinguish two more cases. +‚ Case 1: 𝑒 Ď y, 𝑞 R 𝑄p𝐷q since 𝜋𝑒𝑞 “ 𝑡1 but 𝑡1 R 𝜋𝑒𝑄p𝐷q before +the insertion of 𝑡. This further indicates 𝑞 R 𝑄. +‚ Case 2: 𝑒 ´y ‰ H and 𝑒 Xy´𝑝p𝑒q ‰ H, 𝑡1 R 𝜋y𝑉𝑠p𝑅𝑒q before +the insertion of 𝑡1. This way, 𝑡1 R 𝜋y𝑄p𝐷q, thus 𝑞 R 𝑄. +Combining the analysis above, we have𝑞 P 𝑄p𝐷`𝑡q, and 𝑞 R 𝑄p𝐷q +i.e., 𝑞 P Δ𝑄p𝐷,𝑡q. Thus, Ţ +𝑡1:a witness of 𝑡 𝑄p𝐷 ˙ 𝑡1q Ď Δ𝑄p𝐷 ˙ 𝑡q. +Direction Ď. We next show that every result in Δ𝑄p𝐷,𝑡q belongs +to 𝑄p𝐷 `𝑡q˙𝑡1 for some witness tuple 𝑡1 of 𝑡. Consider an arbitrary +query result 𝑞 P 𝑄p𝐷 ` 𝑡q ´ 𝑄p𝐷q. +It suffices to show that there exists at least one node 𝑒 P T such +that tuple 𝑡1 “ 𝜋𝑒𝑞 if 𝑒 Ď y, or tuple 𝑡1 P 𝜋y𝑅𝑒 with 𝑡1 “ 𝜋𝑒Xy𝑞 if +𝑒 X y ´ 𝑝p𝑒q ‰ H, must be a witness. An important observation +is that 𝑡1 now belongs to Δ𝜋y𝑉𝑠p𝑅𝑒q; otherwise, 𝑞 P 𝑄p𝐷q, coming +to a contradiction. Now consider the highest node 𝑒1 such that +𝑡1 “ 𝜋𝑒1Xy𝑞 and 𝑡1 P Δ +` +𝜋y𝑉𝑠p𝑅𝑒1q +˘ +. If 𝑒1 is the root, 𝑡1 must be a +witness of 𝑡, implied by the Definition 5.6. Otherwise, 𝑒1 is not the +root. Consider 𝑡2 “ 𝜋𝑝p𝑒1qXy𝑞. As 𝑡2 R Δ𝑉𝑠p𝑅𝑒2q, 𝑡2 must in𝑉𝑠p𝑅𝑒2q +and 𝑡2 P 𝜋𝑒2𝑄p𝐷q before the insertion of 𝑡, which indicates the +𝜋keyp𝑒1q𝑡1 “ 𝜋keyp𝑒1q𝑡2 P 𝑉𝑝p𝑅𝑒1q, and 𝜋keyp𝑒1q𝑡1 P 𝜋keyp𝑒1q𝑄p𝐷q. +In this way, 𝑡1 is a witness of 𝑡 by definition. +Critical Property: Δ𝑄p𝐷,𝑡1q X Δ𝑄p𝐷,𝑡2q “ H holds for any +pair of witness tuples 𝑡1,𝑡2. It remains to show that there is no +duplicate results in Ť +𝑡1:a witness of 𝑡 Δ𝑄p𝐷 ˙ 𝑡1q. By contradiction, +assume that there exists a query result 𝑞 with at least two witness +tuples. Wlog, let 𝑡1,𝑡2 be two distinct witness tuples in 𝑞, where +𝑡1 P 𝑅𝑒1 for some 𝑒1 Ď y or 𝑒1Xy´𝑝p𝑒1q ‰ H, and some 𝑒2 Ď y or +𝑒2 X y ´ 𝑝p𝑒2q ‰ H. First, 𝑒1 ‰ 𝑒2, as 𝑞 contains at most one tuple +in each relation. Note that the insertion of 𝑡 P 𝑅𝑒 can only change +the status of tuples in the ancestors of 𝑒. Without loss of generality, +let 𝑒1 be the ancestor of 𝑒2. Let 𝑒3 be parent node of 𝑒2 (it could be +the case that 𝑒1 “ 𝑒3). Let 𝑡3 “ 𝜋𝑒3Xy𝑞. Implied by the definition of +witness tuples, 𝑡3 P 𝜋y𝑉𝑠p𝑅𝑒3q before the insertion of 𝑡. Implied by +𝑡3 R Δ𝜋y𝑉𝑠p𝑅𝑒3q, 𝑡1 P 𝜋y𝑉𝑠p𝑅𝑒1q before the insertion, contradicting +the fact that 𝑡1 is a witness tuple. This way, each result in Δ𝑄p𝐷,𝑡q +corresponds to one witness tuple, thus there is no duplicates across +the extended query results over different witness tuples. +□ +Proof of Lemma 5.9. We first show the correctness of Algo- +rithm 6. Consider an arbitrary witness tuple 𝑡1 P 𝑅𝑒1. Denote the +nodes lying on the path from 𝑒1 to 𝑟 as 𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘p𝑟q sequentially. +We can first expand 𝑄p𝐷 ˙ 𝑡1q as follows: +𝑡1 1 +´ +1𝑘 +𝑖“1 𝑉𝑟p𝑒𝑖q +¯ +1 𝑄T𝑒1 1 +´ +1𝑘 +𝑖“2 𝑄T𝑒𝑖 ´T𝑒𝑖´1 +¯ +(13) +where 𝑄T represents the query defined over relations in T. Implied +by the join operator and the properties of generalized join tree, we +can further rewrite (13) =: +ď +𝑆P𝑡11p1𝑘 +𝑖“1𝑉𝑟 p𝑒𝑖qq +𝑆 1 𝑄T𝑒1 1 +´ +1𝑘 +𝑖“2 𝑄T𝑒𝑖 ´T𝑒𝑖´1 +¯ +“ +ď +𝑆P𝑡11p1𝑘 +𝑖“1𝑉𝑟 p𝑒𝑖qq +𝑆 ˆ p𝑄T𝑒1 ˙ t𝑆uq ˆ +´ +1𝑘 +𝑖“2 p𝑄T𝑒𝑖 ´T𝑒𝑖´1 ˙ t𝑆uq +¯ +which is exactly followed by Algorithm 6. Together with Lemma 5.7, +all results of Δ𝑄p𝐷,𝑡q are enumerated without duplication. +We next analyze the time complexity. As all witness tuples can +be stored in a data structure (e.g., a linked list) supporting constant- +delay enumeration, every 𝑡1 (line 1) can be retrieved in 𝑂p1q delay. +It then suffices to show that 𝑄p𝐷 ˙ 𝑡1q can be enumerated with +𝑂p1q delay for every 𝑡1. Note that subquery 𝑡1 1 +´ +1𝑘 +𝑖“1 𝑉𝑟p𝑒𝑖q +¯ +(line 4) can be done in 𝑂p1q delay with our hashing index. For the +remaining subquery 𝑄T𝑒1 ˙ t𝑆u or 𝑄T𝑒𝑖 ´T𝑒𝑖´1 ˙ t𝑆u, we invoke +the procedure FullEnum (line 6-8) and all query results can be +enumerated with 𝑂p1q delay, proved by Lemma 5.4. Combing those +subqueries in a form of Cartesian product can yield query results +with 𝑂p1q delay, thus completing the whole proof. +□ +C +MISSING MATERIALS IN SECTION 6 +Proof of Theorem 6.2. Given an instance of OuMv, we encode +the matrix by 𝑅3 and vectors p𝑣𝑖,𝑢𝑖q by 𝑅2 and 𝑅4 separately. We +construct an update sequence 𝑆 for 𝑄 as follows: (1) we add a tuple +𝑡 “ p𝑖, 𝑗q with lifespan 𝐼p𝑡q “ r´𝑘,𝑘s, for each pair p𝑖, 𝑗q P r𝑛sˆr𝑛s +if 𝑀𝑖𝑗 ‰ 0; (2) we add a tuple 𝑡 “ p𝑖q with lifespan 𝐼p𝑡q “ r𝑖 ´2𝑘,𝑖s +into 𝑅1 and 𝑅5; (3) for each pair of vectors p𝑣𝑖,𝑢𝑖q, we add a tuple +𝑡 “ p𝑖, 𝑣𝑖𝑗q with lifespan 𝐼p𝑡q “ r𝑖,𝑖 ` 2𝑘s to 𝑅2 if 𝑣𝑖𝑗 ‰ 0, and add +a tuple 𝑡 “ p𝑢𝑖𝑗,𝑖q with lifespan 𝐼p𝑡q “ r𝑖,𝑖 ` 2𝑘s to 𝑅4 if 𝑢𝑖𝑗 ‰ 0; +(4) if a query result is enumerated, we output true for 𝑣𝑇 +𝑖 𝑀𝑢𝑖, and +false otherwise; (5) we repeat (3)-(4) for the next pair p𝑣𝑖`1,𝑢𝑖`1q, +until 𝑛 pairs of vectors are all processed. Each tuple in 𝑆 has the +same lifespan as 2𝑘, thus it is a FIFO sequence. +16 + +𝑅5p𝑥4q +𝑅4p𝑥3,𝑥4q +𝑅3p𝑥2,𝑥3q +𝑅2p𝑥1,𝑥2q +𝑅1p𝑥1q +(a) T1 +𝑅4p𝑥3,𝑥4q +𝑅2p𝑥1,𝑥2q +𝑅3p𝑥2,𝑥3q +𝑅5p𝑥4q +𝑅1p𝑥1q +(b) T2 +𝑅4p𝑥3,𝑥4q +𝑅2p𝑥1,𝑥2q +𝑅3p𝑥2,𝑥3q +𝑅5p𝑥4q +𝑅1p𝑥1q +(c) T3 +Figure 13: Join trees for 𝑄 +“ +𝑅1p𝑥1q +1 +𝑅2p𝑥1,𝑥2q +1 +𝑅3p𝑥2,𝑥3q 1 𝑅4p𝑥3,𝑥3q 1 𝑅5p𝑥4q. +We note that in any generalized join tree T of 𝑄, there always +exists a subtree in which either 𝑅1 ´ 𝑅2 ´ 𝑅3 or 𝑅5 ´ 𝑅4 ´ 𝑅3 +is a leaf-to-root path. Wlog, assume 𝑅1 ´ 𝑅2 ´ 𝑅3 is a leaf-to- +root path. First, for each tuple 𝑡 P 𝑅1, 𝜆p𝑡q “ 1 as 𝑅1 is a leaf +node. For 𝑡 “ p𝑖, 𝑣𝑖𝑗q P 𝑅2, we observe that ˜𝐼p𝑡q “ r𝑖,𝑖s as 𝐼p𝑡q “ +r𝑖,𝑖 ` 2𝑘s and 𝐼p𝑡1q “ r𝑖 ´ 2𝑘,𝑖s for some tuple 𝑡1 P 𝑅1. But in +this case, 𝜆p𝑡q “ 1 still holds, as there exists no tuple 𝑡1 P 𝑅1 with +˜𝐼p𝑡1q Ď r𝑖,𝑖s. However, for each tuple 𝑡 P 𝑅3, 𝜆p𝑡q “ 𝑛 as there +exists a tuple 𝑡1 P 𝑅2 such that ˜𝐼p𝑡1q “ r𝑖,𝑖s for every 𝑖 P r𝑛s. +Hence, the enclosureness of 𝑆 on every generalized join tree is +𝜆 “ 𝑛2¨𝑛`𝑛2¨1 +𝑛2 +“ 𝑛. +The correctness of this simulation is obvious. This way, if there +is a data structure that can be updated in 𝑂p𝜆1´𝜖q time while sup- +porting 𝑂p𝜆2´𝜖q-delay enumeration for 𝑄 over any FIFO sequence, +then the OuMv problem can be solved in 𝑂p𝑛2 ¨𝜆1´𝜖 `𝑛 ¨𝜆2´𝜖q “ +𝑂p𝑛3´𝜖q time. Note that the construction above requires a database +of size at least 𝑛2 “ 𝜆2, thus 𝜆 ď +a +|𝐷|. +□ +Proof of Theorem 6.6. We next turn to the update cost of our +indexes. As mentioned in the beginning of Section 6, the total update +cost of the entire sequence is asymptotically dominated by that of +P-Update, which is further bounded by the number of times all the +counters countr𝑡s can change. The following lemma connects this +quantity with the enclosureness of the update sequence. +Lemma C.1. For any tuple 𝑡, countr𝑡s changes 𝑂p𝜆p𝑡qq times. +Proof. The status change of tuple 𝑡 P 𝑅𝑒 falls into one of the +following two cases: (1) tuple 𝑡 is being inserted or deleted; (2) some +tuple 𝑡1 P 𝑅𝑒1 for 𝑒1 P T𝑒 is inserted or deleted, and this update +propagates to 𝑡. Note that tuple 𝑡 can be inserted and deleted once +in its lifespan, thus bounded by 𝑂p1q and the cost is reflected in +R-Update. Then, we will focus on the second case. +We start with the case that 𝑒 has one child node in T. In this +case, countr𝑡s has its value changed between 0 and 1. Note that if an +insertion changes𝑡 from𝑅𝑒{𝑉𝑠p𝑅𝑒q to𝑉𝑠p𝑅𝑒q, subsequent insertions +won’t change the status of 𝑡 unless a deletion occurs. Consider a +set of 𝑘 disjoint intervals ˜𝐼1, ˜𝐼2, ¨ ¨ ¨ , ˜𝐼𝑘 in ordering, such that ˜𝐼𝑗 P rI𝑒, +˜𝐼𝑗 Ď 𝐼p𝑡q for each 𝑗 P r𝑘s, and there exists no additional interval ˜𝐼 +such that ˜𝐼 Ĺ ˜𝐼𝑗 or ˜𝐼1 Ď r˜𝐼` +𝑗 , ˜𝐼´ +𝑗`1s for any 𝑗 P t1, 2, ¨ ¨ ¨ ,𝑘u. Each +of the 𝑘 intervals can change the status of 𝑡 at most twice, so they +together can change the status of 𝑡 at most𝑂p𝑘q times. The effective +lifespan of 𝑡 exactly captures such a quantity. +We next consider a case when 𝑒 has two child nodes 𝑒1,𝑒2 P +T. Similarly, consider a set of 𝑘 disjoint intervals ˜𝐼1, ˜𝐼2, ¨ ¨ ¨ , ˜𝐼𝑘 in +ordering, such that ˜𝐼𝑗 P rI𝑒, ˜𝐼𝑗 Ď 𝐼p𝑡q for each 𝑗 P r𝑘s, and there +exists no additional interval ˜𝐼 such that ˜𝐼 Ĺ ˜𝐼𝑗 or ˜𝐼 Ď r˜𝐼` +𝑗 , ˜𝐼´ +𝑗`1s for +any 𝑗 P t1, 2, ¨ ¨ ¨ ,𝑘u. We can make the following two observations: +(1) For any ˜𝐼𝑗, countr𝑡s can change at most 2 times within ˜𝐼. +(2) For any two adjacent intervals ˜𝐼𝑗 and ˜𝐼𝑗`1, countr𝑡s can change +at most 4 times in their gap. +Together, we can conclude that countr𝑡s can change at most 6 ¨𝑘 “ +𝑂p𝑘q times when there are two child nodes. We next go into details +of (1) and (2) separately. +For (1), we assume ˜𝐼𝑗 P rI𝑒1 without loss of generality. By the +definition of effective lifespan, there cannot be any insertion or +deletion in any node of T𝑒1 within ˜𝐼𝑗. Nevertheless, updates may +still exist within ˜𝐼𝑗 on some node of T𝑒2, which might further change +countr𝑡s. We distinguish two more cases. If countr𝑡s changes from +1 to 2, due to an insertion from T𝑒2, a deletion must not exist within +˜𝐼𝑗 on any node of T𝑒2, implied by the fact that there exists no ˜𝐼 such +that ˜𝐼 Ĺ ˜𝐼𝑗. Hence, countr𝑡s can change at most once in ˜𝐼𝑗 for this +case. Otherwise, countr𝑡s changes from 2 to 1, after a deletion from +T𝑒2. We then go into the first case and countr𝑡s can change at most +one more time. In total, countr𝑡s can change at most twice. +For (2), it is clear that at the right endpoint of ˜𝐼𝑗 and the left +endpoint of ˜𝐼𝑗`1, countr𝑡s can change once as the deletion and +insertion of an effective lifespan. In the meantime, there does not +exist another effective lifespan within their gap, so for any 𝑒𝑖 P +t𝑒1,𝑒2u, there exists no deletion on T𝑒𝑖 in the gap following an +insertion in T𝑒𝑖 . This way, countr𝑡s can change at most four times +(i.e. 2 Ñ 1 Ñ 0 Ñ 1 Ñ 2) within their gap. +At last, we consider the general case when 𝑒 has multiple child +nodes in T. In this case, countr𝑡s has its value changed among +0, 1, ¨ ¨ ¨ , 𝑗, where 𝑗 is the number of child nodes of 𝑒. By extend- +ing the previous two observations, we conclude that countr𝑡s can +change at most 3𝑗 ¨𝑘 times, where 𝑗 can be considered as a constant. +With respect to all possible choices of 𝑘, we observe that +𝑘 ď +max +JĎ rI𝑒 +@𝐼p𝑡1qPJ,𝐼p𝑡1qĎ𝐼p𝑡q +@𝐼p𝑡2q,𝐼p𝑡3qPJ,𝐼p𝑡2qX𝐼p𝑡3q“H +1 ` |J| “ 𝜆p𝑡q, +thus countr𝑡s can change at most 𝑂p𝜆p𝑡qq times. +□ +The time cost of Algorithm 3 is determined by the number of iter- +ations of for-loop (line 2 or 8). One can easily observe that countr𝑡s +will be changed for some tuple 𝑡 once in each iteration, there- +fore the running time can be bounded by the number of changes +to countr𝑡s over all tuples 𝑡. Now consider an update sequence +𝑆 with enclosureness 𝜆. Implied by Lemma C.1 the total update +cost is 𝑂 +`ř +𝑡PI 𝜆p𝑡q +˘ +, which is 𝑂 +´ ř +𝑡PI 𝜆p𝑡q +|I| +¯ +“ 𝑂p𝜆q amortized. +Putting everything together, we have completed the proof for The- +orem 6.6. +□ +Proof of Lemma 6.8. The key idea of proving Lemma 6.8 is to +show that for any q-hierarchical query 𝑄, there exists a free-connex +join tree T such that for every relation 𝑅𝑒, 𝑒 is a leaf node in +T, i.e., T is a height-1 generalized join tree. Then by definition, +enclosureness of arbitrary update sequence over T is always 1. +17 + +We construct T for 𝑄 in the following way. In the base case, if +𝑄 only contains one relation 𝑅𝑒, we just return a single node 𝑒. In +general, we distinguish two more cases on 𝑄. If 𝑄 is connected, +we create a generalized relation containing the set of common +attributes that appearing in all relations and set it as the root. Then, +we invoke this procedure recursively for the residual query of 𝑄 +by removing such common attributes from all relations. If 𝑄 is +disconnected, we construct a tree for each connected component +of 𝑄, and add each tree as a child of the root 𝑟 (if such a root does +not exist, we add a super root 𝑟 with attributes as H. +Next, we need to show T is free-connex. Suppose not, there exists +a pair of relations 𝑒,𝑒1 P T such that 𝑒 X y “ H and 𝑒1 X y ‰ H, +but 𝑒1 is a descendent of 𝑒. Based on the recursive procedure above, +it is clear that if 𝑒 is the ancestor of 𝑒1, then 𝑒 Ď 𝑒1. In this way, there +must exist a pair of attributes 𝑥1,𝑥2 such that 𝑥2 P 𝑒, 𝑥1,𝑥2 P 𝑒1, +𝑥1 P y and 𝑥2 R y, contradicting to the fact that 𝑄 is q-hierarchical. +At last, it is easy to check that for every relation 𝑅𝑒, 𝑒 is a leaf +node. This way, we have find such a generalized join tree T, thus +completing the whole proof. +□ +Proof of Lemma 6.9. Given a height-2 generalized join tree T +and consider an arbitrary tuple 𝑡 P 𝑅𝑒. If 𝑒 is a leaf node, ˜𝐼p𝑡q “ +r𝑡`,𝑡´s and 𝜆Tp𝑡q “ 1. If 𝑒 is an internal node, ˜𝐼p𝑡q Ď r𝑡`,𝑡´s. +But here, as the join tree is a height-2 join tree, every 𝑒’s child +node must be a leaf node, hence every tuple 𝑡1 P 𝑅𝑒1 for 𝑒1 P T𝑒 +has ˜𝐼p𝑡1q “ r𝑡` +1 ,𝑡´ +1 s, and there exists no tuple 𝑡2 such that 𝑡` ă +𝑡` +2 and 𝑡´ +2 +ă 𝑡´. As each tuple 𝑡 has 𝜆Tp𝑡q “ 1, by definition, +𝜆Tp𝑆q “ 1. +□ +Proof of Lemma 6.10. As there is no deletion for every tuple 𝑡, +˜𝐼p𝑡q “ r𝑡`, `8s. Hence, for every 𝑡, by definition, 𝜆Tp𝑡q “ 1. +□ +D +SQL QUERIES +3-Hop Full Join Query. +SELECT G1.src as A, G2.src as B, G3.src as C, G3.dst as D +FROM G G1, G G2, G G3 +WHERE G1.dst = G2.src AND G2.dst = G3.src +AND FILTER OVER (G3.dst) +4-Hop Full Join Query. +SELECT G1.src as A, G2.src as B, G3.src as C, G3.dst as D, +G4.dst as E +FROM G G1, G G2, G G3, G G4 +WHERE G1.dst = G2.src AND G2.dst = G3.src AND G3.dst = G4. +src AND FILTER OVER (G4.dst) +3-Hop Join-Project Query. +SELECT G2.src as B, G3.src as C +FROM G G1, G G2, G G3 +WHERE G1.dst = G2.src AND G2.dst = G3.src +4-Hop Join-Project Query. +SELECT G2.src as B, G3.src as C, G3.dst as D +FROM G G1, G G2, G G3, G G4 +WHERE G1.dst = G2.src AND G2.dst = G3.src AND G3.dst = G4. +src +AND FILTER OVER (G4.dst) +2-Comb Query. +SELECT G1.src as A, G2.src as B, G3.src as C, G3.dst as D +FROM G G1, G G2, G G3, V1, V2 +WHERE G1.dst = G2.src AND G2.dst = G3.src AND V1.v = G1.src +and V2.v = G3.dst +Star Query. +SELECT G1.src, COUNT(G1.dst, G2.dst, G3.dst, G4.dst) +FROM G G1, G G2, G G3, G G4 +WHERE G1.src = G2.src AND G1.src = G3.src AND G1.src = G4. +src +GROUP BY G1.src; +Dumbbell Full Join Query. +SELECT * +FROM G G1, G G2, G G3, G G4, G G5, G G6, G G6 +WHERE G1.dst = G2.src +AND G2.dst = G3.src +AND G3.dst = G1.src +AND G5.dst = G6.src +AND G6.dst = G7.src +AND G7.dst = G5.src +AND G4.src = G3.dst +AND G4.dst = G5.src +Dumbbell Join-Project Query. +SELECT G4.src, G4.dst +FROM G G1, G G2, G G3, G G4, G G5, G G6, G G6 +WHERE G1.dst = G2.src +AND G2.dst = G3.src +AND G3.dst = G1.src +AND G5.dst = G6.src +AND G6.dst = G7.src +AND G7.dst = G5.src +AND G4.src = G3.dst +AND G4.dst = G5.src +SNB Query 1. +SELECT p_personid, p_firstname, p_lastname, m_messageid, +k_person1id +FROM person, message, knows +WHERE p_personid = m_creatorid +AND k_person2id = p_personid; +SNB Query 2. +SELECT k1.k_person1id, k1.k_person2id, k2.k_person2id, +t_tagid, m_messageid +FROM tag, message, message_tag, knows1 k1, knows2 k2 +WHERE m_messageid = mt_messageid +AND mt_tagid = t_tagid +AND k1.k_person2id = k2.k_person1id +AND m_creatorid = k2.k_person2id +AND m_c_replyof is NULL +AND FILTER OVER (k1.k_person1id) +SNB Query 3. +18 + +SELECT k1.k_person1id, k1.k_person2id, k2.k_person2id, +t_tagid, m_messageid +FROM tag, message, message_tag, knows1 k1, knows2 k2 +WHERE m_messageid = mt_messageid +AND mt_tagid = t_tagid +AND k1.k_person2id = k2.k_person1id +AND k2.k_person2id <> k1.k_person1id +AND m_creatorid = k2.k_person2id +AND m_c_replyof is NULL +AND FILTER OVER (k1.k_person1id) +SNB Query 4. +SELECT t_name, t_tagid, count(distinct m_messageid) +FROM tag, message, message_tag, knows +WHERE m_messageid = mt_messageid +AND mt_tagid = t_tagid +AND m_creatorid = k_person2id +AND m_c_replyof is NULL +AND FILTER OVER (k_person1id) +GROUP BY t_name, t_tagid +19 + diff --git a/ANE2T4oBgHgl3EQfnAgQ/content/tmp_files/load_file.txt b/ANE2T4oBgHgl3EQfnAgQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2f6ece15e1dc33ea77db7ce64e04f4d4c880e126 --- /dev/null +++ b/ANE2T4oBgHgl3EQfnAgQ/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf,len=1478 +page_content='Change Propagation Without Joins Qichen Wang Hong Kong Baptist University qcwang@hkbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='hk Xiao Hu University of Waterloo xiaohu@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='ca Binyang Dai, Ke Yi HKUST {bdaiab,yike}@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='hk ABSTRACT We revisit the classical change propagation framework for query evaluation under updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The standard framework takes a query plan and materializes the intermediate views, which incurs high polynomial costs in both space and time, with the join operator be- ing the culprit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this paper, we propose a new change propagation framework without joins, thus naturally avoiding this polynomial blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Meanwhile, we show that the new framework still supports constant-delay enumeration of both the deltas and the full query results, the same as in the standard framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, we provide a quantitative analysis of its update cost, which not only recovers many recent theoretical results on the problem, but also yields an effective approach to optimizing the query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The new framework is also easy to be integrated into an existing stream- ing database system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Experimental results show that our system prototype, implemented using Flink DataStream API, significantly outperforms other systems in terms of space, time, and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 INTRODUCTION We study the problem of query evaluation under updates, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' incremental view maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a query 𝑄, a database 𝐷, and a sequence of updates, where each update is either the insertion or deletion of a tuple, the goal is to maintain the query results 𝑄p𝐷q continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, there are two modes to return the updated𝑄p𝐷q to the user (an end user or an upper-level application): full enumeration and delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The former is pull-based, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the system returns 𝑄p𝐷q passively upon request of the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' while in the latter case, we push the delta Δ𝑄p𝐷,𝑡q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the change to 𝑄p𝐷q caused by the insertion/deletion of 𝑡, to the user after each update 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' These two modes are applicable to different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Full enumeration cannot be done too frequently if 𝑄p𝐷q is large, and it may miss some ephemeral events in between two requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Delta enumeration offers real-time responses with low latency, but it requires the user to have the ability to “consume” the deltas in a timely fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It can be considered as a stream-in-stream-out operator, where the input is a stream of updates to the base tables, while the output is a stream of updates to the query result (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', a stream of deltas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If the user wishes to always have a complete and accurate 𝑄p𝐷q, it has to maintain 𝑄p𝐷q and update it with the deltas as they are received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If approximation is acceptable, some more efficient streaming algorithms can be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Change propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Change propagation [12, 28, 35] is a widely used framework in database systems for solving this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It can be instantiated with any query plan, which is a tree where the leaves are the base relations and each internal node is a relational operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' At each internal node, it maintains the results of the sub- query corresponding to the subtree at this internal node, which is often called a materialized view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 1(a) shows a particular 𝑉4 “ 𝑉2 1 𝑉3 𝑉1 “ 𝑅2 1 𝑅3 𝑉3 “ 𝜋𝑥4𝑅4 𝑅4 𝑉2 “ 𝑅1 1 𝑉1 𝑅2 𝑅3 𝑅1 (a) Old plan 𝑉2 “ 𝑅1 1 𝑅2 1 𝑅3 1 𝑉1 𝑅1 𝑅2 𝑅3 𝑉1 “ 𝜋𝑥4𝑅4 (b) Another old plan X : 𝑉𝑠pr𝑥3sq 𝜋 : 𝑉𝑝p𝑅2q 𝜋 : 𝑉𝑝p𝑅3q ˙ : 𝑉𝑠p𝑅2q ˙ : 𝑉𝑠p𝑅3q 𝜋 : 𝑉𝑝p𝑅1q 𝑅2 𝑅3 𝜋 : 𝑉𝑝p𝑅4q 𝑉𝑠p𝑅1q “ 𝑅1 𝑉𝑠p𝑅4q “ 𝑅4 𝑅1 𝑅2 𝑅3 𝑅4 r𝑥3s (c) Our new plan Figure 1: For 𝑄 “ 𝜋𝑥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥4𝑅1p𝑥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥2q 1 𝑅2p𝑥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥3q 1 𝑅3p𝑥3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥4q 1 𝑅4p𝑥4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥5q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1(a) and 1(b) are two plans under the standard change propagation framework and 1(c) is our new plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' query plan for the query 4-Hop query from benchmark [31] 𝑄 :“ 𝜋𝑥1,𝑥2,𝑥3,𝑥4𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q 1 𝑅3p𝑥3,𝑥4q 1 𝑅4p𝑥4,𝑥5q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Under the standard change propagation framework, we maintain four materialized views𝑉1,𝑉2,𝑉3,𝑉4 “ 𝑄 (if only delta enumeration is needed, then 𝑉4 need not be maintained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When a tuple 𝑡 is inserted or deleted in a relation, say 𝑅1, it follows the leaf-to-root path to propagate the deltas to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, it first computes Δ𝑉2 “ Δ𝑅1 1 𝑉1 “ 𝑡 1 𝑉1, then computes Δ𝑄 “ Δ𝑉4 “ Δ𝑉2 1 𝑉3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that with the help of the materialized views, it avoids re-computing some of the sub-queries during updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, the penalty is space: both𝑉1 and𝑉2 can have quadratic size in the worst case [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To avoid space blowup, one can use a different query plan, say, the one shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This query plan does not have any materialized views (except 𝑉1 “ 𝜋𝑥4𝑅4, which has at most linear size), but it has to compute a multi-way join, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝑅1 1 𝑅2 1 𝑅3 1 𝑡 upon each update in 𝑅4, which could take quadratic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Making things worse, this quadratic blowup exacerbates for queries involving more relations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Prior work has designed advanced techniques to address this space or time blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The Dynamic Yannakakis algorithm [21–23] has linear space and linear update time while supporting constant- delay enumeration for free-connex queries1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the update time fur- ther reduces to 𝑂p1q amortized2 for q-hierarchical queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Concur- rently, Berkholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [8] designed a different algorithm for the q-hierarchical case with the same space/time guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, these algorithms have not been integrated into any full-fledged database or data warehouse products, possibly due to the complica- tions of the techniques and the use of non-standard operations not routinely found in existing database systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Change propagation without joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The main contribution of this paper is to achieve (and improve for certain classes of queries 1All technical terms in the introduction are formally defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2All update time bounds are amortized in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='04003v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='DB] 10 Jan 2023 and/or update sequences) the results above, but still under the standard change propagation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Our observation is that the only relational operator that may cause a super-linear blowup is join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, if the query plan has no joins, then both space and update time will be at most linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To avoid joins, our high-level strategy is to replace each join in the query plan by a semi-join (or an intersection) plus a projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, not every query plan is amenable to this replacement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The key technical contribution of this paper, therefore, is the construction of such a query plan for every free-connex conjunctive query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, such a join-free query plan for the earlier query is shown in Figure 1(c), which will be elaborated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Since our query plan has no joins, linear space and linear up- date time follow straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Still, two technical challenges remain: (1) how to support constant-delay enumeration, and (2) how to achieve an update time better than linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (1) is trivial under a traditional query plan where the root corresponds to the query results 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Since our query plan is join-free, no node in the plan corresponds to 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Instead, our query plan can be considered as a compact, linear-size representation of a polynomially sized 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By borrowing ideas from the static case [6], we show how to enumerate 𝑄p𝐷q with constant delay, by appropriately traversing this compact representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Supporting constant-delay enumera- tion of the delta Δ𝑄p𝐷,𝑡q, on the other hand, is quite different from the static case, and we need new techniques which exploit some important properties of our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To address issue (2), Wang and Yi [37] introduced the notion of enclosureness 𝜆 of an update sequence, which captures the hardness of the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It is linear in the worst case, but is often a constant in many common cases, such as any first-in-first-out (FIFO) update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' They also designed an algorithm with update cost 𝑂p𝜆q for foreign-key acyclic queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Such queries are relatively easy to handle since their result size is at most linear, so they are immune to the polynomial blowup problem caused by non-key joins, such as free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Indeed, we show (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2) that there is a simple free-connex query for which it is impossible to achieve 𝑂p|𝐷|1{2´𝜀q update time even over FIFO update sequences, which implies that the previous definition of 𝜆 is not achievable for free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Nevertheless, we show that, after a simple relaxation of the definition, 𝜆 is still an appropriate measure of the update complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' in particular, we show that change propagation under our query plan achieves 𝑂p𝜆q update time for every free- connex query under the new definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To further illustrate the usefulness of our new definition of 𝜆, we show that for certain queries (such as q-hierarchical queries) and/or update sequences (such as FIFO or insertion-only), 𝜆 is indeed a small constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For general queries, 𝜆 also provides guidance on what would constitute a good query plan for change propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Specifically, this paper achieves the following results: (1) We show how to construct a change propagation query plan without joins for any free-connex conjunctive query, such that the space needed by the query plan is linear and the update time is 𝑂p𝜆q, for an appropriately defined notion of enclosureness 𝜆 of the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) We show how to support constant-delay enumeration of both full query results and each delta in our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (3) We show that 𝜆 is a constant for certain classes of conjunctive queries (such as q-hierarchical queries) and/or special update sequences (such as FIFO or insertion-only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' These results not only recover the prior known result of [8, 21] on q-hierarchical queries, but also extend it to cover many other cases commonly encountered in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (4) We show how our framework can handle various extensions such as selections, aggregations, and non-free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (5) We demonstrate the practicality of our new framework by im- plementing it on top of Flink and comparing it with state-of- the-art view maintenance and SQL-over-stream systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2 RELATED WORK Our new change propagation framework is inspired by several lines of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In the static case, the classical Yannakakis algorithm [38] has runtime 𝑂p|𝐷| ` |𝑄p𝐷q|q for every free-connex query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It consists of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The first stage uses a series of semi-joins to remove all the dangling tuples in 𝑂p|𝐷|q time, and the second stage performs pairwise joins to compute 𝑄p𝐷q in 𝑂p|𝑄p𝐷q|q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The Dynamic Yannakakis algorithm [21] extends the algorithm to the dynamic case, but it deviates from the change propagation framework, making it harder to integrate into existing database systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Our algorithm can also be viewed as a dynamic version of the Yannakakis algorithm, but it strictly follows the standard change propagation framework while achieving a better runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The Dynamic Yannakakis algorithm has an update cost of 𝑂p|𝐷|q for free-connex queries, while our algorithm achieves 𝑂p𝜆q up- date time, where 𝜆 is the enclosureness of the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We have 𝜆 ď |𝐷| for all update sequences, while the former is usually much smaller on real-world update sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, Dy- namic Yannakakis achieves𝑂p1q update time only for q-hierarchical queries, while our algorithm also achieves 𝑂p1q update time for non-q-hierarchical queries if the update sequences enjoy some spe- cial properties, such as first-in-first-out or insertion-only (formally defined in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The gap between Dynamic Yannakakis and our algorithm can be as large as Θp|𝐷|q on some non-q-hierarchical queries (see Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Bagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [6] observe that, in the static case, the second stage of the Yannakakis algorithm can be enhanced to support constant- delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We adapt their ideas to support enumeration in the dynamic case for our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, as there is no notion of delta in the static case, we need some new ideas to support delta enumeration with constant delay, which non-trivially relies on some nice features of our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [27] show that it is possible to increase the enumera- tion delay in exchange for faster update time, on hierarchical (but non-q-hierarchical) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We have not considered this trade-off, as we believe the constant delay is important, and our update cost 𝜆 is low enough for most queries and update sequences already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, their trade-off only applies to full enumeration, not delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Nevertheless, for cases where 𝜆 is high, it would be an interesting direction to explore such a trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In the standard change propagation framework, a single update to a base relation may incur many changes in the intermediate views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Higher-Order Incremental View Maintenance (HIVM) [4] has been proposed to remedy this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It takes the changes to a 2 view as another query (delta query) and maintains this delta query recursively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' HIVM improves upon IVM for many complex queries in practice, and it can also extend to accelerate several machine learning tasks [33, 34], but there is no theoretical guarantee on its update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, HIVM still uses super-linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The problem is also related to stream joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In particular, a cash- register stream corresponds to an insertion-only update sequence, while a turnstile stream is an update sequence with arbitrary inser- tions and deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The sliding-window stream model is a special case of a FIFO update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Most stream processing systems like Flink [9] and Trill [11] use standard change propagation for multi-way stream joins, which we will compare against in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Some specialized systems are designed for two-way stream joins [14, 17, 25, 30, 36], but they do not extend to multi-way joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3 PRELIMINARIES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 Problem Definition Conjunctive queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We focus on conjunctive queries (CQ) of the following form: 𝑄 :“ 𝜋y p𝑅1p𝑒1q 1 𝑅2p𝑒2q 1 ¨ ¨ ¨ 1 𝑅𝑛p𝑒𝑛qq, (1) where each 𝑅𝑖 is a relation with a set of attributes/variables 𝑒𝑖, 𝑖 “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each tuple 𝑡 P 𝑅𝑖 assigns a value to each attribute in 𝑒𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any 𝑥 P 𝑒𝑖, 𝑡r𝑥s “ 𝜋𝑥𝑡 denotes the value of 𝑡 on attribute 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Similarly, for a subset of attributes 𝑒 Ď 𝑒𝑖, 𝑡r𝑒s “ 𝜋𝑒𝑡 denotes the tuple formed by the values of 𝑡 on the attributes in 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let V “ 𝑒1 Y ¨ ¨ ¨ Y 𝑒𝑛 be the set of all attributes in the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We call y Ď V the output attributes, while ¯y “ V ´ y are the non- output attributes, also known as the existential variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If y “ V, such a query is known as a full join query;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' otherwise, it is said to be join-project query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For simplicity, we assume that each 𝑅𝑖 in 𝑄 is distinct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the query does not have self-joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Nevertheless, self-joins can be taken care of easily: Suppose a relation 𝑅 appears twice in the query (with different attribute renamings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then we consider them as two identical copies of 𝑅, and for any update to 𝑅, we apply the update to both copies of 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a database 𝐷, we write 𝑄p𝐷q for the query results of 𝑄 on 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We use 𝑄p𝐷 ˙ 𝑡q to denote the query results that depend on a given tuple 𝑡, and call 𝑄p𝐷 ˙ 𝑡q the query results witnessed by 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Such a witness query will be frequently used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a query 𝑄 in the form of (1) and a tuple 𝑡 P 𝑅𝑖, it is clear that 𝑄p𝐷 ˙ 𝑡q “ 𝜋y p𝑅1 1 ¨ ¨ ¨ 1 𝑅𝑖´1 1 t𝑡u 1 𝑅𝑖`1 1 ¨ ¨ ¨ 1 𝑅𝑛q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that for a full join CQ, we have 𝑄p𝐷 ˙ 𝑡q “ 𝑄p𝐷 ` 𝑡q ˙ 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' for join-project queries, 𝑡 itself may not appear in 𝑄p𝐷 ˙ 𝑡q due to the projection on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When analyzing the costs of algorithms, we adopt the notion of data complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the size of the query 𝑄 is taken as a constant while |𝐷| is an asymptotic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Semi-joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The semi-join 𝑅𝑖p𝑥𝑖q ˙ 𝑅𝑗p𝑥𝑗q is defined as 𝑅𝑖p𝑥𝑖q ˙ 𝑅𝑗p𝑥𝑗q “ t𝑡|𝑡 P 𝜋𝑥𝑖𝑅𝑖 1 𝑅𝑗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Updates and Deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' An update to a database 𝐷 is either the inser- tion or deletion of a tuple 𝑡 in some relation 𝑅𝑖 of 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this paper, we adopt set semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We denote 𝐷 ` 𝑡 as the database after inserting 𝑡 and 𝐷 ´ 𝑡 as the database after deleting 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In particular, this means that if 𝑅𝑖 already contains 𝑡, then inserting 𝑡 into 𝑅𝑖 will not change 𝑅𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' if 𝑅𝑖 does not contain 𝑡, deleting 𝑡 from 𝑅𝑖 has no effect, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We ignore these non-effective updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The delta of an update to 𝑄 is defined as Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷 ` 𝑡q ´ 𝑄p𝐷q in case of the insertion of 𝑡 and Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷q ´ 𝑄p𝐷 ´ 𝑡q in the case of deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For a full join query, Δ𝑄p𝐷,𝑡q “ 𝑄p𝐷 ˙ 𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For join-project queries, Δ𝑄p𝐷,𝑡q Ď 𝑄p𝐷 ˙ 𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In particular, it is possible to have Δ𝑄p𝐷,𝑡q “ H even if 𝑄p𝐷 ˙ 𝑡q ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We target constant delay [6] for both full and delta enumeration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the time between the start of the enumeration process to the first tuple in 𝑄p𝐷q (or Δ𝑄p𝐷,𝑡q), the time between any consecutive pair of tuples, and the time between the last tuple and the termination of the enumeration process should all be bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Classification of CQs Acyclic queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The acyclicity of a CQ 𝑄 is defined by the acyclic- ity of the hypergraph pV, t𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑒𝑛uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, 𝑄 is acyclic if there exists a join tree T, whose nodes are the relations in 𝑄 such that, for each attribute 𝑥 P V, all nodes of T containing 𝑥 form a connected component of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, Figures 2(a) and 2(b) are two possible join trees for the query 𝑄1 :“ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We will often not distinguish between a node in T and the relation it represents, or its set of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this paper, we use an equivalent definition based on generalized relations [13, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Different from previous definition of generalized relation, it now can be a proper subset of any 𝑒𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We can show that the following is an equivalent definition of acyclic queries3: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 (Acyclic queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A CQ 𝑄 is acyclic if there exists a rooted join tree T satisfying the following properties: (1) each input relation in𝑄 corresponds to a unique node in T, each leaf of T corresponds to an input relation, and each internal node in T corresponds to either an input relation in 𝑄 or one of its generalized relations (some generalized relations may not appear in T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) for each attribute 𝑥, all nodes of T containing 𝑥 form a con- nected component of T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (3) a node corresponding to a generalized relation must appear above any node corresponding to an input relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and (4) if 𝑒 is the parent of 𝑒1 in T and 𝑒 is a generalized relation, 𝑒 Ď 𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' An example is given in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In a generalized join tree T, we use 𝑟 to denote the root, and T𝑒 for the subtree rooted at node 𝑒, C𝑒 for the set of children of node 𝑒 and 𝑝p𝑒q for the parent of node 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let keyp𝑒q “ 𝑒 X 𝑝p𝑒q be the join key between node 𝑒 and 𝑝p𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The height of T is defined as the maximum number of relations on any leaf-to-root path, without counting generalized relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A CQ 𝑄 is free-connex if the hypergraphs pV, t𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑒𝑛uq and pV, t𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑒𝑛, yuq are both acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By defi- nition, any free-connex query must by acyclic, and an acyclic full join query must be free-connex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For our development, we need the following equivalent definition of free-connex queries: 3Proof of equivalence is given in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3 𝑅2p𝑥2,𝑥3q 𝑅1p𝑥1,𝑥2q (a) T1 𝑅1p𝑥1,𝑥2q 𝑅2p𝑥2,𝑥3q (b) T2 𝑅2p𝑥2,𝑥3q 𝑅1p𝑥1,𝑥2q r𝑥2s (c) T3 Figure 2: Three (generalized) join trees for 𝑄1 “ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In 2(c), node r𝑥2s is a generalized relation with one attribute 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The height of T1, T2 is 2 and that of T3 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 (Free-connex queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A CQ 𝑄 is free-connex if it has a generalized join tree T, such that 𝑟 Ď y, and for every 𝑥1 P y and every 𝑥2 P V ´ y, topp𝑥2q is not an ancestor of topp𝑥1q in T, where topp𝑥q is the highest node in T that contains 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Such a T is called a free-connex join tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, for the query 𝑄1 1 :“ 𝜋𝑥2𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q, all three join trees in Figure 2 are valid free-connex join trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If the output attribute is 𝑥1, then only Figure 2(a) is a valid free-connex join tree (so it does not have a height-1 free-connex join tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If the output attributes are p𝑥1,𝑥3q, then the query is not free-connex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Q-hierarchical queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A more restricted subclass of free-connex queries is q-hierarchical query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let E𝑥 denote the set of relations containing attribute 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 (Q-hierarchical queries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A CQ 𝑄 is q-hierarchical if (1) for every pair of attributes 𝑥1,𝑥2, either E𝑥1 Ď E𝑥2 or E𝑥2 Ď E𝑥1 or E𝑥1 X E𝑥2 “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and (2) for every pair of attributes 𝑥1,𝑥2, if 𝑥1 P y and E𝑥1 Ĺ E𝑥2, then 𝑥2 P y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' These classifications capture the hardness of evaluation or enu- meration for a CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Firstly, a full join query can be evaluated in linear time in terms of input and output size if and only if it is acyclic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' for join-project CQs, this complete class extends to free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, free-connex and q-hierarchical CQs have played important roles in query enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [6] showed that in static settings, constant-delay enumeration after a linear-time pre- processing step is possible for a CQ if and only if it is free-connex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Berkholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [8] showed that in dynamic settings, constant-delay enumeration is possible for a CQ from a data structure that can be updated in constant time if and only if it is q-hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4 CHANGE PROPAGATION WITHOUT JOINS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 A New Query Plan Given a free-connex query𝑄, our new query plan is guided by a free- connex (generalized) join tree T of 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We illustrate the construction using the query in Figure 1 with the join tree highlighted in red (note that the join tree is not unique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A normal query plan following this join tree would compute a series of joins p𝑅1 1 𝑅2q 1 p𝑅3 1 𝜋𝑥4𝑅4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In our new query plan, we replace each join with a semi-join followed by a projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, we maintain two views for each node 𝑒 P T, a semi-join view 𝑉𝑠p𝑅𝑒q and a projection view 𝑉𝑝p𝑅𝑒q, defined recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Every non-root node 𝑒 P T has a projection view 𝑉𝑝p𝑅𝑒q :“ 𝜋keyp𝑒q𝑉𝑠p𝑅𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) Noted that the root node does not have a projection view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To define the semi-join view 𝑉𝑠p𝑅𝑒q, we distinguish three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithm 1: PlanGenerationp𝑄,𝑇q Input :A generalized join tree 𝑇 for query 𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output:A new query plan 𝑇 for 𝑄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 foreach node 𝑒 in a postorder traversal of 𝑇 do 2 Replace node 𝑒 with 𝑉𝑠p𝑅𝑒q in 𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3 if 𝑒 is not the root of 𝑇 then 4 Add 𝑉𝑝p𝑅𝑒q between 𝑉𝑠p𝑅𝑒q and the parent of 𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5 Return 𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (i) If 𝑒 is a leaf, 𝑅𝑒 is an input relation, and 𝑉𝑠p𝑅𝑒q :“ 𝑅p𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (ii) If 𝑒 is an internal node and 𝑅𝑒 is an input relation, then 𝑉𝑠p𝑅𝑒q :“ 𝑅𝑒 ˙ 𝑉𝑝p𝑅𝑒1q ˙ ¨ ¨ ¨ ˙ 𝑉𝑝p𝑅𝑒𝑘 q, (3) where C𝑒 “ t𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑒𝑘u are the children of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (iii) If 𝑒 is an internal node that corresponds to a generalized virtual relation 𝑅𝑒, since all the 𝑉𝑝p𝑅𝑒𝑖 q’s have the same attributes keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 “ 𝑒 for every 𝑖 (by the last property in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1), (3) simplifies to an intersection: 𝑉𝑠p𝑅𝑒q :“ 𝑉𝑝p𝑅𝑒1q X ¨ ¨ ¨ X 𝑉𝑝p𝑅𝑒𝑘 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (4) Our query plan simply connects these views together using the formulae above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithm 1 takes as input a generalized join tree, and outputs a new query plan under our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 1(c) shows the new query plan for the example query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that 𝑅2 and 𝑅4 fall into case (ii), while the root node r𝑥3s is under case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As neither projection nor semi-join (including intersection as a special case) enlarges the input relations, the following is straight- forward: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' All views in our query plan have size 𝑂p|𝐷|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 3(a) shows the initial index built for the query in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For𝑅1 and𝑅4, both semi-join and projection views are defined as themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑉𝑠p𝑅2q contains tuples in 𝑅2 that can join with 𝑉𝑝p𝑅1q, which include p2, 2q and p2, 4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑉𝑠p𝑅3q is defined similarly including 4 tuples from 𝑅3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the generalized node r𝑥3s, we define the virtual relation 𝑅pr𝑥3sq “ 𝑉𝑝p𝑅2q Y 𝑉𝑝p𝑅3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Only tuple p4q belongs to 𝑅pr𝑥3sq, since every other tuple in 𝑅pr𝑥3sq fails to join with 𝑉𝑝p𝑅2q and 𝑉𝑝p𝑅3q: their counters need to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Change propagation Change propagation using our new query plan can be done using standard (actually, even simpler for certain operators) propagation formulae [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For completeness, we briefly describe them below, which are also needed to understand the algorithms in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' S-Update When there is an update to 𝑉𝑠p𝑅𝑒q for some 𝑒, we use an S-Update to update 𝑉𝑝p𝑅𝑒q by formula (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This can be done in 𝑂p1q time by derivation counting [12], a standard technique to propagate changes through a projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Specifically, we associate a counter countr𝑡1s for each tuple 𝑡1 P 𝑉𝑝p𝑅𝑒q that stores the number of tuples 𝑡 P 𝑉𝑠p𝑅𝑒q such that 𝑡rkeyp𝑒qs “ 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The detailed process, which needs to distinguish between an insertion and a deletion, is given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that for the algorithm to run in 𝑂p1q time, we need a hash index on 𝑉𝑝p𝑅𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' P-Update Let 𝑒𝑖 be a child of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When there is an update to some 𝑉𝑝p𝑅𝑒𝑖 q, we use a P-Update to update 𝑉𝑠p𝑅𝑒q by formula (3) in the 4 Algorithm 2: S-Updatep𝑒,𝑡q Input :An update 𝑡 from 𝑉𝑠p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output:Updated 𝑉𝑝p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 𝑡1 Ð 𝑡rkeyp𝑒qs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2 if 𝑡 is an insertion into 𝑉𝑠p𝑅𝑒q then 3 if 𝑡1 P 𝑉𝑝p𝑅𝑒q then countr𝑡1s Ð countr𝑡1s ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4 else 5 𝑉𝑝p𝑅𝑒q Ð 𝑉𝑝p𝑅𝑒q Y t𝑡1u, countr𝑡1s Ð 1, P-Update(𝑝p𝑒q,𝑡1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6 else 7 if countr𝑡1s “ 1 then 8 𝑉𝑝p𝑅𝑒q Ð 𝑉𝑝p𝑅𝑒q ´ t𝑡1u, P-Update(𝑝p𝑒q,𝑡1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 9 else countr𝑡1s Ð countr𝑡1s ´ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithm 3: P-Update(𝑒,𝑡) Input :An update 𝑡 from 𝑉𝑝p𝑅𝑒𝑖 q for some 𝑒𝑖 P C𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output:Updated 𝑉𝑠p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 if 𝑡 is an insertion into 𝑉𝑝p𝑅𝑒𝑖 q then 2 foreach 𝑡1 P 𝑅𝑒 with 𝑡1rkeyp𝑒𝑖qs “ 𝑡 do 3 countr𝑡1s Ð countr𝑡1s ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4 if countr𝑡1s “ |C𝑒| then 5 𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q Y t𝑡1u, S-Updatep𝑒,𝑡1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6 else 7 foreach 𝑡1 P 𝑅𝑒 with 𝑡1rkeyp𝑒𝑖qs “ 𝑡 do 8 countr𝑡1s Ð countr𝑡1s ´ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 9 if countr𝑡1s “ |𝐶𝑒| ´ 1 then 10 𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q ´ t𝑡1u, S-Updatep𝑒,𝑡1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' case where 𝑒 is an input relation or (4) in case 𝑒 is a generalized relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We consider the former case first;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the latter case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The standard change propagation formula for a semi-join [19] rewrites it as a join followed by a projection, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝑅𝑒 ˙ 𝑅𝑒𝑖 :“ 𝜋𝑒p𝑅𝑒 1 𝑅𝑒𝑖 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This defeats the whole purpose of avoiding joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, observe that in our query plan, 𝑅𝑒𝑖 has already been projected onto keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 Ď 𝑒 before the semi-join, thus this allows a very simple and efficient way to maintain the whole multi- way semi-join (3) as one operator, which can also be considered as a “horizontal” version of derivation counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, we maintain a counter countr𝑡1s for every tuple 𝑡1 in 𝑅𝑒, storing the number of child nodes 𝑒𝑖 P C𝑒 such that 𝑡1rkeyp𝑒𝑖qs P 𝑉𝑝p𝑅𝑒𝑖 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A tuple 𝑡1 appears in 𝑉𝑠p𝑅𝑒q if and only if countr𝑡1s “ |C𝑒|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The algorithm is then immediate, as shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We also need a hash index (that needs to support 𝑒 X 𝑒𝑖 as the key for each 𝑒𝑖 P C𝑒) on 𝑅𝑒 so that each counter change can be done in 𝑂p1q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, unlike the S-Update, a P-Update may take more than constant time since multiple tuples may change their counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In fact, this is the only place where the update time blows up during change propagation in our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' R-Update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The last case is when there is an update in an input relation 𝑅𝑒, we also need to update 𝑉𝑠p𝑅𝑒q by formula (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We call this an R-Update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The detailed procedure, given in Algorithm 4, Algorithm 4: R-Update(𝑒,𝑡) Input :An update 𝑡 from an input relation 𝑅𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output:Updated 𝑉𝑠p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 if 𝑡 is an insertion into 𝑅𝑒 then 2 countr𝑡s Ð 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3 foreach 𝑒𝑖 P C𝑒 do 4 if 𝑡rkeyp𝑒𝑖qs P 𝑉𝑝p𝑒𝑖q then countr𝑡s Ð countr𝑡s ` 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5 if countr𝑡s “ |C𝑒| then 6 𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q Y t𝑡u, S-Updatep𝑒,𝑡q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7 else 8 if countr𝑡s “ |C𝑒| then 9 𝑉𝑠p𝑅𝑒q Ð 𝑉𝑠p𝑅𝑒q ´ t𝑡u, S-Updatep𝑒,𝑡q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' simply maintains the counters in 𝑅𝑒, and then 𝑉𝑠p𝑅𝑒q, in a straight- forward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It is obvious that an R-Update takes 𝑂p1q time (also using the hash index on 𝑉𝑝p𝑅𝑒𝑖 q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 3(b) shows the index after inserting p1, 1q into 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This new tuple first triggers an insertion to 𝑉𝑝p𝑅1q, which further increments counters of the three tuples in 𝑉𝑠p𝑅2q with 𝑥2 “ 1, which are then brought into 𝑉𝑠p𝑅2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' From here, the propa- gation diverges into three paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Tuple p1, 2q P 𝑅2 increments the counter of p1q P 𝑉𝑝p𝑅2q but this propagation path stops here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Tuple p1, 1q P 𝑉𝑠p𝑅2q first inserts a new tuple p1q to 𝑉𝑝p𝑅2q, which then further increments the counter of tuple p1q in the root, bringing it to 𝑉𝑠pr𝑥3sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Tuple p1, 4q P 𝑅2 increments the counter of p4q P 𝑉𝑝p𝑅2q and the propagation stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 3(c) shows the index after deleting p1, 1q from𝑅4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This dele- tion first decrements the counter of tuple p1q P 𝑉𝑝p𝑅4q, removing it from 𝑉𝑝p𝑅4q, and further decrements the counter of p1, 1q P 𝑉𝑠p𝑅3q, removing it from𝑉𝑠p𝑅3q as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, the counter of p1q P 𝑉𝑝p𝑅3q decreases from 2 to 1, and the propagation stops here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' All projection and semi-join views in our query plan can be updated in 𝑂p|𝐷|q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5 ENUMERATION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 Full Result Enumeration We first consider how to perform constant-delay enumeration of 𝑄p𝐷q from our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We need the following lemma: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any node 𝑒, 𝑉𝑠p𝑅𝑒q “ 𝜋𝑒p1𝑒1PT𝑒 𝑅𝑒1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We prove it by the induction on the height of generalized join tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, it holds for any leaf node 𝑒, since 𝑉𝑠p𝑅𝑒q “ 𝑅𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next consider an arbitrary internal node 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let 𝐶𝑒 “ t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u be the set of children of node 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By hypothesis, we assume this lemma holds for every 𝑒𝑖 P 𝐶𝑒, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑉𝑠p𝑅𝑒𝑖 q “ 𝜋𝑒𝑖 p1𝑒1PT𝑒𝑖 𝑅𝑒1q 5 Algorithm 5: FullEnumpT,𝑒,𝑡q Input: A free-connex generalized join tree T, a node 𝑒 P T and a tuple 𝑡 P 𝑅𝑒;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output: Query results over T𝑒 that can be joined with 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 if 𝑒 ´ y ‰ H then 2 if 𝑒 X y ´ 𝑝p𝑒q “ H then Yield xy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 3 else Yield 𝜋yX𝑒p𝑅𝑒 ˙ 𝑡q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4 else 5 Let C𝑒 “ t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6 foreach 𝑡1 P FullEnumpT,𝑒1,𝑡rkeyp𝑒1qsq do 7 foreach 𝑡2 P FullEnumpT,𝑒2,𝑡rkeyp𝑒2qs do 8 ¨ ¨ ¨ 9 foreach 𝑡𝑘 P FullEnumpT,𝑒𝑘,𝑡rkeyp𝑒𝑘qsq do 10 Yield 𝑡 1 𝑡1 1 𝑡2 1 ¨ ¨ ¨ 1 𝑡𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' we can rewrite 𝑉𝑠p𝑅𝑒q as follows: ô 𝑅𝑒 ˙ 𝑉𝑝p𝑅𝑒1q ˙ ¨ ¨ ¨ ˙ 𝑉𝑝p𝑅𝑒𝑘 q ô 𝑅𝑒 1 𝑉𝑝p𝑅𝑒1q 1 ¨ ¨ ¨ 1 𝑉𝑝p𝑅𝑒𝑘 q ô 𝑅𝑒 1 ´ 𝜋keyp𝑒1q𝑉𝑠p𝑅𝑒1q ¯ 1 ¨ ¨ ¨ 1 ´ 𝜋keyp𝑒𝑘q𝑉𝑠p𝑅𝑒𝑘 q ¯ ô 𝑅𝑒 1 ´ 𝜋keyp𝑒1qp1𝑒1PT𝑒1 𝑅𝑒1q ¯ 1 ¨ ¨ ¨ 1 ´ 𝜋keyp𝑒𝑘qp1𝑒1PT𝑒𝑛 𝑅𝑒1q ¯ ô 𝜋𝑒p𝑅 1𝑒1PT𝑒 𝑅𝑒1qq where the first equation follows the definition of semi-join views,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the second equation follows the fact that keyp𝑒𝑖q “ 𝑒𝑖 X 𝑒 Ď 𝑒,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the third equation follows the definition of projection views,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the fourth equation follows the hypothesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and the last equation follows the facts that T𝑒 “ t𝑒u Y T𝑒1 Y T𝑒2 Y ¨ ¨ ¨ Y T𝑒𝑘 and keyp𝑒𝑖q is exactly the set of join attributes shared by 𝑅𝑒 and p1𝑒1PT𝑒𝑖 𝑅𝑒1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ In plain language, the semi-join view of node 𝑒 is essentially the projection of the join results of relations in the subtree rooted at 𝑒, to attributes in 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' An immediate corollary is Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑉𝑠p𝑅𝑟q “ 𝜋𝑟𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This means that the semi-join view at the root 𝑟 (recall that 𝑟 does not have a projection view) contains precisely all the query results projected onto 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Using the notion of a witness query, this leads to the following useful fact for full enumeration, where Ţ denotes disjoint union: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑄p𝐷q “ Ţ 𝑡P𝑉𝑠p𝑅𝑟 q 𝑄p𝐷 ˙ 𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2, and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 allow us to use essen- tially the same algorithm from Bagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [6] to achieve constant- delay enumeration of 𝑄p𝐷q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' see Algorithm 5, which takes as input a node 𝑒 P T and a tuple 𝑡 P 𝑅𝑒, and yields the query results over T𝑒 that can be joined with 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To enumerate 𝑄p𝐷q, we simply invoke FullEnumpT,𝑟,𝑡q for every tuple 𝑡 P 𝑉𝑠p𝑅𝑟q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithm 5 enumerates 𝑄p𝐷q with 𝑂p1q delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We prove it by induction on the height of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The algorithm stops if the root contains no output attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For ease of expression, assume any node 𝑒 with 𝑒 X y “ H is removed from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We first establish a based case, in which T contains only one node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The algorithm returns 𝜋yX𝑒𝑉𝑠p𝑅𝑒q in 𝑂p|𝜋yX𝑒𝑉𝑠p𝑅𝑒q|q time, since all tuples in 𝜋yX𝑒𝑉𝑠p𝑅𝑒q can be enumerated in 𝑂p1q delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, this base case can be handled with 𝑂p1q delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In general, we have the hypothesis holds on all child nodes 𝑒𝑖 of 𝑒 (line 6-9): Algorithm 5 can enumerate all join results that agree with values 𝜋keyp𝑒𝑖q𝑡1 over attributes keyp𝑒𝑖q in the subtree T𝑒𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let 𝑡𝑖 be a join result returned from T𝑒𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' From the properties of join tree, line 10 will return a valid join result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Emitting every combination of join results over all subtrees of 𝑒1 just takes 𝑂p1q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Delta Enumeration Delta enumeration is straightforward in a standard query plan, as the root node corresponds to 𝑄p𝐷q, so all changes propagated to the root are precisely Δ𝑄p𝐷,𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, it becomes tricky in our new query plan, as no node corresponds to 𝑄p𝐷q, which is necessarily the case if a linear-size representation of 𝑄p𝐷q is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In our query plan, one cannot just inspect the root, because not every change propagates to the root, and many propagations stop mid- way, which is actually the main reason why our query plan is not only space-efficient but also time-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Recall that the full enumeration algorithm relies on Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then the key question is, can we have an analogy of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 for the delta Δ𝑄p𝐷,𝑡q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In other words, can we identify a set of witness tuples 𝑡1 for 𝑡 such that the delta Δ𝑄p𝐷,𝑡q is the disjoint union of 𝑄p𝐷 ˙ 𝑡1q?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Fortunately, the answer is yes, but the answer is not as simple as Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let’s first consider the insertion case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When we insert 𝑡 into some 𝑅𝑒, the propagation follows the path from 𝑒 to 𝑟, by (possibly) ap- plying an R-Update first, then an S-Update, P-Update, S-Update, P-Update, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='. Recall that both S-update and R-update only propa- gate a single change upward (see line 8, 12 in Algorithm 2 and Algo- rithm 4), but P-update may propagate multiple changes upward (see line 6, 12 in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, there could be multiple propagation paths starting from 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To be more precise, we denote the nodes lying on the path from 𝑒 to 𝑟 as 𝑒0 “ 𝑒,𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘 “ 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Every propaga- tion path inserts a tuple into each of the views on the path, and we denote the inserted tuples on such a path as p𝑡,𝑡𝑠 0,𝑡𝑝 0 ,𝑡𝑠 1,𝑡𝑝 1 , ¨ ¨ ¨ , q, where 𝑡𝑠 𝑖 P 𝑉𝑠p𝑅𝑖q and 𝑡𝑝 𝑖 P 𝑉𝑝p𝑅𝑖q for 𝑖 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Now, we distinguish three cases of a propagation path with respect to its ending tuple: (1) 𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) 𝑡𝑝 𝑗 for some 𝑗 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (3) 𝑡𝑠 𝑖 for some 𝑖 P t0, 1, 2, ¨ ¨ ¨ ,𝑘u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Case (1) happens when the first update is an R-Update and does not propagate any further change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This means that in Algorithm 4, there exists some child node 𝑒1 of 𝑒 such that 𝑡rkeyp𝑒1qs R 𝑉𝑝p𝑅𝑒1q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=',𝑡 cannot join with T𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this case,𝑡 will not produce any change to 𝑄p𝐷q, thus can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Case (2) happens when P-Update(𝑒𝑗,𝑡𝑝 𝑗 ) does not propagate any further change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Putting it into Algorithm 3, this means that either there exists no tuple 𝑡1 P 𝑅𝑝p𝑒𝑗 q that can join with 𝑡𝑝 𝑗 , or if such a tuple exists, but it cannot join with any query result over T 1𝑒 for some child node 𝑒1 of 𝑝p𝑒𝑗q, since its counter is smaller than |C𝑝p𝑒𝑗 q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In either case, this propagation path will not cause any change to 𝑄p𝐷q, thus can also be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Case (3) happens when S-update(𝑒𝑖,𝑡𝑠 𝑖 ) does not propagate any further change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Putting it into Algorithm 2, this means that either we have reached the root, or there exists some other tuple 𝑡1 P 𝑉𝑝p𝑅𝑖q such that 𝑡1 ‰ 𝑡𝑠 𝑖 and 𝑡𝑠 𝑖 rkeyp𝑒𝑖qs “ 𝑡1rkeyp𝑒𝑖qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is the 6 Algorithm 6: DeltaEnum(T,𝑡) Input: A free-connex generalized join tree T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' an updated tuple 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Output: Delta results induced by 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1 Let 𝑒0,𝑒1, ¨ ¨ ¨ ,𝑒𝑘 “ 𝑟 be the nodes on 𝑡’s propagation path;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2 foreach witness tuple 𝑡1 of 𝑡 do 3 Let 𝑒𝑖 be the node such that 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒𝑖,𝑡q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4 𝑆 Ð 𝑡1 1 𝑉𝑙p𝑅𝑒𝑖`1q 1 ¨ ¨ ¨ 1 𝑉𝑙p𝑅𝑒𝑘 q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5 foreach 𝑞 P 𝑆 do 6 𝑆𝑖 ÐFullEnumpT𝑒𝑖,𝑒𝑖,𝑞r𝑒𝑖sq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7 𝑆𝑗 ÐFullEnumpT𝑒𝑗 ´ T𝑒𝑗´1,𝑒𝑗,𝑞r𝑒𝑗sq, 𝑗 P r𝑖 ` 1,𝑘s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 8 Yield 𝑆𝑖 1 𝑆𝑖`1 1 ¨ ¨ ¨ 1 𝑆𝑘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' only case where changes to 𝑄p𝐷q can possibly happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We will give a more detailed characterization of this case later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Live views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To support constant-delay delta enumeration, we maintain a live view for each node 𝑒 such that 𝑒 X y ‰ H: 𝑉𝑙p𝑅𝑒q :“ 𝜋𝑒𝑄p𝐷q, which are the “live” tuples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', appearing in the query results) projected onto 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that 𝑉𝑙p𝑅𝑒q Ď 𝜋y𝑉𝑠p𝑅𝑒q, which means for 𝑒 Ď y, it can be implemented by simply adding an extra bit in 𝑉𝑠p𝑅𝑒q, indicating if the corresponding tuple is in 𝑉𝑙p𝑅𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the root 𝑟, there is no need to maintain 𝑉𝑙p𝑅𝑟q separately since𝑉𝑙p𝑅𝑟q “ 𝑉𝑠p𝑅𝑟q by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the leaf nodes, their live views need not be maintained, either, since they will not be needed by delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The other live views can be maintained by the following observation: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any non-root node 𝑒 such that 𝑒 Xy ‰ H and any tuple 𝑡 P 𝜋y𝑉𝑠p𝑅𝑒q, 𝑡 P 𝑉𝑙p𝑅𝑒q if and only if 𝑡 1 𝑉𝑙p𝑅𝑝p𝑒qq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Based on the Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5, the maintenance of 𝑉𝑙p𝑅𝑒q can pig- gyback on the delta enumeration: After enumerating a result 𝑡1 P Δ𝑄p𝐷,𝑡q, we update the live views in a top-down fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For every non-root 𝑒 such that 𝑒 Xy ‰ H, if the update is insertion, then we always add 𝑡1r𝑒s to𝑉𝑙p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' if the update is deletion, then we delete 𝑡1r𝑒s from 𝑉𝑙p𝑅𝑒q if 𝑡1r𝑒s cannot join with 𝑉𝑙p𝑅𝑝p𝑒qq, which can be done in 𝑂p1q time with a hash index on 𝑉𝑙p𝑅𝑝p𝑒qq (which is physically the same hash index on 𝑉𝑠p𝑅𝑝p𝑒qq for 𝑒 Ď y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This only adds another constant to the delay of delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Witness tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We now are ready to give a more precise char- acterization of the ending tuples falling into Case (3) that actually cause changes to 𝑄p𝐷q, called witness tuples: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6 (Witness tuple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Suppose 𝑡 is inserted into or deleted from 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A tuple 𝑡1 is a witness of 𝑡 if 𝑡1 P Δ𝑉𝑠p𝑅𝑟,𝑡q, or (5) 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q ˙ 𝑉𝑙p𝑅𝑝p𝑒qq (6) for some non-root 𝑒 such that 𝑒 X y ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Here Δ𝑉𝑠p𝑅𝑒,𝑡q denotes the tuples to be inserted into (or deleted from) 𝑉𝑠p𝑅𝑒q due to 𝑡 and 𝑉𝑙p𝑅𝑝p𝑒qq is the live view before the update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We give some intuition behind Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, (5) is the counterpart of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 for delta enumeration and such a 𝑡1 is guaranteed to generate changes to 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (6) is specific for delta enumeration, addressing the situation mentioned earlier, where the propagation stops mid-way yet still causes changes to 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that in this case, the attributes of 𝑡1 are 𝑒 X y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then (6) implies that 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q and 𝑡1rkeyp𝑒qs P 𝜋keyp𝑒q𝑉𝑙p𝑅𝑝p𝑒qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Since 𝑡1rkeyp𝑒qs P 𝜋keyp𝑒q𝑉𝑙p𝑅𝑝p𝑒qq, it must have 𝑡1rkeyp𝑒qs P 𝑉𝑝p𝑅𝑒q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑡1rkeyp𝑒qs R Δ𝑉𝑝p𝑅𝑒,𝑡q, which means that the propagation stops at node 𝑒 under case (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In addition, each witness tuple 𝑡1 should (i) contribute to the delta over T𝑒 induced by 𝑡, and (ii) join with tuples from the remaining relations in T ´ T𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For (i), it suffices to require 𝑡1 P Δ ` 𝜋y𝑉𝑠p𝑅𝑒q ˘ “ 𝜋yΔ𝑉𝑠p𝑅𝑒,𝑡q, since Δ ` 𝜋𝑒Xyp1𝑒1PT𝑒 𝑅𝑒1q ˘ “ Δ ` 𝜋y𝑉𝑠p𝑅𝑒q ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For (ii), it suffices to require 𝑡1 ˙𝑉𝑙p𝑅𝑝p𝑒qq ‰ H, and this is exactly the reason we introduced 𝑉𝑙p𝑅𝑒q in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Δ𝑄p𝐷,𝑡q “ Ţ 𝑡1:a witness of 𝑡 𝑄p𝐷 ˙ 𝑡1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We are now ready to state the counterpart of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 for delta enumeration, in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Unlike Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3, the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7 is nontrivial, and the details are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To perform delta enumeration using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7, we still need to address two issues: (1) how to find all witness tuples 𝑡1, and (2) how to enumerate 𝑄p𝐷 ˙ 𝑡1q with constant delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To find all the witness tuples, we consider the two cases in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6: (5) can be computed easily after updating 𝑉𝑠p𝑅𝑟q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' for (6), just an extra check with 𝑉𝑙p𝑅𝑝p𝑒qq is needed, which can be done in 𝑂p1q time using the hash index on 𝑉𝑙p𝑅𝑝p𝑒qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' These steps only increase the update cost by a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It remains to describe how to enumerate 𝑄p𝐷 ˙ 𝑡1q for each witness 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As before, let 𝑒0,𝑒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ,𝑒𝑘 “ 𝑟 be the nodes on the propagation path, and suppose we are given a witness tuple 𝑡1 P 𝜋yΔ𝑉𝑠p𝑅𝑒𝑖,𝑡q for some 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We first enumerate the query results par- ticipated by 𝑡1 together with relations on the path from 𝑒𝑖`1 to the root 𝑟, denoted as 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This can be done by joining 𝑡 with the live views associated with these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For each such result 𝑞 P 𝑆, we enumerate the query results that participated by 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This enumer- ation is done by partitioning the whole generalized join tree into disjoint subtrees T𝑒𝑖, T𝑒𝑖`1 ´T𝑒𝑖, ¨ ¨ ¨ , T𝑒𝑘 ´T𝑒𝑘´1, and invoking Ful- lEnum for each subtree separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, we join these subtrees together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The detailed process is given in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that, as written, the algorithm does not achieve constant-delay enumer- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, this can be easily fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, the join in line 4 can be enumerated with constant delay using (a variant of) FullEnum starting from 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then we interleave the two enumeration processes: After enumerating each 𝑞 P 𝑆, we immediately call line 6–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, line 6–8 can be rewritten into nested loops so as to enumerate the join 𝑆𝑖 1 ¨ ¨ ¨ 1 𝑆𝑘 with constant delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In fact, this join is more like a cross product (common attributes must have the same value, the same as those in 𝑞), and a total of Π𝑘 𝑗“𝑖|𝑆𝑗| results will be yielded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In figure 3(a), there are two query results p1, 2, 4, 4q and p2, 2, 4, 4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In figure 3(b), when the propagation stops, we have ‚ Tuple p1, 2q P 𝑅2 is not a witness as it cannot join with any tuple in 𝑉𝑙pr𝑥3sq, thus no delta is produced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ Tuple p1q P r𝑥3s is a witness, which triggers delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For a witness in the root, DeltaEnum simply degenerates to FullEnumpT𝑟,𝑟, p1qq, which outputs tp1, 1, 1, 1q, p1, 1, 1, 2qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ Tuple p1, 4q P 𝑅2 is a witness, which triggers delta enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' DeltaEnum finds 𝑆 “ p1, 4q 1 𝑉𝑙pr𝑥3sq “ tp1, 4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For p1, 4q P 𝑆, 7 r𝑥3s 𝑉𝑠pr𝑥3sq 𝑥3 𝑐r𝑡s 1 1 2 1 3 1 4 2 𝑅2 𝑉𝑝p𝑅2q 𝑥3 𝑐r𝑡s 4 1 2 1 Ô 𝑉𝑠p𝑅2q 𝑥2 𝑥3 𝑐r𝑡s 1 2 0 2 2 1 4 3 0 1 1 0 2 4 1 1 4 0 𝑅3 𝑉𝑝p𝑥3q 𝑥3 𝑐r𝑡s 1 2 3 1 4 1 Ô 𝑉𝑠p𝑅3q 𝑥3 𝑥4 𝑐r𝑡s 1 1 1 2 5 0 3 3 1 1 2 1 4 4 1 𝑅1 𝑉𝑝p𝑅1q 𝑥2 𝑐r𝑡s 2 2 3 1 Ò 𝑉𝑠p𝑅1q 𝑥1 𝑥2 1 2 2 2 3 3 𝑅4 𝑉𝑝p𝑅4q 𝑥4 𝑐r𝑡s 1 1 2 1 3 1 4 1 Ò 𝑉𝑠p𝑅4q 𝑥4 𝑥5 1 1 2 2 3 3 4 4 (a) Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' r𝑥3s 𝑉𝑠pr𝑥3sq 𝑥3 𝑐r𝑡s 1* 2 2 1 3 1 4 2 𝑅2 𝑉𝑝p𝑅2q 𝑥3 𝑐r𝑡s 4 2 2 2 1 1 Ô 𝑉𝑠p𝑅2q 𝑥2 𝑥3 𝑐r𝑡s 1 2 1 2 2 1 4 3 0 1 1 1 2 4 1 1* 4* 1 𝑅3 𝑉𝑝p𝑥3q 𝑥3 𝑐r𝑡s 1 2 3 1 4 1 Ô 𝑉𝑠p𝑅3q 𝑥3 𝑥4 𝑐r𝑡s 1 1 1 2 5 0 3 3 1 1 2 1 4 4 1 𝑅1 𝑉𝑝p𝑅1q 𝑥2 𝑐r𝑡s 2 2 3 1 1 1 Ò 𝑉𝑠p𝑅1q 𝑥1 𝑥2 1 2 2 2 3 3 1 1 𝑅4 𝑉𝑝p𝑅4q 𝑥4 𝑐r𝑡s 1 1 2 1 3 1 4 1 Ò 𝑉𝑠p𝑅4q 𝑥4 𝑥5 1 1 2 2 3 3 4 4 (b) After the insertion of p1, 1q into 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' r𝑥3s 𝑉𝑠pr𝑥3sq 𝑥3 𝑐r𝑡s 1 2 2 1 3 1 4 2 𝑅2 𝑉𝑝p𝑅2q 𝑥3 𝑐r𝑡s 4 2 2 2 1 1 Ô 𝑉𝑠p𝑅2q 𝑥2 𝑥3 𝑐r𝑡s 1 2 1 2 2 1 4 3 0 1 1 1 2 4 1 1 4 1 𝑅3 𝑉𝑝p𝑥3q 𝑥3 𝑐r𝑡s 1 1 3 1 4 1 Ô 𝑉𝑠p𝑅3q 𝑥3 𝑥4 𝑐r𝑡s 1* 1* 0 2 5 0 3 3 1 1 2 1 4 4 1 𝑅1 𝑉𝑝p𝑅1q 𝑥2 𝑐r𝑡s 2 2 3 1 1 1 Ò 𝑉𝑠p𝑅1q 𝑥1 𝑥2 1 2 2 2 3 3 1 1 𝑅4 𝑉𝑝p𝑅4q 𝑥4 𝑐r𝑡s \x1a 1 \x1a 1 2 1 3 1 4 1 Ò 𝑉𝑠p𝑅4q 𝑥4 𝑥5 �1 �1 2 2 3 3 4 4 (c) After the deletion of p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1q from 𝑅4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' r𝑥3s 𝑉𝑠pr𝑥3sq 𝑥3 𝑐r𝑡s 1 2 2 1 3 1 4 2 𝑅2 𝑉𝑝p𝑅2q 𝑥3 𝑐r𝑡s 4 2 2 2 1 1 Ô 𝑉𝑠p𝑅2q 𝑥2 𝑥3 𝑐r𝑡s 1 2 1 2 2 1 4 3 0 1 1 1 2 4 1 1 4 1 𝑅3 𝑉𝑝p𝑥3q 𝑥3 𝑐r𝑡s 1 1 3 1 4 1 Ô 𝑉𝑠p𝑅3q 𝑥3 𝑥4 𝑐r𝑡s 1 1 0 2 5 0 3 3 1 1 2 1 4 4 1 𝑅1 𝑉𝑝p𝑅1q 𝑥2 𝑐r𝑡s 2 2 3 1 1 2 Ò 𝑉𝑠p𝑅1q 𝑥1 𝑥2 1 2 2 2 3 3 1 1 4* 1* 𝑅4 𝑉𝑝p𝑅4q 𝑥4 𝑐r𝑡s 2 1 3 1 4 1 Ò 𝑉𝑠p𝑅4q 𝑥4 𝑥5 2 2 3 3 4 4 (d) After the insertion of p4, 1q into 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 3: A running instance for query in Figure 1 using the plan in Figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Tuples in white are in 𝑉𝑠p𝑅q, in grey are in 𝑅z𝑉𝑠p𝑅q, in cyan are in 𝑉𝑙p𝑅q (live views for leaf nodes are not needed, but we still show them for clarity), with star symbols are the witness tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Changes in each step are marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' it invokes FullEnumpT𝑟 ´ T𝑅2,𝑟, p4qq with tp4, 4qu returned and FullEnumpT𝑅2, 𝑅2, p1, 4qq with tp1, 1, 4qu returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Joining them yields the delta tp1, 1, 4, 4qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, as each new result is enumerated, we update the live views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In figure 3(c), tuple p1, 1q P Δ𝑉𝑠p𝑅3q is a witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' DeltaEnum first finds 𝑆 “ tp1, 1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For p1, 1q P 𝑆, it invokes FullEnumpT𝑟 ´ T𝑅3,𝑟, p1qq with tp1, 1, 1qu returned, and FullEnumpT𝑅3, 𝑅3, p1, 1qq with tp1, 1, 4qu returned (delta enumeration upon a deletion is done before the tuple deletion so as to find the delta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Joining them yields the delta tp1, 1, 1, 1qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, we update live views with the delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithm 6 enumerates Δ𝑄p𝐷,𝑡q with constant delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We have now closed the loop: while enumerating Δ𝑄p𝐷,𝑡q, we update the live views as described earlier, which are needed for enumerating the next delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6 UPDATE COST ANALYSIS We have shown that the enumeration delay of both full query results and deltas is a constant, and this holds for the query plan defined by any free-connex join tree as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, the update cost differs for different query plans and can be as large as 𝑂p|𝐷|q in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is caused by P-Update, which may trigger an S-Update to every tuple in its parent node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, such a worst-case behavior only happens on contrived update sequences, and the actual update cost can be much better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Characterizing the update cost will be important for constructing a good query plan, as there can be many free-connex join trees for a given free-connex query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As we will see, the height of the join tree is an important parameter, and this is precisely the reason why we make our framework applicable to any generalized join tree, as the height of a generalized join tree can be lower than that of any standard join tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, the query in Figure 2 has a generalized join tree of height 1 while the two standard join trees have height 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the query in Figure 1 has a generalized join tree of height 2 while any standard join tree has height as least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 Enclosureness Update sequences and lifespans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given an update sequence 𝑆𝐷, the lifespan of tuple 𝑡 is an interval 𝐼p𝑡q “ r𝑡`,𝑡´s, where 𝑡` denotes the timestamp when 𝑡 is inserted into 𝐷 and 𝑡´ denotes the timestamp when 𝑡 is deleted from 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We set 𝑡` “ ´8 to indicate that 𝑡 exists in the initial 𝐷 and 𝑡´ “ `8 indicates that 𝑡 still exists in 𝐷 after the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that if a tuple is repeatedly inserted and deleted, it will be treated as multiple tuples, which have the same values but disjoint lifespans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Although our algorithms will be able to handle arbitrary update sequences, their performance can be better if the update sequences possess some nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In particular, the following two re- strictive classes of update sequences are of practical importance: ‚ First-in-first-out (FIFO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A update sequence 𝑆𝐷 is FIFO if for any two tuples 𝑡1,𝑡2 P 𝑆𝐷, 𝑡` 1 ă 𝑡` 2 implies 𝑡´ 1 ă 𝑡´ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' FIFO sequences are commonly used in practice, such as sliding-window or tumbling-window models over streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ Insertion-only or deletion-only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A update sequence 𝑆𝐷 is insertion-only (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' deletion-only) if for any tuple 𝑡 P 𝑆𝐷, 𝑡´ “ `8 (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑡` “ ´8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The two cases are symmetric, so we will only discuss the insertion-only case in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 8 The notion of enclosureness was first introduced in [37] to give an instance-specific characterization of the hardness of the update sequence, which we briefly review next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 (Enclosureness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given an update sequence 𝑆𝐷, the enclosureness of a tuple 𝑡 P 𝑆𝐷 is 𝜆p𝑡q :“ max JĎ𝑆𝐷 @𝑡1PJ,𝐼p𝑡1qĂ𝐼p𝑡q @𝑡2,𝑡3PJ,𝐼p𝑡2qX𝐼p𝑡3q“H |J|, (7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the largest number of disjoint lifespans in 𝑆𝐷 contained in 𝐼p𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then the enclosureness of the update sequence is the average enclosureness of all the tuples (but at least 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝜆p𝑆𝐷q :“ max ˜ř 𝑡P𝑆𝐷 𝜆p𝑡q |𝑆𝐷| , 1 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (8) We often omit 𝑆𝐷 and simply write 𝜆 :“ 𝜆p𝑆𝐷q for the enclosure- ness of an update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then, they give an algorithm that can update any foreign-key acyclic query in 𝑂p𝜆q time for any 𝑆𝐷 while supporting 𝑂p1q-delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is appealing, since while 𝜆 can be as large as 𝑂p|𝑆𝐷|q in the worst case, it is often a small constant for many com- mon update sequences, including FIFO, FILO (first-in-last-out), and insertion-/deletion-only sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The worst-case situation only happens when there are many tuples with long lifespans joining with many tuples with short lifespans, something that is uncommon in practice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', many big but ephemeral changes to the query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, their analysis crucially relies on the nice property of foreign-key acyclic queries, that their result size is at most linear, which is not the case for non-key joins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In fact, we show below that the 𝑂p𝜆q update time is unachievable for free-connex queries, which follows from the negative result that we prove below: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider the query 𝑄 “ 𝑅1p𝑥1q 1 𝑅2p𝑥1,𝑥2q 1 𝑅3p𝑥2,𝑥3q 1 𝑅4p𝑥3,𝑥4q 1 𝑅5p𝑥4q over a FIFO update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If there is an algorithm for 𝑄 with update time 𝑂p|𝐷|1{2´𝜖q while sup- porting 𝑂p|𝐷|1´𝜖q-delay enumeration of full results for any constant 𝜖 ą 0, then the OuMv conjecture4 fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that this theorem separates the difficulty of (at least one of) free-connex queries from foreign-key acyclic queries, for which 𝑂p1q update time is possible for FIFO sequences [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Join-tree-specific Enclosureness Hope is not all lost despite the negative result above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 only holds for a particular free-connex query;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' other queries may still be updated in 𝑂p1q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Secondly, the definition of enclosure- ness in [37] only considers the time dimension while ignoring the structure dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', which relation each update is applied to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' These observations motivate a more refined definition of enclosure- ness that also depends on the join tree (which nodes the updates are applied to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As we will see, a hard query like the one in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 can still be solved efficiently, when information from both the structural dimension and the time dimension is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 4The OuMv conjecture [20] is that the following problem cannot be solved in𝑂p𝑛3´𝜖q time for any constant𝜖 ą 0: Given an𝑛ˆ𝑛 matrix 𝑀 and a sequence of𝑛-dimensional vectors𝑢1, 𝑣1,𝑢2, 𝑣2, ¨ ¨ ¨ ,𝑢𝑛, 𝑣𝑛, compute𝑢𝑖𝑀𝑣𝑖 for each𝑖 over the Boolean semiring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The algorithm must return 𝑢𝑖𝑀𝑣𝑖 before 𝑢𝑖`1, 𝑣𝑖`1 are revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (1, 1) · · (n, 1) (1, 2) · · Time (n, 2) (1, 1) · · (1, n) (2, 1) · · (2, n) R2 n/1/1 n/1/1 n/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/n/1 1/n/1 1/n/1 R1 Figure 4: As update sequence in Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each interval is the lifespan of a tuple, and three numbers above each in- terval are its enclosureness over T1, T2 and T3 in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 (Effective lifespan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a free-connex query 𝑄, a free-connex generalized join tree T of 𝑄, a database 𝐷, and an update sequence 𝑆𝐷, the two effective lifespans of a tuple 𝑡1 P 𝑅𝑒 with lifespan 𝐼p𝑡1q “ r𝑡` 1 ,𝑡´ 1 s are p𝐼p𝑡1q “ « 𝑡` 1 , min ˜ 𝑡´ 1 , min 𝑡2P𝑅𝑒1:𝑒1PT𝑒´t𝑒u,𝑡´ 2 ą𝑡` 1 𝑡´ 2 ¸ff ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' q𝐼p𝑡1q “ « max ˜ 𝑡` 1 , max 𝑡2P𝑅𝑒1:𝑒1PT𝑒´t𝑒u,𝑡` 2 ă𝑡´ 1 𝑡` 2 ¸ ,𝑡´ 1 ff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In plain language, p𝐼p𝑡1q is obtained from 𝐼p𝑡1q by moving forward its ending time to the first deletion of a tuple from any descendent of 𝑒, while to obtain q𝐼p𝑡1q, we move its starting to the last insertion from any descendent of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We can now define the join-tree-specific enclosureness of a tuple: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a free-connex query 𝑄, a free-connex gen- eralized join tree T of 𝑄, a database 𝐷, and an update sequence 𝑆𝐷, for a node 𝑒 P T and a tuple 𝑡 P 𝑅𝑒, its enclosureness is 𝜆Tp𝑡q “ max @𝑡1PJ,D𝑒1PT𝑒´t𝑒u,𝑡1P𝑅𝑒1 @𝑡1PJ,9𝐼p𝑡1qĎ𝐼p𝑡q @𝑡2,𝑡3PJ,9𝐼p𝑡2qX9𝐼p𝑡3q“H |J|, (9) where each 9𝐼 is either p𝐼 or q𝐼, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the largest number of disjoint effective lifespans of tuples in the descendants of 𝑒, which are contained in the lifespan of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then the enclosureness of the update sequence is still the average: 𝜆Tp𝑆𝐷q :“ max ˜ř 𝑡P𝑆𝐷 𝜆Tp𝑡q |𝑆𝐷| , 1 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We often write 𝜆T :“ 𝜆Tp𝑆𝐷q for the enclosureness of an update sequence with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider 𝑄 :“ 𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q in Figure 2 with T1, T2, T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the update sequence in Figure 4, 𝜆T1 “ 𝜆T2 “ 𝑛 and 𝜆T3 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In fact, 𝜆T3 “ 1 for any update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The main analytical result of this paper is the following theorem, whose proof is quite technical given in Appendix C: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any free-connex query 𝑄, the update cost of the query plan in Section 4 induced by any given free-connex generalized join tree T of 𝑄 is 𝑂p𝜆Tq under any update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 9 This result is complemented with a matching lower bound, for at least one particular query: 𝑄 “ 𝜋𝑥1p𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2qq, which has one join tree as shown in Figure 2(a) (one could add a general- ized relation r𝑥1s at the top, but it does not change the enclosure- ness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, for this query, 𝜆T does not really depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [37] Suppose there is an algorithm for the query 𝑄 “ 𝜋𝑥1p𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2qq with update time 𝑂p𝜆1´𝜖q while supporting𝑂p𝜆1´𝜖q-delay enumeration of full results for any constant 𝜖 ą 0, then the OMv conjecture5 fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 Implications of Enclosureness We present some implications of our join-tree-specific enclosure- ness and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6, exhibiting an interesting trade-off between the hardness of update sequences and the complexity of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Arbitrary update sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For arbitrary update sequences, prior work [8, 21] has shown how to achieve 𝑂p1q update time while supporting 𝑂p1q-delay enumeration for any q-hierarchical query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It turns out that this is an easy consequence of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6, plus the following structural property of q-hierarchical queries, as well as the simple fact that 𝜆T “ 1 if the height of T is 1: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Every q-hierarchical query has a free-connex general- ized join tree of height 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For arbitrary update sequences, q-hierarchical queries are pre- cisely the class of queries for which 𝑂p1q update time is possible [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, for queries outside this class, we must restrict the update sequence in order to achieve 𝑂p1q update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We consider the following two classes of update sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' FIFO sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The update time is shown to be 𝑂p1q for foreign- key acyclic joins over FIFO sequences [37], but nothing is known for non-key joins (except for q-hierarchical queries which do not rely on FIFO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We present the first extension in this direction: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any free-connex query 𝑄 with a free-connex gener- alized join tree T of height at most 2, 𝜆T “ 1 for any FIFO sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that the height limit of 2 is the best one can hope for, since the query in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 has a join tree of height 3 and the theorem shows that it cannot be updated in 𝑂p1q time over FIFO sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Although the height-2 limitation restricts the class of queries, this already includes some fairly complex queries, such as the one in Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' more examples can be found in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Insertion-only sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As we restrict the update sequence further, we can handle more queries in 𝑂p1q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For simplicity, the following result only considers insertion-only sequences, but the same result holds for deletion-only or FILO sequences as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any free-connex query 𝑄 and any join tree T, 𝜆T “ 1 for any insertion-only update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Combining Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10, the following theorem is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For a free-connex query 𝑄, there is an index that can be updated in 𝑂p1q amortized time under any insertion-only update sequence, while supporting 𝑂p1q-delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5The OMv conjecture is similar to the OuMv conjecture, except that the algorithm needs to compute 𝑀𝑣𝑖 for every 𝑣𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10 incorporates the static result [6] as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a static database 𝐷, we can simply insert every tuple from 𝐷 into our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10, this builds a data structure in 𝑂p|𝐷|q time that supports 𝑂p1q-delay enumeration of 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Also, the dichotomy result of [6] states that 𝑂p|𝐷|q-time preprocessing and 𝑂p1q-delay enumeration are possible only for free-connex queries, thus Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10 cannot be extended to beyond free-connex queries, either, even over insertion-only sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider an insertion-only update sequence for the query in Figure 1: (1) tuples p𝑖, 𝑗q P r𝑛s ˆ r𝑛s are inserted into 𝑅2, 𝑅3 and 𝑅4 initially;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) tuples p𝑖, 𝑗q P r𝑛s ˆ r𝑛s are inserted into 𝑅1 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Standard change propagation or HIVM needs to materi- alize Δp𝑅1 1 𝑅2 1 𝑅3q, hence incurs 𝑂p𝑛3q cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' the Dynamic Yannakakis algorithm [21] needs to scan all tuples p𝑖1, 𝑗1q P 𝑅2 for 𝑖1 “ 𝑗, once p𝑖, 𝑗q is inserted into 𝑅1, hence incurs Θp𝑛q cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and our framework only incurs 𝑂p1q cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Query plan optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If the given query and/or the update sequence do not fall into any of the three cases above where 𝑂p1q update time can be guaranteed, our enclosureness analysis still yields an effective heuristic for choosing a good T, which in turn determines the query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, it is clear that T with a smaller height is always preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Furthermore, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4 suggests that we should put nodes with more updates higher in T, as a tuple in a node might increase the enclosureness of tuples in its ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, in our implementation, we construct all join trees and use the one that minimizes ÿ 𝑒PT 𝑑p𝑒q𝑁p𝑒q, where 𝑑p𝑒q is the depth of 𝑒 in T (not counting generalized relations and itself) and 𝑁p𝑒q is the number of updates to 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑁p𝑒q is unavailable, we can estimate it by observing (and buffering) the first few updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7 EXTENSIONS TO GENERAL QUERIES 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 General CQs Acyclic but non-free-connex queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider such a query 𝜋𝑥1,𝑥3𝑅1p𝑥1,𝑥2q 1 𝑅2p𝑥2,𝑥3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We simply add 𝑥2 as an output at- tribute to turn it into a free-connex query, and then do a projection over 𝑥1,𝑥3 during enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that enumeration may con- tain duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, if a DISTINCT keyword is declared explicitly, duplicates need to be removed, hence making the delay more than constant, but this is inevitable due to the lower bound [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Cyclic queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Cyclic queries can also be easily handled in our framework by resorting to Generalized Hypertree Decomposition (GHD) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More specifically, by grouping several relations into a bag, an arbitrary CQ can be converted into a free-connex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, Figure 5(a) shows a GHD for the “dumbbell” query with 3 bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We can use standard change propagation within each bag, and apply our framework across the bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This results in the query plan in Figure 5(b), which has 𝑂p𝑁 2q space and 𝑂p𝑁q update time while supporting constant-delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, the standard change propagation framework would use a query plan like the one in Figure 5(c), which has 𝑂p𝑁 3q space and update time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Of course, all these are worst-case bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' on realistic inputs, the costs are lower, but our new query plan is still order-of-magnitude better than the old plan, as shown in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 10 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑅1 𝑅2 𝑅3 𝑅4 𝑅5 𝑅6 𝑅7 (a) Hypergraph and a GHD ˙ : 𝑉𝑠p𝑅7q 𝜋 : 𝑉𝑝p𝐵1q 𝑅7 𝜋 : 𝑉𝑝p𝐵2q 𝑉𝑠p𝐵1q : 𝑅3 1 𝑉1 𝑉𝑠p𝐵2q : 𝑅6 1 𝑉2 𝑅3 𝑉1 : 𝑅2 1 𝑅1 𝑉2 : 𝑅4 1 𝑅5 𝑅6 𝑅2 𝑅1 𝑅4 𝑅5 𝐵1 𝐵2 (b) Query plan under new change propagation 𝑉6 “ 𝑉4 1 𝑉5 𝑉4 “ 𝑉2 1 𝑅7 𝑉5 “ 𝑉3 1 𝑅6 𝑉2 “ 𝑉1 1 𝑅3 𝑅7 𝑉3 “ 𝑅4 1 𝑅5 𝑅6 𝑅3 𝑉1 “ 𝑅2 1 𝑅3 𝑅5 𝑅4 𝑅1 𝑅2 (c) Query plan under standard change propagation Figure 5: 5(a) is the hypergraph of the “dumbbell” query 𝑄 “ 𝑅1p𝑥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥2q 1 𝑅2p𝑥1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥3q 1 𝑅3p𝑥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥3q 1 𝑅4p𝑥5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥6q 1 𝑅5p𝑥4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥5q 1 𝑅6p𝑥4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥6q 1 𝑅7p𝑥3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='𝑥4q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' with GHD illustrated in red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 5(c),5(b) are query plans under the standard, new change propaga- tion framework respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In 5(c), 𝐵1, 𝐵2 are treated as two basic relations, on which projection and semi-join views are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If one is interested in further improving the theoretical bounds, the algorithm for maintaining the query results inside each bag can be replaced by a better algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For example, Kara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [26] present an algorithm for maintaining the triangle join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Replacing with the new algorithm can improve the space usage from 𝑂p𝑁 2q to 𝑂p𝑁 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑁q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other side, although the algorithm [26] can improve the update cost for each bag to 𝑂p ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑁q, the “dumbbell” query still suffers from 𝑂p𝑁q update cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is indeed unavoid- able as a single tuple update can change as large as 𝑂p𝑁q results materialized for one bag, which further propagates to the overall framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, the update cost for this GHD-based change propagation framework is determined by updates not only inside each bag but also across bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Beyond the triangle join, not many results are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is still an actively researched problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' any improvement here will also improve general CQs when plugged into our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a CQ 𝑄 with a free-connex GHD of width6 𝑤, there is an index of 𝑂p𝑁 𝑤q size that can be updated in Ωp𝑁 𝑤q time while supporting 𝑂p1q-delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Maintaining any bag of relations requires Ωp𝑁 𝑤q time, and it needs 𝑂p𝑁 𝑤q space to store all query results in the bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' After maintaining each bag of relations, the algorithms proposed in Section 4 can use to maintain between each bag, which takes 𝑂p|𝐷|q time for maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Noted that current database size |𝐷| is bounded by the largest bag size, which will be 𝑁 𝑤, makes the total maintenance time to Ωp𝑁 𝑤q and space cost to 𝑂p𝑁 𝑤q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ The following lemma can be easily derived from the above theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the “dumbbell” query, there is an index of 𝑂p𝑁 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5q size that can be updated in 𝑂p𝑁q time per tuple update, while sup- porting 𝑂p1q-delay enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 6The definition of width depends on the algorithm used for maintaining query results inside each bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If adopting the standard change propagation framework, the width is defined as the maximum width over all bags, where the width of a bag is the optimal integral edge covering number of the corresponding subquery derived for this bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Selection, union, and set difference The query plan in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 works for CQs with joins and projec- tions, but it can be equipped with other operators easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ If there is a selection 𝜎𝜙 on an input relation 𝑅𝑒 where 𝜙 is a predicate on 𝑒, then for an update with tuple 𝑡 P 𝑅𝑒, we simply check if 𝜙p𝑡q is true, and discard this update if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This only adds 𝑂p1q time to the update cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ For the union of CQs 𝑄 “ 𝑄1 Y ¨ ¨ ¨ Y 𝑄𝑘, we just maintain each 𝑄𝑖 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Full enumeration can be supported with 𝑂p1q delay using the technique in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We note that [10] assumes that the data structure on each 𝑄𝑖p𝐷q can check if 𝑡 P 𝑄p𝐷𝑖q in 𝑂p1q time for any given 𝑡, which is indeed supported by our query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For delta enumeration, we can use the same technique to enumerate Δ𝑄1p𝐷q Y ¨ ¨ ¨ Y Δ𝑄𝑘p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, this is not the same as Δp𝑄1 Y ¨ ¨ ¨ Y 𝑄𝑘q, and we need to check, say, if some new result 𝑡 P Δp𝑄1q already exists in 𝑄2p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, while the technique of [10] is still correct for delta enumeration, the delay is not bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' How to support 𝑂p1q-delay delta enumeration for UCQs remains an interesting open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ For a query like 𝑄 “ 𝑄1 ´ 𝑄2, we can as above maintain 𝑄1 and 𝑄2 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For enumeration, we enumerate every 𝑡 P 𝑄1p𝐷q and check if 𝑡 P 𝑄2p𝐷q, although this does not guarantee con- stant delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In fact, even in the static case, it is an open question whether 𝑄1p𝐷q ´ 𝑄2p𝐷q can be enumerated in 𝑂p1q delay after linear-time preprocessing where 𝑄1 and 𝑄2 are both free-connex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 Aggregations Standard relational algebra can be extended to support aggregations, and we adopt the following formalism [2, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let p𝑆, ‘, bq be a commutative ring7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Every tuple 𝑡 P 𝑅𝑒 has an annotation 𝑣p𝑡q P 𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For a full CQ 𝑄 in the form of p1q, the annotation for any join result 𝑡 P 𝑄p𝐷q is defined as 𝑣p𝑡q :“ b 𝑒P𝑄 𝑣p𝑡r𝑒sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For a non-full query 𝜋y𝑄, the projection becomes GROUP BY y, and the annotation for each result 𝑡 P 𝜋y𝑄p𝐷q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the aggregate of each group) is 𝑣p𝑡q :“ ‘ 𝑡1P𝑄p𝐷q:𝜋y𝑡1“𝑡 𝑣p𝑡1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 7In the static case, p𝑆, ‘, bq is only required to be a semi-ring, but we need additive inverses to support deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 11 Our new change propagation framework can support aggrega- tions easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any relation 𝑒, let 𝑣p𝑡q be the annotation for 𝑡 P 𝑅𝑒, 𝑣𝑠p𝑡q be the annotation for 𝑡 P 𝑉𝑠p𝑅𝑒q and 𝑣𝑝p𝑡q be the annotation for 𝑡 P 𝑉𝑝p𝑅𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Following the definitions of semi-join view 𝑉𝑠 and projection view 𝑉𝑝 (see Section 4), the annotation of 𝑡 P 𝑉𝑠p𝑅𝑒q can be written as 𝑣𝑠p𝑡q :“ ˜ b 𝑒𝑖Pt𝑒1,¨¨¨,𝑒𝑘u:𝑒𝑖Xy“H 𝑣𝑝p𝜋𝑒𝑖𝑡q ¸ b " 𝑣p𝑡q 𝑒 X y “ H 1 𝑒 X y ‰ H (10) where t𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘u are the children nodes of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The annotation of 𝑡 P 𝑉𝑝p𝑅𝑒q can be written as 𝑣𝑝p𝑡q :“ ‘ 𝑡1P𝑉𝑠p𝑅𝑒q:𝜋keyp𝑒q𝑡1“𝑡 𝑣𝑠p𝑡1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (11) Now, we can rewrite 𝑣p𝑡q for each query result 𝑡 P 𝑄 as: 𝑣p𝑡q :“ b 𝑒P𝑄 p𝑣p𝜋𝑒𝑡q b 𝑣𝑠p𝜋𝑒𝑡qq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (12) We store these annotations alongside their counters in the query plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then, the change propagation and enumeration procedures should be modified according to the formulas above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' More precisely, whenever the counter of a tuple is updated, we also update its annotation by (10) or (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' One technical difference is that, during an S-Update, we need to keep propagating the change upwards whenever the counter changes, not just when the counter changes from 0 to 1 or vice versa as in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, after enumerating a result, we compute its annotation by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 8 EXPERIMENTS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 Setup Prototype implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We have implemented our algo- rithms and built a system prototype called CROWN (Change pROpagation Without joiNs) on top of Flink DataStream API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' All of our algorithms are implemented as DataStream functions, which take as input an update stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each tuple in the update stream is associated with a flag indicating whether the update is an inser- tion or deletion, as well as the name of the updated relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' After processing an update, the DataStream function outputs the deltas triggered by this update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Enumeration of full query results can be invoked upon the user’s request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implementing the prototype over Flink allows us to inherit all the benefits of Flink, such as fault- tolerance and the ability to work with a variety of data sources and sinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To dispatch tuples in a load-balanced fashion, we borrow a similar idea from massively parallel algorithms, such as HyperCube [3, 7, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We have evaluated our algorithms in both centralized and dis- tributed settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The centralized version runs on a single machine with a single thread, where we disable certain Flink features such as false tolerance, serialization, and dispatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is for a fair comparison with other centralized systems (DBToaster and Trill) that do not support these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The distributed version has all these features enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It runs over two machines, each equipped with two Intel Xeon 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1GHz processors with 48 cores and 416 GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The machine runs Linux, with Scala 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='12, dotnet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='403, Flink 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5, and Spark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each query is evaluated 10 times on each engine and we report the average runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We set a 4-hour time limit for each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' CROWN Flink DBToaster DBToaster Trill Spark Distributed ✓ ✓ ✓ Full ✓ ✓ ✓ ✓ enumeration Delta ✓ ✓ enumeration Updates Arbitrary FIFO Arbitrary Batch Arbitrary Internal This Standard HIVM HIVM Standard paper change change propagation propagation Table 1: Comparison of different query processing engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Query processing engines compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We compare CROWN with (1) DBToaster [4], the best HIVM engine that supports multi- way joins over arbitrary update streams in centralized settings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) DBToaster Spark [32], which can support IVM with batch updates in a distributed/parallel setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (3) Trill [11], a continuous query evaluation system over streaming data using the standard change propagation framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and (4) the native Flink SQL engine over streaming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Table 1 summarizes various features of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that only CROWN supports both full enumeration and delta enumera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Flink can support insertion-only update streams or window streams, but not arbitrary update streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We run every experiment twice: one for delta enumeration, and the other for full enumera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For full enumeration, we request the full query results after processing every 10% of the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As Trill does not support full enumeration, we ask Trill to report the entire delta stream for full enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Queries and updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We evaluate all systems over two classes of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The first class contains graph pattern queries from the benchmark by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [31], over the SNAP dataset (Stanford Network Analysis Project) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Such a benchmark evaluates the performance of each system for join queries over static data, and we modify it to adapt to the dynamic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We test all acyclic queries from the benchmark, such as hop (path) queries, star queries and comb queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We also test the dumbbell query, which is a variant of the lollipop query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The detailed query definition is given in the Appendix D and one example of the 3-Hop query is given below, where we use a filter over to control the output size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as A, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as D FROM G G1, G G2, G G3 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND FILTER OVER (G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst) The second class includes more complex analytical queries over the LDBC Social Network Benchmark (LDBC-SNB) [15], which accesses the neighborhood of a given node in the graph with con- tinuous updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The following shows one example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' which finds the number of distinct messages associated with a particular tag ID,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' while satisfying the filter conditions: SELECT t_name,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' t_tagid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' COUNT(DISTINCT m_messageid) FROM tag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' message,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' message_tag,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' knows WHERE m_messageid = mt_messageid AND mt_tagid = t_tagid AND m_creatorid = k_person2id AND m_c_replyof IS NULL AND FILTER OVER (k_person1id) GROUP BY t_name,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' t_tag_ids 12 𝐴 𝐵 𝐶 𝐷 𝐺1 𝐺2 𝐺3 (a) 3-Hop Query 𝐴 𝐵 𝐶 𝐷 𝐺1 𝐺2 𝐺3 (b) 2-Comb Query 𝐴 𝐵 𝐶 𝐷 𝐸 𝐺1 𝐺2 𝐺4 𝐺3 (c) 4-Hop Query,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SNB Q2 𝐵 𝐶 𝐴 𝐷 𝐸 𝐺1 𝐺2 𝐺4 𝐺3 (d) Star Query 𝑖𝑑2 𝑖𝑑1 𝑖𝑑3 𝑀𝑖𝑑 𝑇𝑖𝑑 𝐾1 𝐾2 𝑀 𝑀𝑇 𝑇 (e) SNB Q3 𝑖𝑑2 𝑖𝑑1 𝑀𝑖𝑑 𝑇𝑖𝑑 𝑛𝑎𝑚𝑒 𝐾 𝑀 𝑇 𝑀𝑇 (f) SNB Q4 Figure 6: The relational hypergraphs of queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The solid dots are output attributes for join-project and aggregation queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 3-Hop 4-Hop 2-Comb SNB Q1 SNB Q2 SNB Q3 dumbbell 3-Hop 4-Hop dumbbell Star SNB Q4 Processing Time (Sec) CROWN Flink DBToaster CPP DBToaster Spark CROWN Delta Trill Aggregate Queries Join-Project Queries Full Join Queries Figure 7: Processing times of CROWN, Flink, DBToaster, and Trill 1e-02 1e-01 1e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 1 3 10 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Processing Time (ms) Scale Factor CROWN Flink DBToaster Trill (a) SNB Q1 1e-02 1e-01 1e+00 1e+01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 1 3 10 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Processing Time (ms) Scale Factor CROWN DBToaster Trill (b) SNB Q2 1e-03 1e-02 1e-01 1e+00 1e+01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='3 1 3 10 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Processing Time (ms) Scale Factor CROWN DBToaster Trill (c) SNB Q4 Figure 8: Average Processing Time v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Scale Factor 1e+00 1e+01 1e+02 1e+03 1e+04 1 2 4 10 20 Processing Time (Sec) λ Epinion Google Bitcoin BerkStan Figure 9: Runtime v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' enclosureness 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1e+01 1e+02 1e+03 1e+04 1 2 4 8 16 32 Processing Time (Sec) Parallelism CROWN 4-Hop CROWN SNB Q3 Flink 4-Hop DBToaster SNB Q3 Figure 10: Runtime v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' parallelism 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1e+01 1e+02 1e+03 1e+04 25% 50% 75% Latency (ms) Percentage of stream being processed CROWN Trill Figure 11: Average latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 6 shows the join hypergraphs of all queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Except for 2-Comb, SNB Q3 and Q4, they have a height-2 free-connex general- ized join tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The star query (figure 6(d)) has a height-1 free-connex generalized join tree, so it is q-hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The 4-Hop query (fig- ure 6(c)) and SNB Q4 query (figure 6(f)) have the same hypergraph structure but different output attributes, and the 4-Hop query has a height-2 free-connex generalized join trees while SNB Q4 query does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We create FIFO streams with a parameter 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For graph queries, we assign a distinct integer 𝑡𝑒 to each edge 𝑒 in the graph, where 𝑒 has its lifespan r𝑡𝑒,𝑡𝑒 ` 𝑤s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For LDBC-SNB queries, each tuple 𝑡 in the benchmark already has an insertion timestamp 𝑡`, and we set its deletion time as 𝑤 days after its insertion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝑡´ “ 𝑡` ` 𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that the sliding window for graph queries is count-based, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the window always contains the same number of tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, the window for LDBC-SNB queries is time-based, so the number of tuples in a window fluctuates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2 Experiment Results Runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 7 shows the total runtime of evaluating each graph query over a mid-sized graph Epinions and each SNB query in the centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The graph contains approximately 500K edges and 76K vertices, as well as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7B 3-Hop paths and 378B 4-Hop paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, we use the default scale factor of 1 for all SNB queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Under the scale factor, the total size of raw data is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5𝐺𝐵, and the largest relation contains 15 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We set a filter 13 condition that only keeps 10% of the designated endpoints for all queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A missing bar in the figure indicates that the corresponding system did not finish within the 4-hour limit or aborted with an error (mostly out-of-memory errors and garbage collection timeout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Only CROWN can finish all queries successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Trill only handles a few graph queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' One possible explanation is that graph queries tend to generate a large number of deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, Flink ran out of memory when evaluating SNB Q2, Q3, and Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For those queries where the systems can finish, we see that CROWN provides a speedup from 2x to 67x compared with Flink, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='8x to 234x compared with DBToaster, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7x to 523x compared with Trill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Moreover, in handling join-project queries, CROWN requires much less time than handling the corresponding full join queries, while Flink requires more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In addition, CROWN performs well for both full and delta enumeration, and different modes of output do not affect the overall performance of CROWN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Enclosureness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To test the influences of enclosureness, we create multiple update sequences with different 𝜆, over different graphs from the SNAP dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We disable the output to see how the update cost would change with different 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The experiment results are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' From the results, we can see the maintenance cost of CROWN increases almost linear as 𝜆 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Distributed processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To compare CROWN with DBToaster Spark and Flink in a distributed setting, we built a small cluster with 32 task slots, and tested 4-Hop as well as SNB Q3 query, on which DBToaster and Flink cannot finish in a centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 10 shows the results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' missing data points or lines indicate the system cannot finish within the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Although we adopt the HyperCube algorithm to dispatch all tuples, CROWN can still obtain linear speedup with 𝑝 ă 16, where 𝑝 is the number of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When more workers are available, the margin gain becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This is as expected, since (1) speedup becomes sublinear when adding more workers implied by HyperCube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) the processing time is already short, causing the system’s overhead to dominate the entire runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For all finished data points, CROWN can provide a speedup from 45x to 324x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As Flink and DBToaster cannot finish all experiments with 128GB memory, so we increase the memory usage for these two systems to 500GB, where these two systems still only complete a tiny por- tion of the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, CROWN can finish all experiments with only 128GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If we further limit the memory usage of CROWN to 16GB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 500MB per worker, CROWN still works well without much change in its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Finally, we tested the latency of delta enumeration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the time between an update being received and its deltas being out- putted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 11 shows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The average latency of CROWN is less than 90ms, while that of Trill is more than 6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In addition, the average latency is stable for CROWN, but it keeps growing for Trill, making it infeasible to process streams for long periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To test the scalability of different platforms, we change the scale factor of the SNB benchmark and compare the average update cost between different platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The experiment results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The results show that the average processing time of CROWN is stable under different data sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In contrast, the data size will affect the average processing time of other platforms, suggesting CROWN has better scalability than the competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1e+01 1e+02 1e+03 1e+04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5% 1% 5% 10% 20% 50% 75% 1 1e+05 1e+06 1e+07 1e+08 1e+09 Processing Time (Sec) Number of Results Percentage of Input Size Compares with no Filter Condition CROWN Flink DBToaster CPP DBToaster Spark Input Output Intermediate Join (a) 3-Hop query 1e+01 1e+02 1e+03 1e+04 1e+05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5% 1% 5% 10% 20% 50% 75% 1 1e+04 1e+05 1e+06 1e+07 1e+08 Processing Time (Sec) Number of Results Percentage of Input Size Compares with no Filter Condition (b) 4-Hop query Figure 12: Runtime v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' selectivity Selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Figure 12(a) shows the runtime when varying selectiv- ity of join conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For standard change propagation and HIVM, the maintenance cost depends not only on the input and output size, but also on the size of intermediate views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the 3-Hop query 𝐺1p𝐴, 𝐵q 1 𝐺2p𝐵,𝐶q 1 𝐺3p𝐶, 𝐷q for 𝐺1 “ 𝐺2 “ 𝐺 and 𝐺3 “ Filterp𝐺q, the maintenance cost will be bounded by the size of the view 𝐺1 1 𝐺2 even when 𝐺3 is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In the meantime, the maintenance cost of CROWN only depends on the input and output size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' To better show such a property, we adjust the filter condition in the 3-Hop query, which only changes |𝐺3| instead of |𝐺1 1 𝐺2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Trill is omitted here as it exceeded the 4-hour limit for all data points except for the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' When |𝐺3| ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5%|𝐺|, the output size exceeds the input size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' and when |𝐺3| ě 20%|𝐺|, the output size exceeds the intermediate join size |𝐺1 1 𝐺2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' From the results, we can see the runtime of CROWN scales almost linearly as |𝐺|`|𝑄|, which is as expected since the update sequence has 𝜆 “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other hand, the runtime of the DBToaster and Flink scales proportionally to |𝐺1 1 𝐺2|`|𝑄|, which leads to poor performance when |𝐺3| ď 20%|𝐺|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A larger gap can be observed in Figure 12(b) when evaluating the 4-Hop query with projection, where the inter- mediate join size exceeds the size of the query results, even without any filter conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The runtime of Flink and DBToaster on the 4-Hop query exceeds the 3-Hop query, even with a small output size.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 21–30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1145/2746539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2746609 [21] Muhammad Idris, Martin Ugarte, and Stijn Vansummeren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2017.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proceedings of the GRADES’15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [32] Milos Nikolic, Mohammad Dashti, and Christoph Koch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' How to win a hot dog eating contest: Distributed incremental view maintenance with batch updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ACM SIGMOD International Conference on Management of Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ACM, 511–526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [33] Milos Nikolic and Dan Olteanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Incremental view maintenance with triple lock factorization benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ACM SIGMOD International Conference on Management of Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ACM, 365–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [34] Milos Nikolic, Haozhe Zhang, Ahmet Kara, and Dan Olteanu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' F-IVM: learn- ing over fast-evolving relational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2773–2776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [35] Kenneth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Ross, Divesh Srivastava, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Sudarshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Materialized View Maintenance and Integrity Constraint Checking: Trading Space for Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proceedings of the 1996 ACM SIGMOD International Conference on Management of Data (Montreal, Quebec, Canada) (SIGMOD ’96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 447–458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1145/233269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='233361 [36] Pratanu Roy, Jens Teubner, and Rainer Gemulla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Low-latency handshake join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proceedings of the VLDB Endowment 7, 9 (2014), 709–720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [37] Qichen Wang and Ke Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Maintaining Acyclic Foreign-Key Joins under Updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proceedings of the 2020 ACM SIGMOD International Conference on Management of Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1225–1239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' [38] Mihalis Yannakakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Algorithms for acyclic database schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' International Conference on Very Large Data Bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 82–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A MISSING PROOFS IN SECTION 3 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 ([1, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' A query is 𝛼-acyclic if it has a join tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a query with a generalized join tree T, there exists a join tree T 1 without generalized relation for the query, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', the query is 𝛼-acyclic query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It suffices to show that any generalized join tree T can be transformed into a tree T 1 satisfying the following properties: (i) there is a one-to-one mapping between input relations and nodes in T 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (ii) for each attribute 𝑥, all nodes of T 1 containing 𝑥 form a connected component of T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If T does not contain any generalized relations, then we are done, since (1)-(2) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 already implies (i)-(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We start with the base case that T only contains one generalized relation, say ˆ𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by (3) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1, ˆ𝑅 must be the root of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this case, we can arbitrarily pick one of its child relations to be the root, say 𝑅𝑐, and make all 𝑅𝑐’s siblings be the child node of 𝑅𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It is clear that the resulted tree T 1 is a join tree, since both (i) and (ii) are preserved automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In general, we choose a subtree T 2 that only contains one gen- eralized relation which must also be the root of T 2, and perform a local transformation on T 2 as we have done in the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then, there is no generalized relation in T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We repeat this procedure until all generalized relations are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It can be easily checked 15 that (ii) is always preserved in the whole procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' After all gen- eralized relations are removed, the resulted tree must be a join tree, as both (i) and (ii) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ B MISSING PROOFS IN SECTION 5 Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We first prove the “only if" direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any 𝑡 P 𝑉𝑙p𝑅𝑒q, there exists a 𝑡1 P 𝑄p𝐷q, such that 𝜋𝑒𝑡1 “ 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Mean- while, it indicates that 𝑡2 “ 𝜋𝑝p𝑒q𝑡1 must satisfy 𝑡2 P 𝑉𝑙p𝑅𝑝p𝑒qq, because 𝑡2 P 𝜋𝑝p𝑒q𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, 𝑡2 can join 𝑡, indicates 𝑡 1 𝑉𝑙p𝑅𝑝p𝑒qq ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the “if" direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let 𝑡1 P 𝑉𝑙p𝑅𝑝p𝑒qq be a tuple that can join with 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We divide the join tree T into two subtrees pT𝑒, TzT𝑒q and divide the output attributes y into two sets py𝑒, yzy𝑒q accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Because 𝑡1 P 𝑉𝑙p𝑅𝑝p𝑒qq, 𝜋yzy𝑒𝑄p𝐷 ˙ 𝑡1q ‰ H and we let 𝑡2 be one tuple from 𝜋yzy𝑒𝑄p𝐷˙𝑡1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' On the other side, since 𝑡 P 𝑉𝑠p𝑅𝑒q, there also exists a tuple 𝑡𝑒 in 𝜋y𝑒 p1𝑒1PT𝑒 𝑅𝑒1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 𝑡𝑒 can join with 𝑡2 as 𝑡 can join with 𝑡1 and 𝜋𝑒𝑡𝑒 “ 𝑡, 𝜋𝑝p𝑒q𝑡2 “ 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, 𝑡𝑒 1 𝑡2 P 𝑄p𝐷q, indicates that 𝑡 P 𝑉𝑙p𝑅𝑒q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='g, we assume that 𝑡 is inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The case that 𝑡 is deleted follows the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Direction Ě.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We show that each result in 𝑄p𝐷 ˙ 𝑡1q also appears in Δ𝑄p𝐷,𝑡q, for every witness tuple 𝑡1 of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Wlog, consider a query result 𝑞 P 𝑄p𝐷 ˙𝑡1q for some witness tuple 𝑡1, where either 𝑡1 P 𝑅𝑒 for 𝑒 Ď y or 𝑡1 P 𝜋y𝑅𝑒 for 𝑒 X y ´ 𝑝p𝑒q ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, 𝑄p𝐷 ˙ 𝑡1q Ď 𝑄p𝐷 ` 𝑡q since we have 𝑡1 P Δ ` 𝜋y𝑉𝑠p𝑅𝑒q ˘ and 𝜋keyp𝑒q𝑡1 P 𝑉𝑝p𝑅𝑒q after the insertion of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, all results witnessed by 𝑡1 appear in 𝑄p𝐷 ` 𝑡q after the insertion of 𝑡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝑞 P 𝑄p𝐷 ` 𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next show 𝑞 R 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Now let’s go back to the timestamp before the insertion of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by the definition of witness tuple, 𝑡1 R 𝜋y𝑉𝑠p𝑅𝑒q then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We distinguish two more cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ Case 1: 𝑒 Ď y, 𝑞 R 𝑄p𝐷q since 𝜋𝑒𝑞 “ 𝑡1 but 𝑡1 R 𝜋𝑒𝑄p𝐷q before the insertion of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This further indicates 𝑞 R 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' ‚ Case 2: 𝑒 ´y ‰ H and 𝑒 Xy´𝑝p𝑒q ‰ H, 𝑡1 R 𝜋y𝑉𝑠p𝑅𝑒q before the insertion of 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This way, 𝑡1 R 𝜋y𝑄p𝐷q, thus 𝑞 R 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Combining the analysis above, we have𝑞 P 𝑄p𝐷`𝑡q, and 𝑞 R 𝑄p𝐷q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', 𝑞 P Δ𝑄p𝐷,𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Thus, Ţ 𝑡1:a witness of 𝑡 𝑄p𝐷 ˙ 𝑡1q Ď Δ𝑄p𝐷 ˙ 𝑡q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Direction Ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next show that every result in Δ𝑄p𝐷,𝑡q belongs to 𝑄p𝐷 `𝑡q˙𝑡1 for some witness tuple 𝑡1 of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider an arbitrary query result 𝑞 P 𝑄p𝐷 ` 𝑡q ´ 𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It suffices to show that there exists at least one node 𝑒 P T such that tuple 𝑡1 “ 𝜋𝑒𝑞 if 𝑒 Ď y, or tuple 𝑡1 P 𝜋y𝑅𝑒 with 𝑡1 “ 𝜋𝑒Xy𝑞 if 𝑒 X y ´ 𝑝p𝑒q ‰ H, must be a witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' An important observation is that 𝑡1 now belongs to Δ𝜋y𝑉𝑠p𝑅𝑒q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' otherwise, 𝑞 P 𝑄p𝐷q, coming to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Now consider the highest node 𝑒1 such that 𝑡1 “ 𝜋𝑒1Xy𝑞 and 𝑡1 P Δ ` 𝜋y𝑉𝑠p𝑅𝑒1q ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑒1 is the root, 𝑡1 must be a witness of 𝑡, implied by the Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Otherwise, 𝑒1 is not the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider 𝑡2 “ 𝜋𝑝p𝑒1qXy𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As 𝑡2 R Δ𝑉𝑠p𝑅𝑒2q, 𝑡2 must in𝑉𝑠p𝑅𝑒2q and 𝑡2 P 𝜋𝑒2𝑄p𝐷q before the insertion of 𝑡, which indicates the 𝜋keyp𝑒1q𝑡1 “ 𝜋keyp𝑒1q𝑡2 P 𝑉𝑝p𝑅𝑒1q, and 𝜋keyp𝑒1q𝑡1 P 𝜋keyp𝑒1q𝑄p𝐷q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this way, 𝑡1 is a witness of 𝑡 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Critical Property: Δ𝑄p𝐷,𝑡1q X Δ𝑄p𝐷,𝑡2q “ H holds for any pair of witness tuples 𝑡1,𝑡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It remains to show that there is no duplicate results in Ť 𝑡1:a witness of 𝑡 Δ𝑄p𝐷 ˙ 𝑡1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By contradiction, assume that there exists a query result 𝑞 with at least two witness tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Wlog, let 𝑡1,𝑡2 be two distinct witness tuples in 𝑞, where 𝑡1 P 𝑅𝑒1 for some 𝑒1 Ď y or 𝑒1Xy´𝑝p𝑒1q ‰ H, and some 𝑒2 Ď y or 𝑒2 X y ´ 𝑝p𝑒2q ‰ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, 𝑒1 ‰ 𝑒2, as 𝑞 contains at most one tuple in each relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that the insertion of 𝑡 P 𝑅𝑒 can only change the status of tuples in the ancestors of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Without loss of generality, let 𝑒1 be the ancestor of 𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let 𝑒3 be parent node of 𝑒2 (it could be the case that 𝑒1 “ 𝑒3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Let 𝑡3 “ 𝜋𝑒3Xy𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by the definition of witness tuples, 𝑡3 P 𝜋y𝑉𝑠p𝑅𝑒3q before the insertion of 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by 𝑡3 R Δ𝜋y𝑉𝑠p𝑅𝑒3q, 𝑡1 P 𝜋y𝑉𝑠p𝑅𝑒1q before the insertion, contradicting the fact that 𝑡1 is a witness tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This way, each result in Δ𝑄p𝐷,𝑡q corresponds to one witness tuple, thus there is no duplicates across the extended query results over different witness tuples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We first show the correctness of Algo- rithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider an arbitrary witness tuple 𝑡1 P 𝑅𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Denote the nodes lying on the path from 𝑒1 to 𝑟 as 𝑒1,𝑒2, ¨ ¨ ¨ ,𝑒𝑘p𝑟q sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We can first expand 𝑄p𝐷 ˙ 𝑡1q as follows: 𝑡1 1 ´ 1𝑘 𝑖“1 𝑉𝑟p𝑒𝑖q ¯ 1 𝑄T𝑒1 1 ´ 1𝑘 𝑖“2 𝑄T𝑒𝑖 ´T𝑒𝑖´1 ¯ (13) where 𝑄T represents the query defined over relations in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by the join operator and the properties of generalized join tree, we can further rewrite (13) =: ď 𝑆P𝑡11p1𝑘 𝑖“1𝑉𝑟 p𝑒𝑖qq 𝑆 1 𝑄T𝑒1 1 ´ 1𝑘 𝑖“2 𝑄T𝑒𝑖 ´T𝑒𝑖´1 ¯ “ ď 𝑆P𝑡11p1𝑘 𝑖“1𝑉𝑟 p𝑒𝑖qq 𝑆 ˆ p𝑄T𝑒1 ˙ t𝑆uq ˆ ´ 1𝑘 𝑖“2 p𝑄T𝑒𝑖 ´T𝑒𝑖´1 ˙ t𝑆uq ¯ which is exactly followed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Together with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='7, all results of Δ𝑄p𝐷,𝑡q are enumerated without duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next analyze the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As all witness tuples can be stored in a data structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', a linked list) supporting constant- delay enumeration, every 𝑡1 (line 1) can be retrieved in 𝑂p1q delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' It then suffices to show that 𝑄p𝐷 ˙ 𝑡1q can be enumerated with 𝑂p1q delay for every 𝑡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that subquery 𝑡1 1 ´ 1𝑘 𝑖“1 𝑉𝑟p𝑒𝑖q ¯ (line 4) can be done in 𝑂p1q delay with our hashing index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For the remaining subquery 𝑄T𝑒1 ˙ t𝑆u or 𝑄T𝑒𝑖 ´T𝑒𝑖´1 ˙ t𝑆u, we invoke the procedure FullEnum (line 6-8) and all query results can be enumerated with 𝑂p1q delay, proved by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Combing those subqueries in a form of Cartesian product can yield query results with 𝑂p1q delay, thus completing the whole proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ C MISSING MATERIALS IN SECTION 6 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given an instance of OuMv, we encode the matrix by 𝑅3 and vectors p𝑣𝑖,𝑢𝑖q by 𝑅2 and 𝑅4 separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We construct an update sequence 𝑆 for 𝑄 as follows: (1) we add a tuple 𝑡 “ p𝑖, 𝑗q with lifespan 𝐼p𝑡q “ r´𝑘,𝑘s, for each pair p𝑖, 𝑗q P r𝑛sˆr𝑛s if 𝑀𝑖𝑗 ‰ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) we add a tuple 𝑡 “ p𝑖q with lifespan 𝐼p𝑡q “ r𝑖 ´2𝑘,𝑖s into 𝑅1 and 𝑅5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (3) for each pair of vectors p𝑣𝑖,𝑢𝑖q, we add a tuple 𝑡 “ p𝑖, 𝑣𝑖𝑗q with lifespan 𝐼p𝑡q “ r𝑖,𝑖 ` 2𝑘s to 𝑅2 if 𝑣𝑖𝑗 ‰ 0, and add a tuple 𝑡 “ p𝑢𝑖𝑗,𝑖q with lifespan 𝐼p𝑡q “ r𝑖,𝑖 ` 2𝑘s to 𝑅4 if 𝑢𝑖𝑗 ‰ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (4) if a query result is enumerated, we output true for 𝑣𝑇 𝑖 𝑀𝑢𝑖, and false otherwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (5) we repeat (3)-(4) for the next pair p𝑣𝑖`1,𝑢𝑖`1q, until 𝑛 pairs of vectors are all processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each tuple in 𝑆 has the same lifespan as 2𝑘, thus it is a FIFO sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 16 𝑅5p𝑥4q 𝑅4p𝑥3,𝑥4q 𝑅3p𝑥2,𝑥3q 𝑅2p𝑥1,𝑥2q 𝑅1p𝑥1q (a) T1 𝑅4p𝑥3,𝑥4q 𝑅2p𝑥1,𝑥2q 𝑅3p𝑥2,𝑥3q 𝑅5p𝑥4q 𝑅1p𝑥1q (b) T2 𝑅4p𝑥3,𝑥4q 𝑅2p𝑥1,𝑥2q 𝑅3p𝑥2,𝑥3q 𝑅5p𝑥4q 𝑅1p𝑥1q (c) T3 Figure 13: Join trees for 𝑄 “ 𝑅1p𝑥1q 1 𝑅2p𝑥1,𝑥2q 1 𝑅3p𝑥2,𝑥3q 1 𝑅4p𝑥3,𝑥3q 1 𝑅5p𝑥4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We note that in any generalized join tree T of 𝑄, there always exists a subtree in which either 𝑅1 ´ 𝑅2 ´ 𝑅3 or 𝑅5 ´ 𝑅4 ´ 𝑅3 is a leaf-to-root path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Wlog, assume 𝑅1 ´ 𝑅2 ´ 𝑅3 is a leaf-to- root path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' First, for each tuple 𝑡 P 𝑅1, 𝜆p𝑡q “ 1 as 𝑅1 is a leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For 𝑡 “ p𝑖, 𝑣𝑖𝑗q P 𝑅2, we observe that ˜𝐼p𝑡q “ r𝑖,𝑖s as 𝐼p𝑡q “ r𝑖,𝑖 ` 2𝑘s and 𝐼p𝑡1q “ r𝑖 ´ 2𝑘,𝑖s for some tuple 𝑡1 P 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' But in this case, 𝜆p𝑡q “ 1 still holds, as there exists no tuple 𝑡1 P 𝑅1 with ˜𝐼p𝑡1q Ď r𝑖,𝑖s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' However, for each tuple 𝑡 P 𝑅3, 𝜆p𝑡q “ 𝑛 as there exists a tuple 𝑡1 P 𝑅2 such that ˜𝐼p𝑡1q “ r𝑖,𝑖s for every 𝑖 P r𝑛s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, the enclosureness of 𝑆 on every generalized join tree is 𝜆 “ 𝑛2¨𝑛`𝑛2¨1 𝑛2 “ 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The correctness of this simulation is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This way, if there is a data structure that can be updated in 𝑂p𝜆1´𝜖q time while sup- porting 𝑂p𝜆2´𝜖q-delay enumeration for 𝑄 over any FIFO sequence, then the OuMv problem can be solved in 𝑂p𝑛2 ¨𝜆1´𝜖 `𝑛 ¨𝜆2´𝜖q “ 𝑂p𝑛3´𝜖q time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that the construction above requires a database of size at least 𝑛2 “ 𝜆2, thus 𝜆 ď a |𝐷|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next turn to the update cost of our indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As mentioned in the beginning of Section 6, the total update cost of the entire sequence is asymptotically dominated by that of P-Update, which is further bounded by the number of times all the counters countr𝑡s can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The following lemma connects this quantity with the enclosureness of the update sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For any tuple 𝑡, countr𝑡s changes 𝑂p𝜆p𝑡qq times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The status change of tuple 𝑡 P 𝑅𝑒 falls into one of the following two cases: (1) tuple 𝑡 is being inserted or deleted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) some tuple 𝑡1 P 𝑅𝑒1 for 𝑒1 P T𝑒 is inserted or deleted, and this update propagates to 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that tuple 𝑡 can be inserted and deleted once in its lifespan, thus bounded by 𝑂p1q and the cost is reflected in R-Update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then, we will focus on the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We start with the case that 𝑒 has one child node in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this case, countr𝑡s has its value changed between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Note that if an insertion changes𝑡 from𝑅𝑒{𝑉𝑠p𝑅𝑒q to𝑉𝑠p𝑅𝑒q, subsequent insertions won’t change the status of 𝑡 unless a deletion occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Consider a set of 𝑘 disjoint intervals ˜𝐼1, ˜𝐼2, ¨ ¨ ¨ , ˜𝐼𝑘 in ordering, such that ˜𝐼𝑗 P rI𝑒, ˜𝐼𝑗 Ď 𝐼p𝑡q for each 𝑗 P r𝑘s, and there exists no additional interval ˜𝐼 such that ˜𝐼 Ĺ ˜𝐼𝑗 or ˜𝐼1 Ď r˜𝐼` 𝑗 , ˜𝐼´ 𝑗`1s for any 𝑗 P t1, 2, ¨ ¨ ¨ ,𝑘u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Each of the 𝑘 intervals can change the status of 𝑡 at most twice, so they together can change the status of 𝑡 at most𝑂p𝑘q times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The effective lifespan of 𝑡 exactly captures such a quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next consider a case when 𝑒 has two child nodes 𝑒1,𝑒2 P T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Similarly, consider a set of 𝑘 disjoint intervals ˜𝐼1, ˜𝐼2, ¨ ¨ ¨ , ˜𝐼𝑘 in ordering, such that ˜𝐼𝑗 P rI𝑒, ˜𝐼𝑗 Ď 𝐼p𝑡q for each 𝑗 P r𝑘s, and there exists no additional interval ˜𝐼 such that ˜𝐼 Ĺ ˜𝐼𝑗 or ˜𝐼 Ď r˜𝐼` 𝑗 , ˜𝐼´ 𝑗`1s for any 𝑗 P t1, 2, ¨ ¨ ¨ ,𝑘u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We can make the following two observations: (1) For any ˜𝐼𝑗, countr𝑡s can change at most 2 times within ˜𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' (2) For any two adjacent intervals ˜𝐼𝑗 and ˜𝐼𝑗`1, countr𝑡s can change at most 4 times in their gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Together, we can conclude that countr𝑡s can change at most 6 ¨𝑘 “ 𝑂p𝑘q times when there are two child nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We next go into details of (1) and (2) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For (1), we assume ˜𝐼𝑗 P rI𝑒1 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By the definition of effective lifespan, there cannot be any insertion or deletion in any node of T𝑒1 within ˜𝐼𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Nevertheless, updates may still exist within ˜𝐼𝑗 on some node of T𝑒2, which might further change countr𝑡s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We distinguish two more cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If countr𝑡s changes from 1 to 2, due to an insertion from T𝑒2, a deletion must not exist within ˜𝐼𝑗 on any node of T𝑒2, implied by the fact that there exists no ˜𝐼 such that ˜𝐼 Ĺ ˜𝐼𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, countr𝑡s can change at most once in ˜𝐼𝑗 for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Otherwise, countr𝑡s changes from 2 to 1, after a deletion from T𝑒2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' We then go into the first case and countr𝑡s can change at most one more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In total, countr𝑡s can change at most twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' For (2), it is clear that at the right endpoint of ˜𝐼𝑗 and the left endpoint of ˜𝐼𝑗`1, countr𝑡s can change once as the deletion and insertion of an effective lifespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In the meantime, there does not exist another effective lifespan within their gap, so for any 𝑒𝑖 P t𝑒1,𝑒2u, there exists no deletion on T𝑒𝑖 in the gap following an insertion in T𝑒𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This way, countr𝑡s can change at most four times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 2 Ñ 1 Ñ 0 Ñ 1 Ñ 2) within their gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' At last, we consider the general case when 𝑒 has multiple child nodes in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this case, countr𝑡s has its value changed among 0, 1, ¨ ¨ ¨ , 𝑗, where 𝑗 is the number of child nodes of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' By extend- ing the previous two observations, we conclude that countr𝑡s can change at most 3𝑗 ¨𝑘 times, where 𝑗 can be considered as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' With respect to all possible choices of 𝑘, we observe that 𝑘 ď max JĎ rI𝑒 @𝐼p𝑡1qPJ,𝐼p𝑡1qĎ𝐼p𝑡q @𝐼p𝑡2q,𝐼p𝑡3qPJ,𝐼p𝑡2qX𝐼p𝑡3q“H 1 ` |J| “ 𝜆p𝑡q, thus countr𝑡s can change at most 𝑂p𝜆p𝑡qq times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ The time cost of Algorithm 3 is determined by the number of iter- ations of for-loop (line 2 or 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' One can easily observe that countr𝑡s will be changed for some tuple 𝑡 once in each iteration, there- fore the running time can be bounded by the number of changes to countr𝑡s over all tuples 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Now consider an update sequence 𝑆 with enclosureness 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Implied by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='1 the total update cost is 𝑂 `ř 𝑡PI 𝜆p𝑡q ˘ , which is 𝑂 ´ ř 𝑡PI 𝜆p𝑡q |I| ¯ “ 𝑂p𝜆q amortized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Putting everything together, we have completed the proof for The- orem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' The key idea of proving Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='8 is to show that for any q-hierarchical query 𝑄, there exists a free-connex join tree T such that for every relation 𝑅𝑒, 𝑒 is a leaf node in T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=', T is a height-1 generalized join tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then by definition, enclosureness of arbitrary update sequence over T is always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 17 We construct T for 𝑄 in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In the base case, if 𝑄 only contains one relation 𝑅𝑒, we just return a single node 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In general, we distinguish two more cases on 𝑄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑄 is connected, we create a generalized relation containing the set of common attributes that appearing in all relations and set it as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Then, we invoke this procedure recursively for the residual query of 𝑄 by removing such common attributes from all relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑄 is disconnected, we construct a tree for each connected component of 𝑄, and add each tree as a child of the root 𝑟 (if such a root does not exist, we add a super root 𝑟 with attributes as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Next, we need to show T is free-connex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Suppose not, there exists a pair of relations 𝑒,𝑒1 P T such that 𝑒 X y “ H and 𝑒1 X y ‰ H, but 𝑒1 is a descendent of 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Based on the recursive procedure above, it is clear that if 𝑒 is the ancestor of 𝑒1, then 𝑒 Ď 𝑒1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' In this way, there must exist a pair of attributes 𝑥1,𝑥2 such that 𝑥2 P 𝑒, 𝑥1,𝑥2 P 𝑒1, 𝑥1 P y and 𝑥2 R y, contradicting to the fact that 𝑄 is q-hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' At last, it is easy to check that for every relation 𝑅𝑒, 𝑒 is a leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' This way, we have find such a generalized join tree T, thus completing the whole proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Given a height-2 generalized join tree T and consider an arbitrary tuple 𝑡 P 𝑅𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑒 is a leaf node, ˜𝐼p𝑡q “ r𝑡`,𝑡´s and 𝜆Tp𝑡q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' If 𝑒 is an internal node, ˜𝐼p𝑡q Ď r𝑡`,𝑡´s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' But here, as the join tree is a height-2 join tree, every 𝑒’s child node must be a leaf node, hence every tuple 𝑡1 P 𝑅𝑒1 for 𝑒1 P T𝑒 has ˜𝐼p𝑡1q “ r𝑡` 1 ,𝑡´ 1 s, and there exists no tuple 𝑡2 such that 𝑡` ă 𝑡` 2 and 𝑡´ 2 ă 𝑡´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As each tuple 𝑡 has 𝜆Tp𝑡q “ 1, by definition, 𝜆Tp𝑆q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' As there is no deletion for every tuple 𝑡, ˜𝐼p𝑡q “ r𝑡`, `8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Hence, for every 𝑡, by definition, 𝜆Tp𝑡q “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' □ D SQL QUERIES 3-Hop Full Join Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as A, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as D FROM G G1, G G2, G G3 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND FILTER OVER (G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst) 4-Hop Full Join Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as A, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as D, G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as E FROM G G1, G G2, G G3, G G4 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' src AND FILTER OVER (G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst) 3-Hop Join-Project Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C FROM G G1, G G2, G G3 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src 4-Hop Join-Project Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as D FROM G G1, G G2, G G3, G G4 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' src AND FILTER OVER (G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst) 2-Comb Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as A, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as B, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src as C, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst as D FROM G G1, G G2, G G3, V1, V2 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='v = G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='v = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst Star Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src, COUNT(G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst, G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst) FROM G G1, G G2, G G3, G G4 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src = G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' src GROUP BY G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' Dumbbell Full Join Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT * FROM G G1, G G2, G G3, G G4, G G5, G G6, G G6 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst AND G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src Dumbbell Join-Project Query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src, G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst FROM G G1, G G2, G G3, G G4, G G5, G G6, G G6 WHERE G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src AND G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src = G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst AND G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='dst = G5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='src SNB Query 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT p_personid, p_firstname, p_lastname, m_messageid, k_person1id FROM person, message, knows WHERE p_personid = m_creatorid AND k_person2id = p_personid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SNB Query 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id, k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id, k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id, t_tagid, m_messageid FROM tag, message, message_tag, knows1 k1, knows2 k2 WHERE m_messageid = mt_messageid AND mt_tagid = t_tagid AND k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id AND m_creatorid = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id AND m_c_replyof is NULL AND FILTER OVER (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id) SNB Query 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' 18 SELECT k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id, k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id, k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id, t_tagid, m_messageid FROM tag, message, message_tag, knows1 k1, knows2 k2 WHERE m_messageid = mt_messageid AND mt_tagid = t_tagid AND k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id AND k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id <> k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id AND m_creatorid = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person2id AND m_c_replyof is NULL AND FILTER OVER (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content='k_person1id) SNB Query 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} +page_content=' SELECT t_name, t_tagid, count(distinct m_messageid) FROM tag, message, message_tag, knows WHERE m_messageid = mt_messageid AND mt_tagid = t_tagid AND m_creatorid = k_person2id AND m_c_replyof is NULL AND FILTER OVER (k_person1id) GROUP BY t_name, t_tagid 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE2T4oBgHgl3EQfnAgQ/content/2301.04003v1.pdf'} diff --git a/B9AyT4oBgHgl3EQfePhg/content/tmp_files/2301.00317v1.pdf.txt b/B9AyT4oBgHgl3EQfePhg/content/tmp_files/2301.00317v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b245dd7225a957b61e2f7f3842f351999a7d32e --- /dev/null +++ b/B9AyT4oBgHgl3EQfePhg/content/tmp_files/2301.00317v1.pdf.txt @@ -0,0 +1,1145 @@ +A Note On Acyclic Token Sliding Reconfiguration +Graphs of Independent Sets +David Avis1 +Duc A. Hoang2 +1 Graduate School of Informatics, Kyoto University, Japan +School of Computer Science, McGill University, Canada +avis@cs.mcgill.ca +2 Graduate School of Informatics, Kyoto University, Japan +hoang.duc.8r@kyoto-u.ac.jp +Abstract +We continue the study of token sliding reconfiguration graphs of independent sets initiated by +the authors in an earlier paper (arXiv:2203.16861). Two of the topics in that paper were to study +which graphs G are token sliding graphs and which properties of a graph are inherited by a token +sliding graph. In this paper we continue this study specializing on the case of when G and/or its +token sliding graph TSk(G) is a tree or forest, where k is the size of the independent sets considered. +We consider two problems. The first is to find necessary and sufficient conditions on G for TSk(G) +to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be +a token sliding graph. For the first problem we give a forbidden subgraph characterization for the +cases of k = 2, 3. For the second problem we show that for every k-ary tree T there is a graph G for +which TSk+1(G) is isomorphic to T. A number of other results are given along with a join operation +that aids in the construction of TSk(G)-graphs. +1 +Introduction +In a reconfiguration variant of a computational problem (e.g., Satisfiability, Independent Set, +Vertex-Coloring, etc.), a transformation rule that describes an adjacency relation between feasi- +ble solutions (e.g., satisfying truth assignments, independent sets, proper vertex-colorings, etc.) of the +problem is given. One of the main goals is to decide whether there is a sequence of adjacent feasible +solutions that “reconfigures” one given solution into another. Another way of looking at these reconfig- +uration problems is via the so-called reconfiguration graph—a graph whose nodes are feasible solutions +and two nodes are adjacent if one can be obtained from the other by applying the given rule exactly once. +The mentioned question now becomes deciding whether there is a path between two given nodes in the +reconfiguration graph. Recently, reconfiguration problems have been intensively studied from different +perspectives [2, 6–8]. +One of the most well-studied reconfiguration variants of Independent Set is the so-called Token +Sliding problem, which was first introduced by Hearn and Demaine [4] in 2005. +We refer readers +to [2, 7, 8] and the references therein for more details. Surprisingly, though Token Sliding has been +well-investigated, the realizability and structural properties of its corresponding reconfiguration graph— +the one which we will refer to as the TSk-graph (which stands for Token Sliding (Reconfiguration) +graph)—have not been studied until recently [1]. On the other hand, when considering either general +vertex subsets, dominating sets, or proper vertex-colorings of a graph as the “input feasible solutions”, +their corresponding reconfiguration graphs have been very well-characterized [5, 6]. +For any graph-theoretic terminology and notation not defined here, we refer readers to [3]. Given a +graph G = (V, E) and an integer k ≥ 2. For two sets X, Y , we sometimes use X + Y and X − Y to +indicate X ∪ Y and X \ Y . We abbreviate X ∪ {u} (resp., X \ {u}) by X + u (resp., X − u). We use +NG(u), or simply just N(u) when the graph G is clear from the context, to denote the (open) neighbors +of u, i.e., set of all vertices in G that are adjacent to u. The closed neighbors of u, denoted by NG[u] +or simply N[u], is the set NG(u) + u. The degree of u, denoted by degG(u), is nothing but the size of +NG(u). An independent set (or stable set) of G is a vertex subset I such that for every u, v ∈ I we +have uv /∈ E(G). The TSk-graph of G, denoted by TSk(G), takes all size-k independent sets of G as its +1 +arXiv:2301.00317v1 [math.CO] 1 Jan 2023 + +nodes and two nodes I, J are adjacent (under Token Sliding (TS)) if there exist two vertices u, v ∈ V (G) +such that I − J = {u}, J − I = {v}, and uv ∈ E(G). Two graphs G and H are isomorphic, denoted +by G ≃ H, if there exists a bijective mapping f : V (G) → V (H) such that uv ∈ E(G) if and only if +f(u)f(v) ∈ E(H). A graph G is called a TSk-graph if there exists a graph H such that G ≃ TSk(H). A +forest is a graph having no cycles (i.e., it is acyclic) and a connected forest is a tree. A TSk-tree/forest +is a TSk-graph which is also a tree/forest. Figure 1 illustrates a TS2-tree on six vertices (right). In [1], +ab +ac +bd +ae +ef +ce +TS2(G) +a +b +c +d +e +f +G +Figure 1: A graph G with TS2(G) = D1,3,2. Each node ab represents a size-2 stable set of G. +the authors studied various properties of the family of TSk-graphs. For a graph G, two of the questions +studied were: +(Q1) What are necessary and sufficient conditions for G so that TSk(G) is a forest? +(Q2) What are necessary and sufficient conditions for G to be a TSk-graph? +In this paper, we study these two questions for the case when G is a tree or a forest. +The union G ∪ H of two (labelled) graphs G and H is the graph with V (G ∪ H) = V (G) ∪ V (H) and +E(G ∪ H) = E(G) ∪ E(H). When vertices and edges of G and H are considered distinct regardless of +their labels, we say that G ∪ H is the disjoint union of G and H, and write G + H instead of G ∪ H to +distinguish from their union. We respectively denote by Kn, Pn, and Cn the complete graph, path, and +cycle on n vertices. Km,n (m ≤ n) is the complete bipartite graph whose two partite sets are of sizes m +and n respectively. K1,n is also called a star—a tree obtained by attaching n leaves to a central vertex. +A family of graphs that we will use in the sequel generalizes stars and paths. For fix integers n, r, s ≥ 1, +let Dr,n,s be the tree obtained from Pn by appending r leaves at one end and s leaves at the other. Note +that D1,1,s is the star K1,s+1 and D1,n,1 is the path Pn+2. Figure 1 illustrates D1,3,2 (right). An n-ary +tree is a rooted tree in which each node has at most n children. Any tree with maximum degree at most +n+1 can be rooted at a vertex with degree at most n (e.g., a leaf) to produce a n-ary tree. In particular, +a 2-ary tree is nothing but the well-known binary tree. +In the next section, we begin by partially answering (Q1) when G is a tree/forest and k ∈ {2, 3} and +conclude the section by conjecturing for k ≥ 4. Then, before addressing (Q2) for some trees/forests, in +particular k-ary trees and Dr,n,s, we define an important graph operation which, under certain conditions, +can be used for combining two TSk-graphs by taking their union to obtain a new one. The final section +of the paper gives some concluding remarks. +2 +Results on (Q1) +In this section, we prove the necessary and sufficient conditions on a tree/forest G such that TSk(G) is +acyclic for k ∈ {2, 3}, partially answering (Q1). +We begin with some definitions and observations. The complement G of a graph G is the graph +with V (G) = V (G) and E(G) = {uv : uv /∈ E(G)}. The size-m matching, denoted by mK2, is the +graph obtained by taking the disjoint union of m copies of K2. Observe that TS2(2K2) ≃ C4. We +label vertices in a Dr,n,s (r, s ≥ 1) as follows: Vertices of Pn are labelled p1, . . . , pn. +The r leaves +attached to p1 are u1, . . . , ur and the s leaves attached to pn are v1, . . . , vn. D2,2,2 is shaped like an +2 + +G +TS2(G) +Figure 2: A list G of n-vertex graphs G (4 ≤ n ≤ 7) excluding Cn (n ≥ 5) such that if TS2(G′) has no +cycle then G′ does not contain any member G of G as an induced subgraph. +3 + +H and TS2(D2,2,2) contains a cycle C8 whose vertex-set is {u1v1, u1p2, u1v2, p1v2, u2v2, u2p2, u2v1, p1v1}. +Indeed, respectively from Lemma 1 of [1] and Figure 2, if a n-vertex graph G is either Cn (n ≥ 5) or a +graph in the list G described in Figure 2 (which includes 2K2 and D2,2,2), the graph TS2(G) contains a +cycle. Additionally, we have: +Lemma 1. +(a) For k ≥ 2, TSk(2K2 + nK1) contains a cycle C4 if n ≥ k − 2 otherwise it is acyclic. +(b) For k ∈ {2, 3}, s ≥ 1, TSk(D1,n,s) contains a cycle C4 if n ≥ 2k − 1 otherwise it is acyclic. +(c) For k ∈ {2, 3} and r, s ≥ 2, TSk(Dr,n,s) contains a cycle C8 if n ≥ 2k − 2 otherwise it is acyclic. +Proof. +(a) If n < k − 2, there is no size-k independent set in 2K2 + nK1, thus its TSk-graph is +obviously acyclic. Otherwise, let I ⊆ V (nK1) be an arbitrary independent set of size k − 2, and let +E(2K2) = {ab, cd}. Then, {I +a+c, I +a+d, I +b+c, I +b+d} induce a C4 in TSk(2K2 +nK1). +(b) Observe that if n ≥ 2k − 1, D1,n,s contains an induced 2K2 + (k − 2)K1, which can be obtained +by taking u1p1 and pnv1 as edges of 2K2 and the remaining k − 2 independent vertices from the +path D1,n,s − {u1, p1, p2, pn−1, pn, v1, . . . , vs} on n − 4 vertices. (Since n ≥ 2k − 1, this path has +an independent set of size at least ⌈(n − 4)/2⌉ ≥ ⌈(2k − 5)/2⌉ = k − 2.) Then, using a similar +argument as in (a) we have TSk(D1,n,s) contains a C4. +On the other hand, if n ≤ 2k − 2 for k ∈ {2, 3}, since D1,n−1,s is always an induced subgraph of +D1,n,s for n ≥ 2, it follows that if TS2(D1,n−1,s) has a cycle then so is TS2(D1,n,s). Therefore, +it suffices to show that TSk(D1,2k−2,s) is acyclic for k ∈ {2, 3}. Indeed, based on the number of +tokens placed on the path u1p1 . . . pn (which is at most three), one can verify that each component +of TSk(D1,2k−2,s) is either an isolated vertex, a path, or a star. +(c) Observe that if n ≥ 2k −2, Dr,n,s contains the independent sets I +u1 +v1, I +u1 +pn, I +u1 +vs, +I + p1 + v1, I + p1 + vs, I + ur + v1, I + ur + pn, and I + ur + vs, where I = ∅ when n = 2 and +otherwise I is an independent set of the path p2 . . . pn−1 of size k − 2. (Note that p2 . . . pn−1 has +an independent set of size at most ⌈(n − 2)/2⌉ ≥ k − 2.) They indeed induce a C8 in TSk(Dr,n,s). +On the other hand, if n ≤ 2k−3 for k ∈ {2, 3}, using a similar case-analysis as in (b), one can verify +that each component of TSk(Dr,n,s) is either an isolated vertex, a path, or a star, and therefore it +is acyclic. +We are now ready to show the necessary and sufficient conditions for a tree/forest G such that TSk(G) +is acyclic, where k ∈ {2, 3}. +Proposition 2. Let T be a tree. Then TS2(T) is acyclic if and only if T is {2K2, D2,2,2}-free. +Proof. (⇒) Suppose to the contrary that either 2K2 or D2,2,2 is an induced subgraph of T. In the first +case it follows from the discussion above that TS2(T) contains a C4 and in the second case that it +contains a C8. +(⇐) We assume that TS2(T) contains a cycle and show that it must contain one of the two forbidden +subgraphs. Firstly, suppose that T is a path Pn. Since TS2(T) contains a cycle, it follows from +Lemma 1(b) that n ≥ 5 and so T contains an induced 2K2. +We now assume T has a vertex of at least degree 3. We will construct a copy T ′ of T by initially +choosing a vertex a of maximum degree in T and letting T ′ = N[a]. Note that TS2(T ′) is acyclic. +We add edges from T to T ′ and show after each addition that either T ′ contains a forbidden +subgraph, so we are done, or that TS2(T ′) remains acyclic so that T ̸= T ′. +Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other +child. Since TS2(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children. Let e be a child of b with +maximum degree. We add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2. If +r ≥ 2, we have the required forbidden induced subgraph. If r = 1 then by Lemma 1(b) TS2(T ′) +is acyclic, so there must be extra edges to add to T ′. If c has a child y then {b, c, e, y} induce a +2K2. Otherwise, e must have at least one child g. Adding eg to T ′ we obtain 2K2 as an induced +subgraph on {a, d, e, g}. This completes the proof. +4 + +a +b +c +d +e +a +d +e +b +c +y +a +d +b +c +e +g +r ≥ 2 +r = 1 +Figure 3: Illustration for Proposition 2: Some trees T ′ containing N[b] whose TS2-graphs have a cycle. +Here r is the number of children of b. Copies of 2K2 and D2,2,2 are marked by red color. +Corollary 3. Let T be a tree. Then TS2(T) is acyclic if and only if T is either K1,s or D1,2,s for some +positive integer s. +Proof. The proof of Proposition 2 can be viewed as an algorithm that takes a tree T and either terminates +with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T. +Corollary 4. Let F be a forest. Then TS2(F) is a acyclic if and only if F is {2K2, D2,2,2}-free. +Proof. We prove that TS2(F) contains a cycle if and only if F contains one of the graphs in {2K2, D2,2,2} +as an induced subgraph. +Suppose that TS2(F) contains a cycle. Since the independent sets have size two, both vertices of +each independent set must lie in the same connected component T of F. By Proposition 2, the tree T +must have either 2K2 or D2,2,2 as an induced subgraph. +Conversely if F contains 2K2 or D2,2,2 as an induced subgraph then TS2(F) contains respectively a +C4 or a C8. +Moving to the case of stable sets of size three, the conditions for trees and forests differ slightly. We +deal with the tree case first. +Proposition 5. Let T be a tree. Then TS3(T) is acyclic if and only if T is {2K2 + K1, D2,4,2}-free. +Proof. The structure of the proof is the same as for Proposition 2. However, there are more cases to +consider. +(⇒) Suppose to the contrary that either 2K2 + K1 or D2,4,2 is an induced subgraph of T. In the first +case it follows that TS3(T) contains a C4 and in the second case that it contains a C8. +(⇐) We assume that TS3(T) contains a cycle and show that it must contain one of the two forbidden +subgraphs. The first part of the proof is essentially the same as for Proposition 2 with minor +modifications. Firstly suppose that T is a path Pn. Since TS3(T) contains a cycle it follows from +Lemma 1(b) that n ≥ 7 and so T contains an induced 2K2 + K1. +We now assume T has a vertex of at least degree 3. We will construct a copy T ′ of T by initially +choosing a vertex a of maximum degree in T and letting T ′ = N[a]. Note that TS3(T ′) is acyclic. +We add edges from T to T ′ showing after each addition that either T ′ contains a forbidden subgraph, +so we are done, or that TS3(T ′) remains acyclic so that T ̸= T ′. +Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other child. +Since TS3(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children. Let e be a child of b with maximum +degree. If c has a child y then {b, c, d, e, y} induce a 2K2 + K1 and we are done. Otherwise we +5 + +add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2. By Lemma 1(c), TS3(T ′) is +acyclic and so T ̸= T ′. There are two cases: +(r ≥ 2) Let f be a second child of b and let g be a child of e. Adding eg to T ′ we obtain 2K2 + K1 +as an induced subgraph on {a, d, e, f, g}. +(r = 1) Since e is the only child of b it must have children. Let t ≥ 1 be the number of children of e +and let h be the child of e of maximum degree. We add N[e] to T ′ obtaining a copy of Dt,3,s +and TS3(T ′) is acyclic by Lemma 1(c). There are two subcases: +(t ≥ 2) Let i be any other child of e. Since TS3(T ′) is acyclic h must have at least one child j. +We have now constructed an induced 2K2 + K1 on {a, d, h, i, j}. +(t = 1) If h has a single child k add hk to T ′ which is a copy of D1,4,s and again by Lemma 1(c) +TS3(T ′) is acyclic. So k has a child l. Adding kl to T ′ it contains an induced P7 and we +find the forbidden subgraph 2K2 + K1 on vertices {a, d, e, k, l}. Otherwise, h has at least +two children including vertices k and m. Adding edges hk and hm to T ′ we obtain the +forbidden subgraph D2,4,2. This completes the proof. +a +b +c +d +e +f +g +a +c +d +b +e +h +i +j +a +c +d +b +e +h +k +l +a +c +d +b +h +i +j +r ≥ 2 +r = 1, t ≥ 2 +r = 1, t = 1 +e +Figure 4: Illustration for Proposition 5: Some trees T ′ containing N[b] whose TS3-graphs have a cycle. +Here r and t are respectively the number of children of b and its child e. Copies of 2K2 + K1 and D2,4,2 +are marked by red color. +Corollary 6. Let T be a tree. Then TS3(T) is a acyclic if and only if for some positive integer s, T is +either K1,s, D1,n,s where n ≤ 4, or Dr,n,s where r ≥ 2 and n ≤ 3. +Proof. The proof of Proposition 5 can be viewed as an algorithm that takes a tree T and either terminates +with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T showing that +TS3(T) has a cycle. +6 + +Corollary 7. Let F be a forest. +Then TS3(F) is a forest if and only if F is {2K2 + K1, D2,2,2 + +K1, D2,4,2}-free. +Proof. We prove that TS3(F) contains a cycle if and only if F contains one of the graphs in {2K2 + +K1, D2,2,2 + K1, D2,4,2} as an induced subgraph. +Suppose that TS3(F) contains a cycle C. Since the independent sets have size three, there are three +cases to consider. Firstly, if the three vertices of each independent set in C lie in the same connected +component T of F, by Proposition 5, the tree T must have either 2K2 + K1 or D2,4,2 as an induced +subgraph. Secondly, suppose two of the vertices of each stable set lie in the same connected component +T of F, which must have at least two connected components. Thus, C induces a cycle in TS2(T). So by +Proposition 2, the tree T must have either 2K2 or D2,2,2 as an induced subgraph. Since F has at least +two components, F contains 2K2 + K1 or D2,2,2 + K1. Finally, suppose each vertex of each stable set +lies in a different component of F, which therefore has at least three components. At least two of these +components must be non-trivial, i.e., contain an edge. Therefore, F contains an induced 2K2 + K1. +Conversely, suppose F contains 2K2 + K1, D2,2,2 + K1 or D2,4,2 as an induced subgraph. Then +TS3(F) contains a C4 in the first instance or a C8 in the other two. +For k ≥ 4, we have the following proposition. +Proposition 8. Let F be a forest. For k ≥ 4, if F contains either 2K2+(k−2)K1, or D2,2,2+(k−2)K1, +or D2,4,2 + (k − 3)K1 as an induced subgraph, TSk(F) has a cycle. +Proof. One can verify that TS2(2K2) contains a C4, and TS2(D2,2,2) and TS3(D2,4,2) both contain a +C8. As a result, so do TSk(2K2 + (k − 2)K1), TSk(D2,2,2 + (k − 2)K1), and TSk(D2,4,2 + (k − 3)K1), +respectively. Consequently, TSk(F) has a cycle, as desired. +We conclude this section with the following conjecture for k ≥ 4. +Conjecture 9. Let F be a forest. For k ≥ 4, if TSk(F) is a forest, F is {2K2 + (k − 2)K1, D2,2,2 + (k − +2)K1, D2,4,2 + (k − 3)K1}-free. +3 +H-join and H-decomposition +Before considering (Q2), in this section, we describe an operation for combining TSk-graphs to produce +new ones. We first define a family of base graphs as follows. Let V be a set of k + 1 vertices including +two labelled u and v. Then Bk(V, uv) is the graph with vertex set V and single edge uv. We have +TSk(Bk(V, uv)) = K2 whose two vertices are labelled by the independent sets V − u and V − v. Next, +we define the H-join operation and its inverse. +Definition 10. Vertex-labelled graphs G1 and G2 are H-consistent if the (possibly empty) intersection +of their vertex sets define the same (possibly empty) common induced subgraph H. The H-join of H- +consistent graphs G1 and G2 is the graph H(G1, G2) with V (H(G1, G2)) = V (G1) ∪ V (G2). The edges +E(H(G1, G2)) consist of E(G1)∪E(G2) plus all edges vw with v ∈ V (G1)\V (H) and w ∈ V (G2)\V (H). +Recall that a (vertex) cut-set in a connected graph G is a vertex set W such that G−W is disconnected. +We extend this definition to the case where G is disconnected by allowing W = ∅. We say that W +decomposes G into two (not necessarily connected) induced subgraphs G1 and G2 for which V (G1) ∩ +V (G2) = W and V (G1) ∪ V (G2) = V (G). If G − W has more than two (connected) components, the +decomposition is not unique. +Definition 11. Let G be a vertex-labelled graph. Let W ⊂ V (G) = V (G) decompose the complement +G into G1 and G2. Let H be the subgraph of G induced by W. We say that G can be H-decomposed +into G1 and G2. +It follows from the definitions that if G = H(G1, G2) then G can be H-decomposed into G1 and G2, +and vice versa. It is easy to verify that the size-k independent sets of H(G1, G2) are the union of those +of G1 and those of G2. +As an example consider the two 4-vertex graphs G1 and G2 that are paths with edge sets E(G1) = +{ad, bc, cd} and E(G2) = {ad, ae, eb}. These share a common induced subgraph H with V (H) = {a, b, d} +and E(H) = {ad}. We have V (H(G1, G2)) = {a, b, c, d, e} and E(H(G1, G2)) = {ad, ae, bc, cd, ce, be}. +Note that TS2(G1) is the path with edges {ac − ab, ab − bd} and that TS2(G2) is the path with edges +7 + +a +b +c +d +G1 +a +b +e +d +G2 +c +e +a +b +d +H(G1, G2) +ab +ac +bd +TS2(G1) +ab +de +bd +TS2(G2) +ab +ac +de +bd +TS2(H(G1, G2)) +Figure 5: The graphs G1, G2, H(G1, G2), and their corresponding TS2-graphs. Here TS2(H(G1, G2)) = +TS2(G1) ∪ TS2(G2). +{ab−bd, bd−de}. It can be verified that TS2(H(G1, G2)) is the path with edges {ac−ab, ab−bd, bd−de} +which is the union of two paths TS2(G1) and TS2(G2). (See Figure 5.) +Now consider the graph G3 which is the path with edges {ad, cd, ce}. +G1 and G3 share a com- +mon induced subgraph H with V (H) = {a, c, d} and E(H) = {ad, cd}. +We have E(H(G1, G3)) = +{ad, bc, be, cd, ce}. Note that TS2(G3) is the path with edges {ac−ae, ae−de}. In this case, TS2(H(G1, G3)) +is the graph with edges {ab − ac, ac − ae, ae − de, de − bd, bd − ab, ab − ae} which is the union of TS2(G1), +TS2(G3), and the two additional edges de − bd, ab − ae. (See Figure 6.) +a +b +c +d +G1 +a +e +c +d +G3 +b +e +a +c +d +H(G1, G3) +ab +ac +bd +TS2(G1) +ac +ae +ed +TS2(G3) +ab +ac +ae +bd +ed +TS2(H(G1, G3)) +Figure 6: The graphs G1, G3, H(G1, G3), and their corresponding TS2-graphs. Here TS2(H(G1, G3)) ̸= +TS2(G1) ∪ TS2(G3). +As the last example in this section, consider the graphs G4 and G5 as follows. +G4 is the cycle +with edges {ae, eb, bc, cd, ad} and G5 is the graph with edges {ae, eb, bc, ag, eg, bg}. G4 and G5 shares a +common induced subgraph H with V (H) = {a, e, b, c} and E(H) = {ae, eb, bc}. We have E(H(G4, G5)) = +{ae, eb, bc, cd, ad, ag, eg, bg, dg}. In this case, TS2(H(G4, G5)) is the (non-acyclic) graph with edges {ab− +ac, ac − ce, ce − de, de − bd, ab − bd, ac − cg, ce − cg} which is the union of TS2(G4) and TS2(G5). (See +Figure 7.) +In the next proposition, we show how to compute the TSk-graph of an H-join, generalizing the +examples given above. +Proposition 12. Let k ≥ 2 and let G1 and G2 be two H-consistent graphs. TSk(H(G1, G2)) is the +union of TSk(G1), TSk(G2) and for every pair of k-element independent sets S1 in G1 and S2 in G2 +satisfying +|S1 ∩ V (H)| = |S2 ∩ V (H)| = |S1 ∩ S2| = k − 1, +(1) +the edge between S1 and S2. +8 + +a +e +b +c +d +G4 +a +e +b +c +g +G5 +d +g +a +e +b +c +H(G4, G5) +ab +ac +ce +de +bd +TS2(G4) +ab +ac +ce +cg +TS2(G5) +ab +ac +ce +de +bd +cg +TS2(H(G4, G5)) +Figure 7: The graphs G4, G5, H(G4, G5) and their corresponding (non-acyclic) TS2-graphs. +Here +TS2(G4, G5) = TS2(G4) ∪ TS2(G5). +Proof. As remarked, the k-element independent sets of H(G1, G2) are the same as the union of those +of G1 and G2. Therefore, V (TSk(H(G1, G2))) = V (TSk(G1)) ∪ V (TSk(G2)). Next, consider an edge in +E(TSk(G1)) (respectively, E(TSk(G2))). It is a token-slide between two independent sets S1 and S2 in G1 +(respectively, G2). This remains as a token-slide in H(G1, G2). Therefore, E(TSk(G1)) ∪ E(TSk(G2)) ⊆ +E(TSk(H(G1, G2))). Now, consider an edge in E(TSk(H(G1, G2))) between two independent sets S1 +and S2. If both of these are independent sets are in G1 (respectively, G2) then the edge is also present +in E(TSk(G1)) (respectively, E(TSk(G2))). Otherwise, we may assume the edge in E(TSk(H(G1, G2))) +has as endpoints an independent set S1 in G1 (but not G2) and an independent set S2 in G2 (but not +G1). We have S1 ∩ S2 ⊂ V (H) and since S1 and S2 are adjacent |S1 ∩ S2| = k − 1. It follows that +|S1 ∩ V (H)| = |S2 ∩ V (H)| = k − 1 and so condition (1) is satisfied. We have shown that each edge in +E(TSk(H(G1, G2))) is either in TSk(G1), TSk(G2) or satisfies condition (1), proving the proposition. +For two H-consistent graphs G1 and G2, we say that H(G1, G2) is k-crossing free if there are no +k-element independent sets satisfying condition (1) of Proposition 12. For example, one can verify that +the graphs H(G1, G2) in Figure 5 and H(G4, G5) in Figure 7 are both k-crossing free, while the graph +H(G1, G3) in Figure 6 is not. The following result will be used for constructing TSk-trees/forests. +Corollary 13. Let k ≥ 2 and let G1 and G2 be two H-consistent graphs. H(G1, G2) is k-crossing free +if and only if +TSk(H(G1, G2)) = TSk(G1) ∪ TSk(G2). +(2) +Proof. If H(G1, G2) is k-crossing free then (2) follows from Proposition 12. Otherwise their exist k- +element independent sets S1 is in G1 and S2 is in G2 satisfying (1). This implies that TSk(H(G1, G2)) +contains an additional edge between S1 and S2. +Therefore, if H(G1, G2) is k-crossing free and both TSk(G1) and TSk(G2) are acyclic, then so is +TSk(H(G1, G2)). The reason for allowing H to be empty in defining an H-join is that the corollary +then applies to vertex disjoint graphs G1 and G2, since in this case H(G1, G2) is trivially k-crossing free. +Therefore, we can create reconfiguration graphs that are forests from reconfiguration graphs that are +trees (or forests). +The following result follows from the relationship between H-join and H-decomposition discussed +above. +Corollary 14. If G can be H-decomposed into G1 and G2 and H(G1, G2) is k-crossing free then TSk(G) +can be decomposed into TSk(G1) ∪ TSk(G2). +9 + +4 +Results on (Q2) +We currently have no general necessary and sufficient conditions for when a forest F is a TSk-graph, +but we present some partial results in this section. Firstly, we recall that in [1] it is shown that Pn is +a TSk-graph for all n ≥ 1 and k ≥ 2 and K1,n is a TSk-graph if and only if n ≤ k. In this section, we +show how to construct acyclic TSk-graphs from graphs that have a single edge using the join operation +that was introduced in Section 3. We show that it gives an alternate method of constructing TSk-graphs +which are paths and stars. Moreover, this operation can also be applied to construct more general TSk +trees/forests, especially members of the classes k-ary trees and Dr,n,s. +4.1 +Paths and stars revisited +Using just the base graphs and the H-join operation defined in Section 3, we can obtain large families +of TSk trees/forests. We begin with paths. For any k ≥ 2, let Jk = {b1, . . . , bk} be an independent set +of size k and define the base graph Bi +k = Bk(Jk−2 ∪ {ai, ai+1, ai+2}, aiai+2) and let G2 = Bi +k. +Proposition 15. For i ≥ 2, Gi and Bi +k are H-consistent with H being the independent set Jk−2 ∪ +{ai, ai+1}. Define Gi+1 := H(Gi, Bi +k). Then +TSk(Gi+1) = TSk(Gi) ∪ TSk(Bi +k) ≃ Pi+1. +Proof. We will prove by induction, for i ≥ 2, that TSk(Gi) is the path Pi with vertices labelled Jk−2 ∪ +{aj, aj+1}, j = 1, . . . , i. For the base case i = 2, we observe that indeed TSk(Bi +k) is a P2 with vertices +labelled Jk−2 ∪ {a1, a2} and Jk−2 ∪ {a2, a3}. +For the inductive step we observe that, for i ≥ 2, Gi and Bi +k are H-consistent with H the independent +set Jk−2 ∪ {ai, ai+1}. To verify that H(Gi, Bi +k) is k-crossing free, note that the only independent set we +need to consider in Bi +k is Jk−2∪{ai+1, ai+2}. In the path Pi which is TSk(Gi), the candidate independent +sets are Jk−2 ∪ {aj, aj+1}, j = 1, . . . , i. Their intersection with Bi +k is Jk−2 which has cardinality k − 2. +Therefore condition (1) of Proposition 12 is not satisfied, which indeed confirms that H(Gi, Bi +k) is k- +crossing free. We define Gi+1 := H(Gi, Bi +k). By Corollary 13, TSk(Gi+1) is the union of the above +labelled Pi with a P2 with endpoints Jk−2 ∪ {ai, ai+1} and Jk−2 ∪ {ai+1, ai+2}. This is the required +Pi+1. +An easy inductive argument based on the H-join in the proposition shows that, for i ≥ 2, Gi is +isomorphic to P n+1 ∪ Jk−2, a result proved in Corollary 5(a) of [1]. (Observe that the vertex ai+1 in Gi +is adjacent to every aj for 1 ≤ j ≤ i − 1.) +Next we consider graphs Gi such that TSk(Gi) is the star K1,i. +For k ≥ 2 and 1 ≤ i ≤ k, let +Ik = {a1, . . . , ak} be an independent set of size k, define the base graph Ci +k = Bk(Ik + bi, aibi) and let +G1 = C1 +k. +Proposition 16. For k ≥ 2 and 1 ≤ i ≤ k, Gi and Ci+1 +k +are H-consistent with H being the independent +set Ik. Define Gi+1 := H(Gi, Ci+1 +k +). Then +TSk(Gi+1) = TSk(Gi) ∪ TSk(Ci+1 +k +) ≃ K1,i+1. +Proof. We will prove by induction, for i ≥ 1, that TSk(Gi) is the star K1,i with centre labelled Ik and +leaves labelled Ik + bj − aj, j = 1, . . . , i. For the base case i = 1, we observe that indeed TSk(Ci +k) is a +K1,1 with centre labelled Ik and leaf labelled Ik + b1 − a1. +For the inductive step we observe that, for i ≥ 1, Gi and Ci+1 +k +are H-consistent with H the in- +dependent set Ik. To verify that H(Gi, Ci+1 +k +) is k-crossing free, note that the only independent set +we need to consider in Ci+1 +k +is Ik + bi+1 − ai+1. +In the above labelled K1,i which is TSk(Gi), the +candidate independent sets for condition (1) of Proposition 12 are Ik + bj − aj, j = 1, . . . , i. Their inter- +section with Ik + bi+1 − ai+1 has cardinality k − 2. Therefore, condition (1) is not satisfied. We define +Gi+1 := H(Gi, Ci+1 +k +). By Corollary 13, TSk(Gi+1) is the union of the above labelled K1,i and a K1,1 +with centre also labelled Ik and leaf labelled Ik + bi+1 − ai+1. This is the required K1,i+1. +4.2 +k-ary trees +In this section, we show that for each k ≥ 2, every k-ary tree is a TSk+1-graph (Proposition 19). Next, +we show that any tree T is an induced subgraph of some TS2-forest (Proposition 22). Moreover, we +state and prove the necessary and sufficient conditions for T to be an induced subgraph of some TS2-tree +10 + +(Proposition 23). Additionally, when T = K1,n, we describe a sufficient condition for T to be an induced +subgraph of some TSk-tree (Proposition 24). +We begin by defining a canonical vertex labelling. +In this subsection, for any integer n, define +In := {a1, . . . , an} and Jn := {b1, . . . , bn}. +Definition 17. Let k ≥ 2 and G be a graph for which T := TSk+1(G) is a k-ary tree. We say that G +and T are canonically labelled if +(a) the root of T is labelled Ik+1, +(b) the d ≤ k children of the root are labelled Ik+1 − ai + bi, i = 1, . . . , d, +(c) the labels bj, j = d + 1, . . . , k (if any) are not used, and +(d) all other nodes in T receive a label S such that |Ik+1 ∩ S| ≤ k − 1. +It is clear that labelling K1,d, d ≤ k according to (a) and (b) with root the centre of the star is a +canonical labelling. In this subsection, we will show that every k-ary tree has canonical labelling hence +proving it is a TSk+1-graph. First, we give a lemma that shows how to combine canonically labelled +k-ary trees to get a larger k-ary tree that is canonically labelled. +Lemma 18. For integers k ≥ 2 and 1 ≤ i ≤ d ≤ k, let Gi be a graph for which TSk+1(Gi) a canonically +labelled k-ary tree. We can construct a canonically labelled k-ary tree T isomorphic to the tree formed +by choosing a new root and adjoining it to the root of each Ti. +Proof. The proof consists of showing that we can make a series of H-joins between the leaves of a +canonically labelled K1,d and the roots of the canonically labelled trees Ti, i = 1, . . . , d, after a suitable +relabelling. Suppose the root of Ti has ni ≤ k children. We relabel the vertices in the underlying graphs +as follows: +(i) relabel vertices of the Gi not in Ik+1 ∪ Jk to be distinct, ie, for 1 ≤ i ≤ j ≤ d, we have V (Gi) ∩ +V (Gj) ⊆ Ik+1 ∪ Jk, +(ii) for i = 1, . . . d, j = 1, . . . , ni set bj ← bi +j, where the bi +j were previously unused, and +(iii) for i = 1, . . . d, set ai ← ak+1 and ak+1 ← bi. +By an abuse of notation, for simplicity we let for i = 1, . . . , d, Gi and Ti refer to the relabelled graphs +and trees. Item (i) ensures that the only labels shared between two trees are in Ik+1 ∪ Jk, (ii) ensures +that all labels from Jk in the Ti are given unique labels to avoid clashes, and (iii) gives the root of Ti a +correct label to be a child of a new root labelled Ik. We note that after relabelling bi only appears in +Ti, ai does not appear in Ti and the only labels shared between the Ti are in Ik. Furthermore all tree +vertices have unique labels. +Next take a canonically labelled graph G0 such that TSk+1(G0) ≃ K1,d, with the centre of the star +labelled Ik+1. For i = 1, . . . , d, we claim that the H-join Gi := H(Gi−1, Gi) is well-defined, k-crossing +free, and TSk+1(Gi) is canonically labelled. To see this, note at that iteration i, V (Gi−1) ∩ V (Gi) = +Ik+1 −ai +bi which is the label of the root of Ti and a leaf of TSk+1(Gi−1). Definition 17(d) implies that +condition (1) of Proposition 12 is not satisfied. Therefore by Corollary 13, TSk+1(Gi) is obtained from +TSk+1(Gi−1) by appending Ti to the corresponding leaf in TSk+1(Gi−1). The conditions of Definition +17 are satisfied so TSk+1(Gi) is canonically labelled. At the end of iteration d, T := TSk+1(Gd) is the +required tree. +The construction described in the proof is illustrated in Figure 8. We may now prove the main result +of this section. +Proposition 19. For every k-ary tree T, there is a canonically labelled graph G such that T ≃ TSk+1(G). +Proof. Suppose that the root r of T has d ≤ k children. We prove the proposition by induction on the +height t of T, which is the length of the longest path to a leaf from the root. If t = 1 then T ≃ K1,d +and so has a canonically representation as described following Definition 17. Otherwise, by deleting r +we obtain d subtrees Ti, i = 1, . . . , d, which are also k-ary trees, with height less than t. Therefore, by +induction each Ti can be represented by a canonically labelled graph Gi. It follows from Proposition 18 +that we can perform d H-joins to obtain a canonically labelled graph G for which T ≃ TSk+1(G). +11 + +a1a2a3 +b1a2a3 +a1b2a3 +a1a2a3 +b1a2a3 +a1b2a3 +T1 +T2 +Relabel +a3a2b1 +b1 +1a2a3 +a3b1 +2b1 +a1a3b2 +b2 +1a2a3 +a1b2 +2b2 +k = 2 +a1a2a3 +K1,2 +Figure 8: Construction of D2,3,2 from two K1,2. +As noted in Section 4 of [1], K1,k+1 is an example of a k-ary tree that is not an TSk-graph so the +proposition is tight. Nevertheless, if we add a sufficient number of isolated vertices to K1,t, for t > k, it +becomes a TS2-graph—a result we will now prove in general. We will need a special labelling of a tree +that will be defined next. +Definition 20. A tree T is well-labelled if +(a) the root r of T is labelled ab, +(b) the d children of r have roots labelled ri = bci, i = 1, . . . , d − 1 and rd = acd, +(c) the only labels containing a and b are ab, acd, bci, 1 ≤ i ≤ d − 1, and +(d) for i = 1, . . . , d label ci only occurs in the subtree with root ri. +We note that there is nothing special about the ordering of the subtrees of r. The subtree rooted +at ri can play the role of rd by relabelling those two subtrees with the exchanges a ↔ b and ci ↔ cd, +which leaves T well-labelled. As an example, for d ≥ 1 we can well-label K1,d simply by using (a) and +(b). Consider the graph G defined by V (G) = {a, b} ∪ {ci : 1 ≤ i ≤ d} and E(G) = {aci, cicd : 1 ≤ +i ≤ d − 1} ∪ {bcd}. Furthermore let J = {cicj : 1 ≤ i < j ≤ d − 1}. Then it is not hard to verify that +TS2(G) ≃ K1,d + (d − 1)(d − 2)K1, where the K1,d is well-labelled and the K1 are labelled by the set J. +This motivates the following definition. +Definition 21. A tree T is well-labelled by a labelled graph G if there is an integer n such that TS2(G) ≃ +T + nK1 and T is well-labelled. +We now show the following general result. +Proposition 22. For every tree T there is a graph G and integer n such that T is well-labelled by G +and TS2(G) ≃ T + nK1. +Proof. The proof is by induction on N, the number of nodes in a given tree T. As noted above, the +proposition is true for all stars K1,t and these act as base cases. For the inductive step, assume the +proposition is true for all trees on N nodes and consider a tree T with N + 1 nodes. If T is a star +we are done. Otherwise, let r be the root of T and assume r has degree d with its children ri being +roots of subtrees Ti, 1, . . . , d. We may also assume that Td is a subtree of T with height at least one. +We now construct two trees from T. The first, T 1 consists of T with subtree Td deleted and a pendant +vertex added to its root r. +The second, T 2 consists of Td with a pendant vertex added to its root +rd. By induction, there are integers n1, n2 and graphs G1, G2 which well-label T 1 and T 2 such that +TS2(G1) ≃ T 1 + n1K1 and TS2(G2) ≃ T 2 + n2K1. Apart from the vertex labels used in Definition 20, +we may assume the vertex labels in G1 and G2 are different. +We will show that G1 and a relabelled G2 can be H-joined and that this will identify the pendant +edges added to T 1 and T 2 to give us back T. In T 1 we note that root r is labelled ab, and by relabelling +subtree roots if necessary, that the added pendent vertex can be labelled acd. In T 2 the root rd is also +12 + +labelled ab and we can again assume the added pendant vertex is labelled acd. In T 2 we interchange the +labels b ↔ cd and set ci ← c′ +i, i = 1, . . . , d−1, for labels c′ +i that are unused in either T 1 or T 2. Let G3 and +T 3 denote the relabelled G2 and T 2. Setting H = {a, b, cd}, we have V (G1) ∩ V (G3) = H. H induces +the same subgraph, containing the single edge bcd, in both G1 and G3. G1 and G3 are H-consistent and +since k = 2 and their vertex sets are otherwise disjoint, condition (1) of Proposition 12 is not satisfied. +Let G4 = H(G1, G3). Applying Corollary 13 we have that +T 4 := TS2(G4) ≃ TS2(G1) ∪ TS2(G3) ≃ {T 1 + n1K1} ∪ {T 3 + n2K1} ≃ T + (n1 + n2)K1. +is well-labelled by G4. This proves the proposition. +The proof of the proposition is illustrated in Figure 9. The proposition tells us that for every tree +r +r1 +r2 +r3 +r4 +T +r +r1 +r2 +r3 +r4 +new pendant edges +bc1 +bc2 +bc3 +ab +ac4 +c4c′ +1 +c4c′ +2 +c4c′ +3 +ab +a +c4 +c′ +1 +c′ +2 +c′ +3 +a +b +c1 +c2 +c3 +c4 +(relabelled) G2 +G1 +(relabelled) T 2 +T 1 +ac4 +b +Figure 9: Illustrating Proposition 22. +T there is a graph G for which TS2(G) is forest containing T as an induced subgraph. Therefore, there +can be no forbidden induced subgraph characterization of which forests are TS2-graphs. However, this +does not imply that there can be no forbidden induced subgraph characterization of which trees are +TS2-graphs. Indeed, in the next propositions, we present some of such characterizations. +Proposition 23. Let T be a tree. Then there exists a TS2-tree containing T if and only if T is a 3-ary +tree. +Proof. (⇐) In the proof of Proposition 22, we see that isolated vertices are only added when the base +case of a star appears as a subproblem. Therefore, it suffices to consider only the case T = K1,t, 1 ≤ +t ≤ 4. As we have noted, neither K1,3 nor K1,4 are TS2-graphs. It is not hard to see that there is +a G1 such that TS2(G1) ≃ K1,3 + K1. However, by adding an extra vertex to G1, we can construct +a graph G2 such that TS2(G2) ≃ D1,3,2. Furthermore, we can construct a graph G3 by applying +H-join to two copies of G2 with slightly different vertex-labellings such that TS2(G3) is isomorphic +to a P7 with two pendant vertices attached to the midpoint of the path. (See Figure 10.) Thus, +if follows that when T = K1,t, 1 ≤ t ≤ 4, we can embed it as an induced subgraph of a tree +T ′ = TS2(G), for some graph G (see Figure 10). Our proof of the if direction is complete. +13 + +a +c +d +e +f +b +ab +ae +bd +ac +b +g +c +d +h +a +ab +bd +ac +bg +ce +ef +dh +dg +ab +ae +bd +ac +ce +ef +dh +dg +G2 +TS2(G2) +bg +TS2(G3) +Figure 10: Taking H-join of two copies of G2, where H is the path adcb, results a graph G3 such that +TS2(G3) is isomorphic to a P7 with two pendant vertices attached to the midpoint of the path. +(⇒) We show that if T is a k-ary tree but not a 3-ary tree for k ≥ 4 then there does not exist any +TS2-tree T ′ containing T (as an induced subgraph). (By definition, any k-ary tree is also a ℓ-ary +tree for ℓ ≥ k.) Let x be a vertex of T whose degree is at least five. (Since T is a k-ary tree but +not a 3-ary tree, such a vertex x exists.) +Suppose to the contrary that T ′ exists, i.e., there exists a graph G′ such that T ′ ≃ TS2(G′) contains +T. Without loss of generality, assume that x is labelled by ab, where {a, b} is a size-2 stable set +of G′. By the pigeonhole principle, we may further assume that three neighbors x1, x2, and x3 +of x are labelled ac, ad, and ae, respectively. Since T ′ is a tree, it follows that cd, ce, and de are +respectively the labels of y1, y2, and y3 where yi is not adjacent to any of � +j{xj}+x+� +j̸=i{yj} for +1 ≤ i, j ≤ 3. It follows that T ′ contains the labelled graph F ≃ K1,3 + 3K1 and therefore G′ must +ab +ac +ad +ae +cd +ce +de +F +a +b +c +d +e +G +Figure 11: The graphs F and G in the proof of Proposition 23. +contain the labelled graph G ≃ K1,3 + K1, both described in Figure 11, as an induced subgraph. +Since T ′ ≃ TS2(G′) is a tree and G′ contains G, it follows that G′ has exactly one non-trivial +component C (having more than two vertices) and C contains G, otherwise G′ must contain an +14 + +induced 2K2 and by Proposition 2 its TS2-graph is not a tree, a contradiction. +– Case 1: a ∈ V (C). By definition, the distance from a to any of b, c, d, e in G′ must be at +least two. If there is a path of length at least two between a and one of c, d, e not passing +through b, the graph G′ contains a 2K2, a contradiction. Thus, any path between a and one +of c, d, e must go through b. Moreover, if there is a path of length at least three between a +and b not passing through any of c, d, e, again the graph G′ contains a 2K2, a contradiction. +Since a ∈ V (C), it follows that a and b must have a common neighbor in G′, say f. Observe +that for each y ∈ V (C) − {a, b, c, d, e, f}, y must be adjacent to b in G′, otherwise G′ either +contains 2K2 or D2,2,2 and again by Proposition 2 its TS2-graph is not a tree, a contradiction. +However, this implies that TS2(C) must be a forest and since G′ has exactly one non-trivial +component C, we have TS2(G′) is also a forest, a contradiction. +– Case 2: +a /∈ V (C). +In this case, there are two types of size-2 stable sets of G′: those +containing a and those do not. Since G′ contains G, each type has at least one member. +Moreover, since a is isolated (the only non-trivial component is C and a is not in it), no +member from one type is adjacent to a member from another type in TS2(G′), which means +TS2(G′) is indeed disconnected, a contradiction. +In the above cases, we proved that some contradiction must happen. Our proof is complete. +Indeed, for K1,n, in general we have +Proposition 24. There exists a TSk-ary tree T containing K1,n if n ≤ 2k. +Proof. From either [1] or Proposition 16, the proposition holds for n ≤ k. (Indeed, in this case, T = K1,n.) +Thus, it suffices to consider k + 1 ≤ n ≤ 2k. For each i ∈ {1, . . . , n − k}, let Ai = {1, . . . , k} − i. +Let Ik = {a1, . . . , ak} and Bn = {b1, . . . , bn}. +We construct a graph G0 such that TSk(G0) ≃ +K1,n + (n − k)(k − 1)K1. Let Ik = {a1, . . . , ak} and Bn = {b1, . . . , bn}. Let V (G) = Ik + Bn. Vertices in +Bn form a graph Kn − M where M is the matching that contains bibk+i for 1 ≤ i ≤ n − k. Additionally, +for each i ∈ {1, . . . , k}, we add an edge in G0 between ai and both bi and bk+i. Observe that V (TSk(G0)) +consists of Ik, the sets Ik −ai +bi (1 ≤ i ≤ k), Ik −ai +bk+i (1 ≤ i ≤ n−k), and (Ik −ai +bi)−aj +bk+i +(1 ≤ i ≤ n − k and j ∈ Ai). Moreover, one can verify that the independent sets (Ik − ai + bi) − aj + bk+i +are isolated in TSk(G0) and the remaining independent sets form a K1,n in which Ik is adjacent to every +other set. In short, G0 is indeed our desired graph. +For each i ∈ {1, . . . , n − k}, we construct a graph Gi whose TSk-graph is a star K1,k−1 as follows. +Let V (Gi) = (Ik − ai + bi) + � +j∈{1,...,k}−i{ci +j}. Vertices in � +j∈Ai{ci +j} form a clique in Gi of size k − 1. +We also add an edge in Gi between aj and ci +j for each j ∈ Ai. From either [1] or Proposition 16, one can +verify that TSk(Gi) ≃ K1,k−1 as desired. For each i ∈ {1, . . . , n − k} and j ∈ Ai, we construct a graph +Gi +j whose TSk-graph is a K2 as follows. Let V (Gi +j) = (Ik − ai + bi) − aj + bk+i + ci +j. The only edge in +Gi +j is the one joining ci +j and bk+i. From either [1] or Proposition 15, one can verify that TSk(Gi +j) ≃ K2 +as desired. +Now, we construct a graph G whose TSk-graph is a tree containing K1,n as follows. For convenience, +we assume that for each i ∈ {1, . . . , n−k} the set Ai = {1, . . . , k}−i can be enumerated as {j1, . . . , jk−1}. +We define Ki +j0 = Gi and Ki +jp = Hjp(Ki +jp−1, Gi +jp) for jp ∈ Ai where Hjp is the stable set (Ik − ai + bi) − +ajp +ci +jp for p ∈ {1, . . . , k −1}. Observe that the graphs Ki +jp−1 and Gi +jp are Hjp-consistent, which implies +that Ki +jp are well-defined. Moreover, one can also directly verify that the sets (Ik − ai + bi) − aj + ci +j +and (Ik − ai′ + bi′) − aj′ + ci′ +j′ always differ in at least two members, which means the condition (1) of +Proposition 12 is not satisfied. In short, for each i ∈ {1, . . . , n − k}, we obtain the graph Ki +jk−1 whose +TSk-graph is isomorphic to the one obtained from K1,k−1 by replacing each edge with a P3. Next, we +define K0 = G0 and Ki = Hi(Ki−1, Gi) where i ∈ {1, . . . , n − k} and Hi is the subgraph induced by +(Ik −ai +bi)+bk+i. Observe that the graphs Ki are well-defined because Ki−1 and Gi are Hi-consistent. +Moreover, we have Ik and each (Ik − ai + bi) − aj + ci +j for 1 ≤ i ≤ n − k and j ∈ Ai always differ in +at least two members. It follows that the condition (1) of Proposition 12 is not satisfied. In short, we +finally obtain the graph G = Kn−k whose TSk-graph is indeed a tree containing K1,n as desired. +Unfortunately, we have not been able to show whether the reverse statement of Proposition 24 also +holds. We conclude this section with the following open problems: +15 + +b5a2a3a4 +b1a2a3a4 +a1a2a3b8 +b1c1 +2a3a4 +b1a2c1 +3a4 +a1a2a3b4 +a1c4 +2a3b4 +a1a2a3a4 +TS4(G) +b2 +b1 +b3 +b4 +b5 +b6 +b7 +b8 +c1 +2 +c1 +3 +c1 +4 +c4 +1 +c4 +2 +c4 +3 +a1 +a2 +a3 +a4 +G +b1a2a3c1 +4 +c4 +1a2a3b4 +a1a2c4 +3b4 +b1b5a3a4 +b1a2b5a4 +b1a2a3b5 +b8a2a3b4 +a1b8a2b4 +a1a2b8b4 +Figure 12: Construction of a graph G such that TS4(G) is a tree containing K1,8. Vertices of G in the +yellow box form a clique having all dashed edges removed. The red induced subgraph of G forms a graph +G0 whose TS4(G0) ≃ K1,8 + 12K1. +Problem 25. For every k ≥ 3 and tree T, is there a graph G such that TSk(G) is a forest containing T +as an induced subgraph? +Problem 26. For every k ≥ 3 and (k + 1)-ary tree T, is there a graph G such that TSk(G) is a tree +containing T as an induced subgraph? +Problem 27. Does there exist a TSk-tree T containing K1,n for n > 2k? +4.3 +Dr,n,s +We now consider graphs in the Dr,n,s family for whose TSk-graphs are trees and show how they can +be constructed by the H-join operation. We remark that when n = 1, Dr,n,s is nothing but a star +K1,r+s and this case was considered in [1] and revisited in Proposition 16. Furthermore, it follows from +Proposition 19 that for n, k ≥ 2 and 1 ≤ r ≤ s ≤ k − 1, Dr,n,s is a k-ary tree and so by Proposition 19 +it is a TSk-graph. The reverse statement does not hold in general: there exists a TSk-graph Dr,n,s even +when s ≥ k. For example, one of such graphs, as already proved in [1], is D1,3,2 (r = 1, s = k = 2, and +n = 3). (See also Figure 1.) Indeed, as we will see in Proposition 29, it is the unique TS2-graph among +all trees D1,n,2 for n ≥ 1. Additionally, for the sake of completeness, we will also show in Proposition 30 +that the reverse statement indeed holds when n = 2. +We are now characterizing which D1,n,2-graphs are TS2-graphs and show that this property is non- +hereditary for this simple class of trees. We then consider the Dr,2,s-graphs characterizing those that are +TSk-graphs. +Assume for some G, TS2(G) is a forest containing a K1,3. There are four stable sets in G corresponding +to the vertices of the K1,3. There are two ways of labelling the K1,3 but in each case there are five vertices, +say a, . . . , e, of G involved. Up to permutations of the labels, the corresponding stable sets in G are +either {ab, ac, bd, ae} or {ab, ac, ad, ae}. Using these definitions we have the following lemma. +Lemma 28. Let H be the subgraph of G induced by a, b, . . . , e. The edges of H are +(a) ad, de, eb, bc, cd, if the K1,3 is labelled {ab, ac, bd, ae}, or +(b) bc, bd, be if the K1,3 is labelled {ab, ac, ad, ae}. +Proof. +(a) This labelling of K1,3 immediately gives edges ad, bc, be and non-edges ab, ac, ae, bd. That +leaves three edges of H to be decided: +(i) ce must be a non-edge else there is an edge ae, ac in the K1,3. +(ii) cd is an edge else there is a cycle ab, bd, cd, ad in TS2(G), so it is not a tree. +16 + +ab +ac +bd +ae +ab +ac +ad +ae +a +b +c +d +e +b +c +d +e +(a) +(b) +K1,3 +H +a +Figure 13: If TS2(G) is a forest containing a K1,3 then G must contain one of the induced subgraphs H. +(iii) de is an edge else there is a cycle de, bd, ab, ae in TS2(G). +Note that ce must also be a vertex in TS2(G). +(b) This labelling of K1,3 immediately gives edges bc, bd, be and non-edges ab, ac, ad, ae. There are no +other edges in H as c, d, e form a stable set. This implies that TS2(G) must also contain vertices +cd, ce and de. +Using the lemma we show that precisely one of the D1,n,2-graphs is a TS2-graph, incidentally proving +the non-hereditary property mentioned above for this class of graphs. +Proposition 29. D1,n,2 is a TS2-graph if and only if n = 3. +Proof. We first consider 1 ≤ n ≤ 3 and show that D1,3,2 is a TS2-graph while D1,1,2 = K1,3 and D1,2,2 +are not. (We note that the results for the first two graphs have also been proved in [1].) According to +Lemma 28, if D1,n,2 is a TS2-graph of some graph G, the unique star K1,3 in D1,n,2 can be labelled +in one of two ways. However, we may immediately eliminate the possibility of the labelling in Lemma +28(b). This is because, as pointed out in the proof, there must be additional vertices in D1,n,2 = TS2(G) +labelled cd, ce and de which are non-adjacent since c, d, e form a stable set in G. +This implies that +n ≥ 6. So we may assume that if D1,n,2 is a TS2-graph, the K1,3 must be labelled as in Lemma 28(a) +with corresponding induced subgraph H of D1,n,2. From the proof of Lemma 28(a) there must be an +additional vertex ce in D1,n,2 however this cannot be adjacent to any of the other four vertices. This +implies that n ≥ 3 and so neither D1,1,2 nor D1,2,2 can be TS2-graphs. However we may extend H to +G by adding a vertex f adjacent to all vertices except e, as illustrated in Figure 1. This introduces the +new stable set ef which is adjacent to both ae and ce. Therefore D1,3,2 is isomorphic to TS2(G). We +note that G is the unique graph (up to label permutations) for which this is true, due to the uniqueness +of the labelling of K1,3. +It remains to consider n ≥ 4 and show that D1,n,2 is not a TS2-graph. Suppose to the contrary that +there exists a graph G such that D1,n,2 = TS2(G). Again, D1,n,2 must contain a copy of K1,3 with +exactly two ways of labelling (up to label permutations) by size-2 independent sets of G. +• Case 1: K1,3 is labelled {ab, ac, bd, ae}. Since ac and ae are not adjacent, ce must be a vertex +of D1,n,2 = TS2(G). We consider the following cases: +– Case 1.1: the distance between ce and any vertex of {ac, bd, ae} is at least three. +Since the roles of c and e are equal, we assume without loss of generality that ce is adjacent +to some vertex cf. Observe that a and f are not adjacent in G, otherwise ac and cf are +adjacent, which means the distance between ac and ce is two, a contradiction. Since ce and +cf are adjacent, so are ae and af. Moreover, bf must be a vertex, otherwise there is an edge +between ab and af in D1,n,2 = TS2(G) which creates a C3 having {ab, ae, af} as its vertex-set, +a contradiction. Since ab and ac are adjacent, so are cf and bf. Now, df must be a vertex, +17 + +otherwise bd and bf are adjacent which contradicts D1,n,2 = TS2(G). Since ab and bd are +adjacent, so are af and df. From the proof of Lemma 28(a)(ii) c and d are adjacent in G, so +df and cf are adjacent, which again contradicts D1,n,2 = TS2(G). +– Case 1.2: the distance between ce and one of {ac, bd, ae} is exactly two. Observe +that bd and ce has no common neighbor, otherwise that neighbor must be labelled as one of +{bc, be, dc, de}: the first two can be ignored because ab and ac (resp., ab and ae) are adjacent, +the last two can be ignored because ab and bd are adjacent. Again, since the roles of c and +e are equal, we assume without loss of generality that ae and ce has a common neighbor ef. +Since n ≥ 4, ce must have another neighbor which is different from ef, which can be either +cg or eg for some vertex g of G. +∗ If it is cg then ag must be a vertex, otherwise cg and ac must be adjacent, which creates a +C6 whose vertex-set is {ac, ab, ae, ef, ce, cg}, a contradiction. Since ce and cg are adjacent, +so are ae and ag, which contradicts D1,n,2 = TS2(G). +∗ If it is eg then ag must be a vertex, otherwise eg and ae must be adjacent, which creates +a C4 whose vertex-set is {ae, ef, ce, eg}, a contradiction. Since ce and eg are adjacent, so +are ag and ac, which contradicts D1,n,2 = TS2(G). +• Case 2: K1,3 is labelled {ab, ac, ad, ae}. As before, cd, ce, and de must be vertices in D1,n,2. +Without loss of generality, since the roles of c, d, e are equal, we may assume that only ae is +adjacent to another vertex of D1,n,2. As shown in the proof of Lemma 28(b), D1,n,2 must also +contain vertices cd, ce, de. Let P be the path between ae and cd. Since the roles of c and d are +equal, we can assume without loss of generality that cd is adjacent to a vertex cf in P. Observe +that if af is not a vertex ac and cf are adjacent contradicting the choice of ae. So af is a vertex +and since cd and cf are adjacent so are ad and af, which contradicts D1,n,2 = TS2(G). +We remark that if we add a vertex g to G in Figure 1 joining it to all vertices except d the corresponding +TS2-graph is obtained by adding the edge between bd and dg to TS2(G). Note that this tree is not in +the class Dr,n,s. +In the next proposition we consider two arbitrary stars whose centers are connected by an edge. +Proposition 30. Dr,2,s (1 ≤ r ≤ s) is a TSk-graph if and only if s ≤ k − 1. +Proof. (⇐) It follows directly from Proposition 19. +(⇒) Suppose that Dr,2,s (r ≤ s) is obtained from P2 = p1p2 by attaching r leaves u1, . . . , ur at p1 +and s leaves v1, . . . , vs at p2 for some s ≥ k. We show that this graph is not a TSk-graph for +any fixed k ≥ 2. Suppose to the contrary that there exists a graph G such that Dr,2,s ≃ TSk(G), +i.e., there exists a bijective mapping f : V (Dr,2,s) → V (TSk(G)) such that uv ∈ E(Dr,2,s) if and +only if f(u)f(v) ∈ E(TSk(G)). Without loss of generality, let f(p2) = I = {a1, . . . , ak}, where I +is a size-k independent set of G. Since p2 has s + 1 neighbors, from the pigeonhole principle, it +follows that there must be some i ∈ {1, . . . , k} such that f(u) = I − ai + x and f(v) = I − ai + y, +where u, v ∈ N(p2). Observe that J = (I − ai − aj) + x + y /∈ {f(p2), f(u), f(v)} must be a size-k +independent set of G, where j ∈ {1, . . . , k} − i and therefore there exists z ∈ V (Dr,2,s) − {p2, u, v} +such that f(z) = J. We consider the following cases: +– Neither u nor v is p1. In this case, we must have z /∈ N(p2), otherwise it must be adjacent to +p2, but then f(z) = J and f(p2) = I must be adjacent in TSk(G), a contradiction. It follows +that z ∈ N(p1) − p2 and thus f(p1) must be in {I − ai + x, I − ai + y, I − aj + x, I − aj + y}. +Since neither u nor v is p1, the first two can be ignored. Now, if f(p1) = I − aj + x, the +vertices x and aj must be adjacent in G, which contradicts the fact that f(u) ∈ TSk(G). A +similar contradiction can be derived for the case f(p1) = I − aj + y. Thus, f(p1) cannot be +defined. +– u is p1. Again, z /∈ N(p2). Thus, z ∈ N(p1) − p2, which implies that y and aj must be +adjacent in G. This contradicts f(v) ∈ TSk(G). Thus, f(z) cannot be defined. +In both cases, we showed that some contradiction must occur. Our proof is complete. +18 + +5 +Conclusions +In this paper, we considered two token sliding problems for trees and forests. The two questions studied +seem remarkably complicated, even for this simple class of graphs. For the first question, finding necessary +and sufficient conditions on G for TSk(G) to be a forest, we could only get a complete solution for +k = 2, 3. For the second question, finding necessary and sufficient conditions for a tree or forest to be +a token sliding graph, we could get more general results. Nevertheless, as noted in Section 4 several +interesting important questions remain. We expect the join and decomposition operations introduced +there will be of use for similar questions for more general graphs. +Acknowledgments +Avis’ research is partially supported by the Japan Society for the Promotion of Science (JSPS) KAK- +ENHI Grants JP18H05291, JP20H00579, and JP20H05965 (AFSA) and Hoang’s research by JP20H05964 +(AFSA). +References +[1] David Avis and Duc A. Hoang. On reconfiguration graphs of independent sets under token sliding. +arXiv preprint, 2022. arXiv:2203.16861. +[2] Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the +parameterized complexity of the independent set and (connected) dominating set reconfiguration +problems. arXiv preprint, 2022. arXiv:2204.10526. +[3] Reinhard Diestel. +Graph Theory, volume 173 of Graduate Texts in Mathematics. +Springer, 5th +edition, 2017. +[4] Robert A. Hearn and Erik D. Demaine. PSPACE-completeness of sliding-block puzzles and other +problems through the nondeterministic constraint logic model of computation. Theoretical Computer +Science, 343(1-2):72–96, 2005. +[5] Ruy Fabila Monroy, David Flores-Pe˜naloza, Clemens Huemer, Ferran Hurtado, Jorge Urrutia, and +David R. Wood. Token graphs. Graphs and Combinatorics, 28(3):365–380, 2012. +[6] C.M. Mynhardt and S. Nasserasr. +Reconfiguration of colourings and dominating sets in graphs. +In Fan Chung, Ron Graham, Frederick Hoffman, Ronald C. Mullin, Leslie Hogben, and Douglas B. +West, editors, 50 years of Combinatorics, Graph Theory, and Computing, pages 171–191. CRC Press, +1st edition, 2019. +[7] Naomi Nishimura. Introduction to reconfiguration. Algorithms, 11(4):52, 2018. +[8] Jan van den Heuvel. The complexity of change. In Surveys in Combinatorics, volume 409 of London +Mathematical Society Lecture Note Series, pages 127–160. Cambridge University Press, 2013. +19 + diff --git a/B9AyT4oBgHgl3EQfePhg/content/tmp_files/load_file.txt b/B9AyT4oBgHgl3EQfePhg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..005396f13e5100518e64cf8e86d9e3cc3bbfe9fc --- /dev/null +++ b/B9AyT4oBgHgl3EQfePhg/content/tmp_files/load_file.txt @@ -0,0 +1,830 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf,len=829 +page_content='A Note On Acyclic Token Sliding Reconfiguration Graphs of Independent Sets David Avis1 Duc A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Hoang2 1 Graduate School of Informatics, Kyoto University, Japan School of Computer Science, McGill University, Canada avis@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='ca 2 Graduate School of Informatics, Kyoto University, Japan hoang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='duc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='8r@kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='jp Abstract We continue the study of token sliding reconfiguration graphs of independent sets initiated by the authors in an earlier paper (arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='16861).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Two of the topics in that paper were to study which graphs G are token sliding graphs and which properties of a graph are inherited by a token sliding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this paper we continue this study specializing on the case of when G and/or its token sliding graph TSk(G) is a tree or forest, where k is the size of the independent sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We consider two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The first is to find necessary and sufficient conditions on G for TSk(G) to be a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The second is to find necessary and sufficient conditions for a tree or forest to be a token sliding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the first problem we give a forbidden subgraph characterization for the cases of k = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the second problem we show that for every k-ary tree T there is a graph G for which TSk+1(G) is isomorphic to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A number of other results are given along with a join operation that aids in the construction of TSk(G)-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 1 Introduction In a reconfiguration variant of a computational problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', Satisfiability, Independent Set, Vertex-Coloring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' ), a transformation rule that describes an adjacency relation between feasi- ble solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', satisfying truth assignments, independent sets, proper vertex-colorings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') of the problem is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' One of the main goals is to decide whether there is a sequence of adjacent feasible solutions that “reconfigures” one given solution into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Another way of looking at these reconfig- uration problems is via the so-called reconfiguration graph—a graph whose nodes are feasible solutions and two nodes are adjacent if one can be obtained from the other by applying the given rule exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The mentioned question now becomes deciding whether there is a path between two given nodes in the reconfiguration graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Recently, reconfiguration problems have been intensively studied from different perspectives [2, 6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' One of the most well-studied reconfiguration variants of Independent Set is the so-called Token Sliding problem, which was first introduced by Hearn and Demaine [4] in 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We refer readers to [2, 7, 8] and the references therein for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Surprisingly, though Token Sliding has been well-investigated, the realizability and structural properties of its corresponding reconfiguration graph— the one which we will refer to as the TSk-graph (which stands for Token Sliding (Reconfiguration) graph)—have not been studied until recently [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' On the other hand, when considering either general vertex subsets, dominating sets, or proper vertex-colorings of a graph as the “input feasible solutions”, their corresponding reconfiguration graphs have been very well-characterized [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For any graph-theoretic terminology and notation not defined here, we refer readers to [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Given a graph G = (V, E) and an integer k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For two sets X, Y , we sometimes use X + Y and X − Y to indicate X ∪ Y and X \\ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We abbreviate X ∪ {u} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', X \\ {u}) by X + u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', X − u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We use NG(u), or simply just N(u) when the graph G is clear from the context, to denote the (open) neighbors of u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', set of all vertices in G that are adjacent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The closed neighbors of u, denoted by NG[u] or simply N[u], is the set NG(u) + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The degree of u, denoted by degG(u), is nothing but the size of NG(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' An independent set (or stable set) of G is a vertex subset I such that for every u, v ∈ I we have uv /∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The TSk-graph of G, denoted by TSk(G), takes all size-k independent sets of G as its 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='00317v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='CO] 1 Jan 2023 nodes and two nodes I, J are adjacent (under Token Sliding (TS)) if there exist two vertices u, v ∈ V (G) such that I − J = {u}, J − I = {v}, and uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Two graphs G and H are isomorphic, denoted by G ≃ H, if there exists a bijective mapping f : V (G) → V (H) such that uv ∈ E(G) if and only if f(u)f(v) ∈ E(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A graph G is called a TSk-graph if there exists a graph H such that G ≃ TSk(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A forest is a graph having no cycles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', it is acyclic) and a connected forest is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A TSk-tree/forest is a TSk-graph which is also a tree/forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Figure 1 illustrates a TS2-tree on six vertices (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In [1], ab ac bd ae ef ce TS2(G) a b c d e f G Figure 1: A graph G with TS2(G) = D1,3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Each node ab represents a size-2 stable set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' the authors studied various properties of the family of TSk-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For a graph G, two of the questions studied were: (Q1) What are necessary and sufficient conditions for G so that TSk(G) is a forest?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Q2) What are necessary and sufficient conditions for G to be a TSk-graph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this paper, we study these two questions for the case when G is a tree or a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The union G ∪ H of two (labelled) graphs G and H is the graph with V (G ∪ H) = V (G) ∪ V (H) and E(G ∪ H) = E(G) ∪ E(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' When vertices and edges of G and H are considered distinct regardless of their labels, we say that G ∪ H is the disjoint union of G and H, and write G + H instead of G ∪ H to distinguish from their union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We respectively denote by Kn, Pn, and Cn the complete graph, path, and cycle on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Km,n (m ≤ n) is the complete bipartite graph whose two partite sets are of sizes m and n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' K1,n is also called a star—a tree obtained by attaching n leaves to a central vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A family of graphs that we will use in the sequel generalizes stars and paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For fix integers n, r, s ≥ 1, let Dr,n,s be the tree obtained from Pn by appending r leaves at one end and s leaves at the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that D1,1,s is the star K1,s+1 and D1,n,1 is the path Pn+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Figure 1 illustrates D1,3,2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' An n-ary tree is a rooted tree in which each node has at most n children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Any tree with maximum degree at most n+1 can be rooted at a vertex with degree at most n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', a leaf) to produce a n-ary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In particular, a 2-ary tree is nothing but the well-known binary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the next section, we begin by partially answering (Q1) when G is a tree/forest and k ∈ {2, 3} and conclude the section by conjecturing for k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then, before addressing (Q2) for some trees/forests, in particular k-ary trees and Dr,n,s, we define an important graph operation which, under certain conditions, can be used for combining two TSk-graphs by taking their union to obtain a new one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The final section of the paper gives some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 2 Results on (Q1) In this section, we prove the necessary and sufficient conditions on a tree/forest G such that TSk(G) is acyclic for k ∈ {2, 3}, partially answering (Q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We begin with some definitions and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The complement G of a graph G is the graph with V (G) = V (G) and E(G) = {uv : uv /∈ E(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The size-m matching, denoted by mK2, is the graph obtained by taking the disjoint union of m copies of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that TS2(2K2) ≃ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We label vertices in a Dr,n,s (r, s ≥ 1) as follows: Vertices of Pn are labelled p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The r leaves attached to p1 are u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ur and the s leaves attached to pn are v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' D2,2,2 is shaped like an 2 G TS2(G) Figure 2: A list G of n-vertex graphs G (4 ≤ n ≤ 7) excluding Cn (n ≥ 5) such that if TS2(G′) has no cycle then G′ does not contain any member G of G as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 3 H and TS2(D2,2,2) contains a cycle C8 whose vertex-set is {u1v1, u1p2, u1v2, p1v2, u2v2, u2p2, u2v1, p1v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Indeed, respectively from Lemma 1 of [1] and Figure 2, if a n-vertex graph G is either Cn (n ≥ 5) or a graph in the list G described in Figure 2 (which includes 2K2 and D2,2,2), the graph TS2(G) contains a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Additionally, we have: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (a) For k ≥ 2, TSk(2K2 + nK1) contains a cycle C4 if n ≥ k − 2 otherwise it is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (b) For k ∈ {2, 3}, s ≥ 1, TSk(D1,n,s) contains a cycle C4 if n ≥ 2k − 1 otherwise it is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (c) For k ∈ {2, 3} and r, s ≥ 2, TSk(Dr,n,s) contains a cycle C8 if n ≥ 2k − 2 otherwise it is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (a) If n < k − 2, there is no size-k independent set in 2K2 + nK1, thus its TSk-graph is obviously acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, let I ⊆ V (nK1) be an arbitrary independent set of size k − 2, and let E(2K2) = {ab, cd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then, {I +a+c, I +a+d, I +b+c, I +b+d} induce a C4 in TSk(2K2 +nK1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (b) Observe that if n ≥ 2k − 1, D1,n,s contains an induced 2K2 + (k − 2)K1, which can be obtained by taking u1p1 and pnv1 as edges of 2K2 and the remaining k − 2 independent vertices from the path D1,n,s − {u1, p1, p2, pn−1, pn, v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , vs} on n − 4 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Since n ≥ 2k − 1, this path has an independent set of size at least ⌈(n − 4)/2⌉ ≥ ⌈(2k − 5)/2⌉ = k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Then, using a similar argument as in (a) we have TSk(D1,n,s) contains a C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' On the other hand, if n ≤ 2k − 2 for k ∈ {2, 3}, since D1,n−1,s is always an induced subgraph of D1,n,s for n ≥ 2, it follows that if TS2(D1,n−1,s) has a cycle then so is TS2(D1,n,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, it suffices to show that TSk(D1,2k−2,s) is acyclic for k ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Indeed, based on the number of tokens placed on the path u1p1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' pn (which is at most three), one can verify that each component of TSk(D1,2k−2,s) is either an isolated vertex, a path, or a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (c) Observe that if n ≥ 2k −2, Dr,n,s contains the independent sets I +u1 +v1, I +u1 +pn, I +u1 +vs, I + p1 + v1, I + p1 + vs, I + ur + v1, I + ur + pn, and I + ur + vs, where I = ∅ when n = 2 and otherwise I is an independent set of the path p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' pn−1 of size k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Note that p2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' pn−1 has an independent set of size at most ⌈(n − 2)/2⌉ ≥ k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') They indeed induce a C8 in TSk(Dr,n,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' On the other hand, if n ≤ 2k−3 for k ∈ {2, 3}, using a similar case-analysis as in (b), one can verify that each component of TSk(Dr,n,s) is either an isolated vertex, a path, or a star, and therefore it is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We are now ready to show the necessary and sufficient conditions for a tree/forest G such that TSk(G) is acyclic, where k ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS2(T) is acyclic if and only if T is {2K2, D2,2,2}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇒) Suppose to the contrary that either 2K2 or D2,2,2 is an induced subgraph of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the first case it follows from the discussion above that TS2(T) contains a C4 and in the second case that it contains a C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇐) We assume that TS2(T) contains a cycle and show that it must contain one of the two forbidden subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Firstly, suppose that T is a path Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since TS2(T) contains a cycle, it follows from Lemma 1(b) that n ≥ 5 and so T contains an induced 2K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We now assume T has a vertex of at least degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will construct a copy T ′ of T by initially choosing a vertex a of maximum degree in T and letting T ′ = N[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that TS2(T ′) is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We add edges from T to T ′ and show after each addition that either T ′ contains a forbidden subgraph, so we are done, or that TS2(T ′) remains acyclic so that T ̸= T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since TS2(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let e be a child of b with maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If r ≥ 2, we have the required forbidden induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If r = 1 then by Lemma 1(b) TS2(T ′) is acyclic, so there must be extra edges to add to T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If c has a child y then {b, c, e, y} induce a 2K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, e must have at least one child g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Adding eg to T ′ we obtain 2K2 as an induced subgraph on {a, d, e, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 4 a b c d e a d e b c y a d b c e g r ≥ 2 r = 1 Figure 3: Illustration for Proposition 2: Some trees T ′ containing N[b] whose TS2-graphs have a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Here r is the number of children of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Copies of 2K2 and D2,2,2 are marked by red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS2(T) is acyclic if and only if T is either K1,s or D1,2,s for some positive integer s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proof of Proposition 2 can be viewed as an algorithm that takes a tree T and either terminates with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let F be a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS2(F) is a acyclic if and only if F is {2K2, D2,2,2}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We prove that TS2(F) contains a cycle if and only if F contains one of the graphs in {2K2, D2,2,2} as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose that TS2(F) contains a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since the independent sets have size two, both vertices of each independent set must lie in the same connected component T of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By Proposition 2, the tree T must have either 2K2 or D2,2,2 as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Conversely if F contains 2K2 or D2,2,2 as an induced subgraph then TS2(F) contains respectively a C4 or a C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moving to the case of stable sets of size three, the conditions for trees and forests differ slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We deal with the tree case first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS3(T) is acyclic if and only if T is {2K2 + K1, D2,4,2}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The structure of the proof is the same as for Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However, there are more cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇒) Suppose to the contrary that either 2K2 + K1 or D2,4,2 is an induced subgraph of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the first case it follows that TS3(T) contains a C4 and in the second case that it contains a C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇐) We assume that TS3(T) contains a cycle and show that it must contain one of the two forbidden subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The first part of the proof is essentially the same as for Proposition 2 with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Firstly suppose that T is a path Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since TS3(T) contains a cycle it follows from Lemma 1(b) that n ≥ 7 and so T contains an induced 2K2 + K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We now assume T has a vertex of at least degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will construct a copy T ′ of T by initially choosing a vertex a of maximum degree in T and letting T ′ = N[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that TS3(T ′) is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We add edges from T to T ′ showing after each addition that either T ′ contains a forbidden subgraph, so we are done, or that TS3(T ′) remains acyclic so that T ̸= T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let b be a child of a of highest degree, c be a child of next highest degree, and d be any other child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since TS3(T ′) is acyclic T ̸= T ′ and b must have r ≥ 1 children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let e be a child of b with maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If c has a child y then {b, c, d, e, y} induce a 2K2 + K1 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise we 5 add N[b] to T ′ obtaining a copy of Dr,2,s, where s = degT (a) − 1 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By Lemma 1(c), TS3(T ′) is acyclic and so T ̸= T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There are two cases: (r ≥ 2) Let f be a second child of b and let g be a child of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Adding eg to T ′ we obtain 2K2 + K1 as an induced subgraph on {a, d, e, f, g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (r = 1) Since e is the only child of b it must have children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let t ≥ 1 be the number of children of e and let h be the child of e of maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We add N[e] to T ′ obtaining a copy of Dt,3,s and TS3(T ′) is acyclic by Lemma 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There are two subcases: (t ≥ 2) Let i be any other child of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since TS3(T ′) is acyclic h must have at least one child j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have now constructed an induced 2K2 + K1 on {a, d, h, i, j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (t = 1) If h has a single child k add hk to T ′ which is a copy of D1,4,s and again by Lemma 1(c) TS3(T ′) is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' So k has a child l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Adding kl to T ′ it contains an induced P7 and we find the forbidden subgraph 2K2 + K1 on vertices {a, d, e, k, l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, h has at least two children including vertices k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Adding edges hk and hm to T ′ we obtain the forbidden subgraph D2,4,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' a b c d e f g a c d b e h i j a c d b e h k l a c d b h i j r ≥ 2 r = 1, t ≥ 2 r = 1, t = 1 e Figure 4: Illustration for Proposition 5: Some trees T ′ containing N[b] whose TS3-graphs have a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Here r and t are respectively the number of children of b and its child e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Copies of 2K2 + K1 and D2,4,2 are marked by red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS3(T) is a acyclic if and only if for some positive integer s, T is either K1,s, D1,n,s where n ≤ 4, or Dr,n,s where r ≥ 2 and n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proof of Proposition 5 can be viewed as an algorithm that takes a tree T and either terminates with T = T ′ being one of the trees in the corollary or finds a forbidden induced graph in T showing that TS3(T) has a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 6 Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let F be a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS3(F) is a forest if and only if F is {2K2 + K1, D2,2,2 + K1, D2,4,2}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We prove that TS3(F) contains a cycle if and only if F contains one of the graphs in {2K2 + K1, D2,2,2 + K1, D2,4,2} as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose that TS3(F) contains a cycle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since the independent sets have size three, there are three cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Firstly, if the three vertices of each independent set in C lie in the same connected component T of F, by Proposition 5, the tree T must have either 2K2 + K1 or D2,4,2 as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Secondly, suppose two of the vertices of each stable set lie in the same connected component T of F, which must have at least two connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Thus, C induces a cycle in TS2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' So by Proposition 2, the tree T must have either 2K2 or D2,2,2 as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since F has at least two components, F contains 2K2 + K1 or D2,2,2 + K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Finally, suppose each vertex of each stable set lies in a different component of F, which therefore has at least three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' At least two of these components must be non-trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', contain an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, F contains an induced 2K2 + K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Conversely, suppose F contains 2K2 + K1, D2,2,2 + K1 or D2,4,2 as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TS3(F) contains a C4 in the first instance or a C8 in the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For k ≥ 4, we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let F be a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For k ≥ 4, if F contains either 2K2+(k−2)K1, or D2,2,2+(k−2)K1, or D2,4,2 + (k − 3)K1 as an induced subgraph, TSk(F) has a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' One can verify that TS2(2K2) contains a C4, and TS2(D2,2,2) and TS3(D2,4,2) both contain a C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As a result, so do TSk(2K2 + (k − 2)K1), TSk(D2,2,2 + (k − 2)K1), and TSk(D2,4,2 + (k − 3)K1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Consequently, TSk(F) has a cycle, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We conclude this section with the following conjecture for k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Conjecture 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let F be a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For k ≥ 4, if TSk(F) is a forest, F is {2K2 + (k − 2)K1, D2,2,2 + (k − 2)K1, D2,4,2 + (k − 3)K1}-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 3 H-join and H-decomposition Before considering (Q2), in this section, we describe an operation for combining TSk-graphs to produce new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We first define a family of base graphs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let V be a set of k + 1 vertices including two labelled u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then Bk(V, uv) is the graph with vertex set V and single edge uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have TSk(Bk(V, uv)) = K2 whose two vertices are labelled by the independent sets V − u and V − v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Next, we define the H-join operation and its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Vertex-labelled graphs G1 and G2 are H-consistent if the (possibly empty) intersection of their vertex sets define the same (possibly empty) common induced subgraph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The H-join of H- consistent graphs G1 and G2 is the graph H(G1, G2) with V (H(G1, G2)) = V (G1) ∪ V (G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The edges E(H(G1, G2)) consist of E(G1)∪E(G2) plus all edges vw with v ∈ V (G1)\\V (H) and w ∈ V (G2)\\V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Recall that a (vertex) cut-set in a connected graph G is a vertex set W such that G−W is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We extend this definition to the case where G is disconnected by allowing W = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We say that W decomposes G into two (not necessarily connected) induced subgraphs G1 and G2 for which V (G1) ∩ V (G2) = W and V (G1) ∪ V (G2) = V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If G − W has more than two (connected) components, the decomposition is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let G be a vertex-labelled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let W ⊂ V (G) = V (G) decompose the complement G into G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let H be the subgraph of G induced by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We say that G can be H-decomposed into G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows from the definitions that if G = H(G1, G2) then G can be H-decomposed into G1 and G2, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It is easy to verify that the size-k independent sets of H(G1, G2) are the union of those of G1 and those of G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As an example consider the two 4-vertex graphs G1 and G2 that are paths with edge sets E(G1) = {ad, bc, cd} and E(G2) = {ad, ae, eb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' These share a common induced subgraph H with V (H) = {a, b, d} and E(H) = {ad}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have V (H(G1, G2)) = {a, b, c, d, e} and E(H(G1, G2)) = {ad, ae, bc, cd, ce, be}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that TS2(G1) is the path with edges {ac − ab, ab − bd} and that TS2(G2) is the path with edges 7 a b c d G1 a b e d G2 c e a b d H(G1, G2) ab ac bd TS2(G1) ab de bd TS2(G2) ab ac de bd TS2(H(G1, G2)) Figure 5: The graphs G1, G2, H(G1, G2), and their corresponding TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Here TS2(H(G1, G2)) = TS2(G1) ∪ TS2(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' {ab−bd, bd−de}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It can be verified that TS2(H(G1, G2)) is the path with edges {ac−ab, ab−bd, bd−de} which is the union of two paths TS2(G1) and TS2(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (See Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Now consider the graph G3 which is the path with edges {ad, cd, ce}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' G1 and G3 share a com- mon induced subgraph H with V (H) = {a, c, d} and E(H) = {ad, cd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have E(H(G1, G3)) = {ad, bc, be, cd, ce}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that TS2(G3) is the path with edges {ac−ae, ae−de}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this case, TS2(H(G1, G3)) is the graph with edges {ab − ac, ac − ae, ae − de, de − bd, bd − ab, ab − ae} which is the union of TS2(G1), TS2(G3), and the two additional edges de − bd, ab − ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (See Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') a b c d G1 a e c d G3 b e a c d H(G1, G3) ab ac bd TS2(G1) ac ae ed TS2(G3) ab ac ae bd ed TS2(H(G1, G3)) Figure 6: The graphs G1, G3, H(G1, G3), and their corresponding TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Here TS2(H(G1, G3)) ̸= TS2(G1) ∪ TS2(G3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As the last example in this section, consider the graphs G4 and G5 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' G4 is the cycle with edges {ae, eb, bc, cd, ad} and G5 is the graph with edges {ae, eb, bc, ag, eg, bg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' G4 and G5 shares a common induced subgraph H with V (H) = {a, e, b, c} and E(H) = {ae, eb, bc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have E(H(G4, G5)) = {ae, eb, bc, cd, ad, ag, eg, bg, dg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this case, TS2(H(G4, G5)) is the (non-acyclic) graph with edges {ab− ac, ac − ce, ce − de, de − bd, ab − bd, ac − cg, ce − cg} which is the union of TS2(G4) and TS2(G5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (See Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') In the next proposition, we show how to compute the TSk-graph of an H-join, generalizing the examples given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let k ≥ 2 and let G1 and G2 be two H-consistent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' TSk(H(G1, G2)) is the union of TSk(G1), TSk(G2) and for every pair of k-element independent sets S1 in G1 and S2 in G2 satisfying |S1 ∩ V (H)| = |S2 ∩ V (H)| = |S1 ∩ S2| = k − 1, (1) the edge between S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 8 a e b c d G4 a e b c g G5 d g a e b c H(G4, G5) ab ac ce de bd TS2(G4) ab ac ce cg TS2(G5) ab ac ce de bd cg TS2(H(G4, G5)) Figure 7: The graphs G4, G5, H(G4, G5) and their corresponding (non-acyclic) TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Here TS2(G4, G5) = TS2(G4) ∪ TS2(G5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As remarked, the k-element independent sets of H(G1, G2) are the same as the union of those of G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, V (TSk(H(G1, G2))) = V (TSk(G1)) ∪ V (TSk(G2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Next, consider an edge in E(TSk(G1)) (respectively, E(TSk(G2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It is a token-slide between two independent sets S1 and S2 in G1 (respectively, G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This remains as a token-slide in H(G1, G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, E(TSk(G1)) ∪ E(TSk(G2)) ⊆ E(TSk(H(G1, G2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Now, consider an edge in E(TSk(H(G1, G2))) between two independent sets S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If both of these are independent sets are in G1 (respectively, G2) then the edge is also present in E(TSk(G1)) (respectively, E(TSk(G2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, we may assume the edge in E(TSk(H(G1, G2))) has as endpoints an independent set S1 in G1 (but not G2) and an independent set S2 in G2 (but not G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have S1 ∩ S2 ⊂ V (H) and since S1 and S2 are adjacent |S1 ∩ S2| = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows that |S1 ∩ V (H)| = |S2 ∩ V (H)| = k − 1 and so condition (1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We have shown that each edge in E(TSk(H(G1, G2))) is either in TSk(G1), TSk(G2) or satisfies condition (1), proving the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For two H-consistent graphs G1 and G2, we say that H(G1, G2) is k-crossing free if there are no k-element independent sets satisfying condition (1) of Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For example, one can verify that the graphs H(G1, G2) in Figure 5 and H(G4, G5) in Figure 7 are both k-crossing free, while the graph H(G1, G3) in Figure 6 is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The following result will be used for constructing TSk-trees/forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Corollary 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let k ≥ 2 and let G1 and G2 be two H-consistent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' H(G1, G2) is k-crossing free if and only if TSk(H(G1, G2)) = TSk(G1) ∪ TSk(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If H(G1, G2) is k-crossing free then (2) follows from Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise their exist k- element independent sets S1 is in G1 and S2 is in G2 satisfying (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This implies that TSk(H(G1, G2)) contains an additional edge between S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, if H(G1, G2) is k-crossing free and both TSk(G1) and TSk(G2) are acyclic, then so is TSk(H(G1, G2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The reason for allowing H to be empty in defining an H-join is that the corollary then applies to vertex disjoint graphs G1 and G2, since in this case H(G1, G2) is trivially k-crossing free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, we can create reconfiguration graphs that are forests from reconfiguration graphs that are trees (or forests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The following result follows from the relationship between H-join and H-decomposition discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Corollary 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If G can be H-decomposed into G1 and G2 and H(G1, G2) is k-crossing free then TSk(G) can be decomposed into TSk(G1) ∪ TSk(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 9 4 Results on (Q2) We currently have no general necessary and sufficient conditions for when a forest F is a TSk-graph, but we present some partial results in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Firstly, we recall that in [1] it is shown that Pn is a TSk-graph for all n ≥ 1 and k ≥ 2 and K1,n is a TSk-graph if and only if n ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this section, we show how to construct acyclic TSk-graphs from graphs that have a single edge using the join operation that was introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We show that it gives an alternate method of constructing TSk-graphs which are paths and stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, this operation can also be applied to construct more general TSk trees/forests, especially members of the classes k-ary trees and Dr,n,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='1 Paths and stars revisited Using just the base graphs and the H-join operation defined in Section 3, we can obtain large families of TSk trees/forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We begin with paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For any k ≥ 2, let Jk = {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , bk} be an independent set of size k and define the base graph Bi k = Bk(Jk−2 ∪ {ai, ai+1, ai+2}, aiai+2) and let G2 = Bi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For i ≥ 2, Gi and Bi k are H-consistent with H being the independent set Jk−2 ∪ {ai, ai+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Define Gi+1 := H(Gi, Bi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TSk(Gi+1) = TSk(Gi) ∪ TSk(Bi k) ≃ Pi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will prove by induction, for i ≥ 2, that TSk(Gi) is the path Pi with vertices labelled Jk−2 ∪ {aj, aj+1}, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the base case i = 2, we observe that indeed TSk(Bi k) is a P2 with vertices labelled Jk−2 ∪ {a1, a2} and Jk−2 ∪ {a2, a3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the inductive step we observe that, for i ≥ 2, Gi and Bi k are H-consistent with H the independent set Jk−2 ∪ {ai, ai+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' To verify that H(Gi, Bi k) is k-crossing free, note that the only independent set we need to consider in Bi k is Jk−2∪{ai+1, ai+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the path Pi which is TSk(Gi), the candidate independent sets are Jk−2 ∪ {aj, aj+1}, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Their intersection with Bi k is Jk−2 which has cardinality k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore condition (1) of Proposition 12 is not satisfied, which indeed confirms that H(Gi, Bi k) is k- crossing free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We define Gi+1 := H(Gi, Bi k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By Corollary 13, TSk(Gi+1) is the union of the above labelled Pi with a P2 with endpoints Jk−2 ∪ {ai, ai+1} and Jk−2 ∪ {ai+1, ai+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This is the required Pi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' An easy inductive argument based on the H-join in the proposition shows that, for i ≥ 2, Gi is isomorphic to P n+1 ∪ Jk−2, a result proved in Corollary 5(a) of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Observe that the vertex ai+1 in Gi is adjacent to every aj for 1 ≤ j ≤ i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Next we consider graphs Gi such that TSk(Gi) is the star K1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For k ≥ 2 and 1 ≤ i ≤ k, let Ik = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ak} be an independent set of size k, define the base graph Ci k = Bk(Ik + bi, aibi) and let G1 = C1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For k ≥ 2 and 1 ≤ i ≤ k, Gi and Ci+1 k are H-consistent with H being the independent set Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Define Gi+1 := H(Gi, Ci+1 k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then TSk(Gi+1) = TSk(Gi) ∪ TSk(Ci+1 k ) ≃ K1,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will prove by induction, for i ≥ 1, that TSk(Gi) is the star K1,i with centre labelled Ik and leaves labelled Ik + bj − aj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the base case i = 1, we observe that indeed TSk(Ci k) is a K1,1 with centre labelled Ik and leaf labelled Ik + b1 − a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the inductive step we observe that, for i ≥ 1, Gi and Ci+1 k are H-consistent with H the in- dependent set Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' To verify that H(Gi, Ci+1 k ) is k-crossing free, note that the only independent set we need to consider in Ci+1 k is Ik + bi+1 − ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the above labelled K1,i which is TSk(Gi), the candidate independent sets for condition (1) of Proposition 12 are Ik + bj − aj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Their inter- section with Ik + bi+1 − ai+1 has cardinality k − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, condition (1) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We define Gi+1 := H(Gi, Ci+1 k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By Corollary 13, TSk(Gi+1) is the union of the above labelled K1,i and a K1,1 with centre also labelled Ik and leaf labelled Ik + bi+1 − ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This is the required K1,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='2 k-ary trees In this section, we show that for each k ≥ 2, every k-ary tree is a TSk+1-graph (Proposition 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Next, we show that any tree T is an induced subgraph of some TS2-forest (Proposition 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, we state and prove the necessary and sufficient conditions for T to be an induced subgraph of some TS2-tree 10 (Proposition 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Additionally, when T = K1,n, we describe a sufficient condition for T to be an induced subgraph of some TSk-tree (Proposition 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We begin by defining a canonical vertex labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this subsection, for any integer n, define In := {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , an} and Jn := {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , bn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let k ≥ 2 and G be a graph for which T := TSk+1(G) is a k-ary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We say that G and T are canonically labelled if (a) the root of T is labelled Ik+1, (b) the d ≤ k children of the root are labelled Ik+1 − ai + bi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d, (c) the labels bj, j = d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k (if any) are not used, and (d) all other nodes in T receive a label S such that |Ik+1 ∩ S| ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It is clear that labelling K1,d, d ≤ k according to (a) and (b) with root the centre of the star is a canonical labelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this subsection, we will show that every k-ary tree has canonical labelling hence proving it is a TSk+1-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' First, we give a lemma that shows how to combine canonically labelled k-ary trees to get a larger k-ary tree that is canonically labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Lemma 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For integers k ≥ 2 and 1 ≤ i ≤ d ≤ k, let Gi be a graph for which TSk+1(Gi) a canonically labelled k-ary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We can construct a canonically labelled k-ary tree T isomorphic to the tree formed by choosing a new root and adjoining it to the root of each Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proof consists of showing that we can make a series of H-joins between the leaves of a canonically labelled K1,d and the roots of the canonically labelled trees Ti, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d, after a suitable relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose the root of Ti has ni ≤ k children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We relabel the vertices in the underlying graphs as follows: (i) relabel vertices of the Gi not in Ik+1 ∪ Jk to be distinct, ie, for 1 ≤ i ≤ j ≤ d, we have V (Gi) ∩ V (Gj) ⊆ Ik+1 ∪ Jk, (ii) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' d, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ni set bj ← bi j, where the bi j were previously unused, and (iii) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' d, set ai ← ak+1 and ak+1 ← bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By an abuse of notation, for simplicity we let for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d, Gi and Ti refer to the relabelled graphs and trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Item (i) ensures that the only labels shared between two trees are in Ik+1 ∪ Jk, (ii) ensures that all labels from Jk in the Ti are given unique labels to avoid clashes, and (iii) gives the root of Ti a correct label to be a child of a new root labelled Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We note that after relabelling bi only appears in Ti, ai does not appear in Ti and the only labels shared between the Ti are in Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Furthermore all tree vertices have unique labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Next take a canonically labelled graph G0 such that TSk+1(G0) ≃ K1,d, with the centre of the star labelled Ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d, we claim that the H-join Gi := H(Gi−1, Gi) is well-defined, k-crossing free, and TSk+1(Gi) is canonically labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' To see this, note at that iteration i, V (Gi−1) ∩ V (Gi) = Ik+1 −ai +bi which is the label of the root of Ti and a leaf of TSk+1(Gi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 17(d) implies that condition (1) of Proposition 12 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore by Corollary 13, TSk+1(Gi) is obtained from TSk+1(Gi−1) by appending Ti to the corresponding leaf in TSk+1(Gi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The conditions of Definition 17 are satisfied so TSk+1(Gi) is canonically labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' At the end of iteration d, T := TSk+1(Gd) is the required tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The construction described in the proof is illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We may now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For every k-ary tree T, there is a canonically labelled graph G such that T ≃ TSk+1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose that the root r of T has d ≤ k children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We prove the proposition by induction on the height t of T, which is the length of the longest path to a leaf from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If t = 1 then T ≃ K1,d and so has a canonically representation as described following Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, by deleting r we obtain d subtrees Ti, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d, which are also k-ary trees, with height less than t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, by induction each Ti can be represented by a canonically labelled graph Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows from Proposition 18 that we can perform d H-joins to obtain a canonically labelled graph G for which T ≃ TSk+1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 11 a1a2a3 b1a2a3 a1b2a3 a1a2a3 b1a2a3 a1b2a3 T1 T2 Relabel a3a2b1 b1 1a2a3 a3b1 2b1 a1a3b2 b2 1a2a3 a1b2 2b2 k = 2 a1a2a3 K1,2 Figure 8: Construction of D2,3,2 from two K1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As noted in Section 4 of [1], K1,k+1 is an example of a k-ary tree that is not an TSk-graph so the proposition is tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Nevertheless, if we add a sufficient number of isolated vertices to K1,t, for t > k, it becomes a TS2-graph—a result we will now prove in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will need a special labelling of a tree that will be defined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A tree T is well-labelled if (a) the root r of T is labelled ab, (b) the d children of r have roots labelled ri = bci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d − 1 and rd = acd, (c) the only labels containing a and b are ab, acd, bci, 1 ≤ i ≤ d − 1, and (d) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d label ci only occurs in the subtree with root ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We note that there is nothing special about the ordering of the subtrees of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The subtree rooted at ri can play the role of rd by relabelling those two subtrees with the exchanges a ↔ b and ci ↔ cd, which leaves T well-labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As an example, for d ≥ 1 we can well-label K1,d simply by using (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Consider the graph G defined by V (G) = {a, b} ∪ {ci : 1 ≤ i ≤ d} and E(G) = {aci, cicd : 1 ≤ i ≤ d − 1} ∪ {bcd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Furthermore let J = {cicj : 1 ≤ i < j ≤ d − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then it is not hard to verify that TS2(G) ≃ K1,d + (d − 1)(d − 2)K1, where the K1,d is well-labelled and the K1 are labelled by the set J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This motivates the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A tree T is well-labelled by a labelled graph G if there is an integer n such that TS2(G) ≃ T + nK1 and T is well-labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We now show the following general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For every tree T there is a graph G and integer n such that T is well-labelled by G and TS2(G) ≃ T + nK1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proof is by induction on N, the number of nodes in a given tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As noted above, the proposition is true for all stars K1,t and these act as base cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the inductive step, assume the proposition is true for all trees on N nodes and consider a tree T with N + 1 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If T is a star we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Otherwise, let r be the root of T and assume r has degree d with its children ri being roots of subtrees Ti, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We may also assume that Td is a subtree of T with height at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We now construct two trees from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The first, T 1 consists of T with subtree Td deleted and a pendant vertex added to its root r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The second, T 2 consists of Td with a pendant vertex added to its root rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By induction, there are integers n1, n2 and graphs G1, G2 which well-label T 1 and T 2 such that TS2(G1) ≃ T 1 + n1K1 and TS2(G2) ≃ T 2 + n2K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Apart from the vertex labels used in Definition 20, we may assume the vertex labels in G1 and G2 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We will show that G1 and a relabelled G2 can be H-joined and that this will identify the pendant edges added to T 1 and T 2 to give us back T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In T 1 we note that root r is labelled ab, and by relabelling subtree roots if necessary, that the added pendent vertex can be labelled acd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In T 2 the root rd is also 12 labelled ab and we can again assume the added pendant vertex is labelled acd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In T 2 we interchange the labels b ↔ cd and set ci ← c′ i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , d−1, for labels c′ i that are unused in either T 1 or T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let G3 and T 3 denote the relabelled G2 and T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Setting H = {a, b, cd}, we have V (G1) ∩ V (G3) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' H induces the same subgraph, containing the single edge bcd, in both G1 and G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' G1 and G3 are H-consistent and since k = 2 and their vertex sets are otherwise disjoint, condition (1) of Proposition 12 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let G4 = H(G1, G3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Applying Corollary 13 we have that T 4 := TS2(G4) ≃ TS2(G1) ∪ TS2(G3) ≃ {T 1 + n1K1} ∪ {T 3 + n2K1} ≃ T + (n1 + n2)K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' is well-labelled by G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proof of the proposition is illustrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The proposition tells us that for every tree r r1 r2 r3 r4 T r r1 r2 r3 r4 new pendant edges bc1 bc2 bc3 ab ac4 c4c′ 1 c4c′ 2 c4c′ 3 ab a c4 c′ 1 c′ 2 c′ 3 a b c1 c2 c3 c4 (relabelled) G2 G1 (relabelled) T 2 T 1 ac4 b Figure 9: Illustrating Proposition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' T there is a graph G for which TS2(G) is forest containing T as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, there can be no forbidden induced subgraph characterization of which forests are TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However, this does not imply that there can be no forbidden induced subgraph characterization of which trees are TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Indeed, in the next propositions, we present some of such characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let T be a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Then there exists a TS2-tree containing T if and only if T is a 3-ary tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇐) In the proof of Proposition 22, we see that isolated vertices are only added when the base case of a star appears as a subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore, it suffices to consider only the case T = K1,t, 1 ≤ t ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As we have noted, neither K1,3 nor K1,4 are TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It is not hard to see that there is a G1 such that TS2(G1) ≃ K1,3 + K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However, by adding an extra vertex to G1, we can construct a graph G2 such that TS2(G2) ≃ D1,3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Furthermore, we can construct a graph G3 by applying H-join to two copies of G2 with slightly different vertex-labellings such that TS2(G3) is isomorphic to a P7 with two pendant vertices attached to the midpoint of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (See Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Thus, if follows that when T = K1,t, 1 ≤ t ≤ 4, we can embed it as an induced subgraph of a tree T ′ = TS2(G), for some graph G (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Our proof of the if direction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 13 a c d e f b ab ae bd ac b g c d h a ab bd ac bg ce ef dh dg ab ae bd ac ce ef dh dg G2 TS2(G2) bg TS2(G3) Figure 10: Taking H-join of two copies of G2, where H is the path adcb, results a graph G3 such that TS2(G3) is isomorphic to a P7 with two pendant vertices attached to the midpoint of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇒) We show that if T is a k-ary tree but not a 3-ary tree for k ≥ 4 then there does not exist any TS2-tree T ′ containing T (as an induced subgraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (By definition, any k-ary tree is also a ℓ-ary tree for ℓ ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Let x be a vertex of T whose degree is at least five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Since T is a k-ary tree but not a 3-ary tree, such a vertex x exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Suppose to the contrary that T ′ exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', there exists a graph G′ such that T ′ ≃ TS2(G′) contains T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Without loss of generality, assume that x is labelled by ab, where {a, b} is a size-2 stable set of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By the pigeonhole principle, we may further assume that three neighbors x1, x2, and x3 of x are labelled ac, ad, and ae, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since T ′ is a tree, it follows that cd, ce, and de are respectively the labels of y1, y2, and y3 where yi is not adjacent to any of � j{xj}+x+� j̸=i{yj} for 1 ≤ i, j ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows that T ′ contains the labelled graph F ≃ K1,3 + 3K1 and therefore G′ must ab ac ad ae cd ce de F a b c d e G Figure 11: The graphs F and G in the proof of Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' contain the labelled graph G ≃ K1,3 + K1, both described in Figure 11, as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since T ′ ≃ TS2(G′) is a tree and G′ contains G, it follows that G′ has exactly one non-trivial component C (having more than two vertices) and C contains G, otherwise G′ must contain an 14 induced 2K2 and by Proposition 2 its TS2-graph is not a tree, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' – Case 1: a ∈ V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' By definition, the distance from a to any of b, c, d, e in G′ must be at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' If there is a path of length at least two between a and one of c, d, e not passing through b, the graph G′ contains a 2K2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Thus, any path between a and one of c, d, e must go through b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, if there is a path of length at least three between a and b not passing through any of c, d, e, again the graph G′ contains a 2K2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since a ∈ V (C), it follows that a and b must have a common neighbor in G′, say f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that for each y ∈ V (C) − {a, b, c, d, e, f}, y must be adjacent to b in G′, otherwise G′ either contains 2K2 or D2,2,2 and again by Proposition 2 its TS2-graph is not a tree, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However, this implies that TS2(C) must be a forest and since G′ has exactly one non-trivial component C, we have TS2(G′) is also a forest, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' – Case 2: a /∈ V (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this case, there are two types of size-2 stable sets of G′: those containing a and those do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since G′ contains G, each type has at least one member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, since a is isolated (the only non-trivial component is C and a is not in it), no member from one type is adjacent to a member from another type in TS2(G′), which means TS2(G′) is indeed disconnected, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the above cases, we proved that some contradiction must happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Our proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Indeed, for K1,n, in general we have Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There exists a TSk-ary tree T containing K1,n if n ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' From either [1] or Proposition 16, the proposition holds for n ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (Indeed, in this case, T = K1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Thus, it suffices to consider k + 1 ≤ n ≤ 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n − k}, let Ai = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k} − i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let Ik = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ak} and Bn = {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , bn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We construct a graph G0 such that TSk(G0) ≃ K1,n + (n − k)(k − 1)K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let Ik = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ak} and Bn = {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , bn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let V (G) = Ik + Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Vertices in Bn form a graph Kn − M where M is the matching that contains bibk+i for 1 ≤ i ≤ n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Additionally, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k}, we add an edge in G0 between ai and both bi and bk+i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that V (TSk(G0)) consists of Ik, the sets Ik −ai +bi (1 ≤ i ≤ k), Ik −ai +bk+i (1 ≤ i ≤ n−k), and (Ik −ai +bi)−aj +bk+i (1 ≤ i ≤ n − k and j ∈ Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, one can verify that the independent sets (Ik − ai + bi) − aj + bk+i are isolated in TSk(G0) and the remaining independent sets form a K1,n in which Ik is adjacent to every other set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In short, G0 is indeed our desired graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n − k}, we construct a graph Gi whose TSk-graph is a star K1,k−1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let V (Gi) = (Ik − ai + bi) + � j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=',k}−i{ci j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Vertices in � j∈Ai{ci j} form a clique in Gi of size k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We also add an edge in Gi between aj and ci j for each j ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' From either [1] or Proposition 16, one can verify that TSk(Gi) ≃ K1,k−1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n − k} and j ∈ Ai, we construct a graph Gi j whose TSk-graph is a K2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let V (Gi j) = (Ik − ai + bi) − aj + bk+i + ci j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The only edge in Gi j is the one joining ci j and bk+i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' From either [1] or Proposition 15, one can verify that TSk(Gi j) ≃ K2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Now, we construct a graph G whose TSk-graph is a tree containing K1,n as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For convenience, we assume that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n−k} the set Ai = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k}−i can be enumerated as {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , jk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We define Ki j0 = Gi and Ki jp = Hjp(Ki jp−1, Gi jp) for jp ∈ Ai where Hjp is the stable set (Ik − ai + bi) − ajp +ci jp for p ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that the graphs Ki jp−1 and Gi jp are Hjp-consistent, which implies that Ki jp are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, one can also directly verify that the sets (Ik − ai + bi) − aj + ci j and (Ik − ai′ + bi′) − aj′ + ci′ j′ always differ in at least two members, which means the condition (1) of Proposition 12 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In short, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n − k}, we obtain the graph Ki jk−1 whose TSk-graph is isomorphic to the one obtained from K1,k−1 by replacing each edge with a P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Next, we define K0 = G0 and Ki = Hi(Ki−1, Gi) where i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , n − k} and Hi is the subgraph induced by (Ik −ai +bi)+bk+i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that the graphs Ki are well-defined because Ki−1 and Gi are Hi-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, we have Ik and each (Ik − ai + bi) − aj + ci j for 1 ≤ i ≤ n − k and j ∈ Ai always differ in at least two members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows that the condition (1) of Proposition 12 is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In short, we finally obtain the graph G = Kn−k whose TSk-graph is indeed a tree containing K1,n as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Unfortunately, we have not been able to show whether the reverse statement of Proposition 24 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We conclude this section with the following open problems: 15 b5a2a3a4 b1a2a3a4 a1a2a3b8 b1c1 2a3a4 b1a2c1 3a4 a1a2a3b4 a1c4 2a3b4 a1a2a3a4 TS4(G) b2 b1 b3 b4 b5 b6 b7 b8 c1 2 c1 3 c1 4 c4 1 c4 2 c4 3 a1 a2 a3 a4 G b1a2a3c1 4 c4 1a2a3b4 a1a2c4 3b4 b1b5a3a4 b1a2b5a4 b1a2a3b5 b8a2a3b4 a1b8a2b4 a1a2b8b4 Figure 12: Construction of a graph G such that TS4(G) is a tree containing K1,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Vertices of G in the yellow box form a clique having all dashed edges removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The red induced subgraph of G forms a graph G0 whose TS4(G0) ≃ K1,8 + 12K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Problem 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For every k ≥ 3 and tree T, is there a graph G such that TSk(G) is a forest containing T as an induced subgraph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Problem 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For every k ≥ 3 and (k + 1)-ary tree T, is there a graph G such that TSk(G) is a tree containing T as an induced subgraph?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Problem 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Does there exist a TSk-tree T containing K1,n for n > 2k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='3 Dr,n,s We now consider graphs in the Dr,n,s family for whose TSk-graphs are trees and show how they can be constructed by the H-join operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We remark that when n = 1, Dr,n,s is nothing but a star K1,r+s and this case was considered in [1] and revisited in Proposition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Furthermore, it follows from Proposition 19 that for n, k ≥ 2 and 1 ≤ r ≤ s ≤ k − 1, Dr,n,s is a k-ary tree and so by Proposition 19 it is a TSk-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The reverse statement does not hold in general: there exists a TSk-graph Dr,n,s even when s ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For example, one of such graphs, as already proved in [1], is D1,3,2 (r = 1, s = k = 2, and n = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (See also Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') Indeed, as we will see in Proposition 29, it is the unique TS2-graph among all trees D1,n,2 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Additionally, for the sake of completeness, we will also show in Proposition 30 that the reverse statement indeed holds when n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We are now characterizing which D1,n,2-graphs are TS2-graphs and show that this property is non- hereditary for this simple class of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We then consider the Dr,2,s-graphs characterizing those that are TSk-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Assume for some G, TS2(G) is a forest containing a K1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There are four stable sets in G corresponding to the vertices of the K1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There are two ways of labelling the K1,3 but in each case there are five vertices, say a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , e, of G involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Up to permutations of the labels, the corresponding stable sets in G are either {ab, ac, bd, ae} or {ab, ac, ad, ae}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Using these definitions we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let H be the subgraph of G induced by a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The edges of H are (a) ad, de, eb, bc, cd, if the K1,3 is labelled {ab, ac, bd, ae}, or (b) bc, bd, be if the K1,3 is labelled {ab, ac, ad, ae}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (a) This labelling of K1,3 immediately gives edges ad, bc, be and non-edges ab, ac, ae, bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' That leaves three edges of H to be decided: (i) ce must be a non-edge else there is an edge ae, ac in the K1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (ii) cd is an edge else there is a cycle ab, bd, cd, ad in TS2(G), so it is not a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 16 ab ac bd ae ab ac ad ae a b c d e b c d e (a) (b) K1,3 H a Figure 13: If TS2(G) is a forest containing a K1,3 then G must contain one of the induced subgraphs H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (iii) de is an edge else there is a cycle de, bd, ab, ae in TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that ce must also be a vertex in TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (b) This labelling of K1,3 immediately gives edges bc, bd, be and non-edges ab, ac, ad, ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' There are no other edges in H as c, d, e form a stable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This implies that TS2(G) must also contain vertices cd, ce and de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Using the lemma we show that precisely one of the D1,n,2-graphs is a TS2-graph, incidentally proving the non-hereditary property mentioned above for this class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' D1,n,2 is a TS2-graph if and only if n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We first consider 1 ≤ n ≤ 3 and show that D1,3,2 is a TS2-graph while D1,1,2 = K1,3 and D1,2,2 are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (We note that the results for the first two graphs have also been proved in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=') According to Lemma 28, if D1,n,2 is a TS2-graph of some graph G, the unique star K1,3 in D1,n,2 can be labelled in one of two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However, we may immediately eliminate the possibility of the labelling in Lemma 28(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This is because, as pointed out in the proof, there must be additional vertices in D1,n,2 = TS2(G) labelled cd, ce and de which are non-adjacent since c, d, e form a stable set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This implies that n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' So we may assume that if D1,n,2 is a TS2-graph, the K1,3 must be labelled as in Lemma 28(a) with corresponding induced subgraph H of D1,n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' From the proof of Lemma 28(a) there must be an additional vertex ce in D1,n,2 however this cannot be adjacent to any of the other four vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This implies that n ≥ 3 and so neither D1,1,2 nor D1,2,2 can be TS2-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' However we may extend H to G by adding a vertex f adjacent to all vertices except e, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This introduces the new stable set ef which is adjacent to both ae and ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Therefore D1,3,2 is isomorphic to TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We note that G is the unique graph (up to label permutations) for which this is true, due to the uniqueness of the labelling of K1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It remains to consider n ≥ 4 and show that D1,n,2 is not a TS2-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose to the contrary that there exists a graph G such that D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Again, D1,n,2 must contain a copy of K1,3 with exactly two ways of labelling (up to label permutations) by size-2 independent sets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Case 1: K1,3 is labelled {ab, ac, bd, ae}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ac and ae are not adjacent, ce must be a vertex of D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We consider the following cases: – Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='1: the distance between ce and any vertex of {ac, bd, ae} is at least three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since the roles of c and e are equal, we assume without loss of generality that ce is adjacent to some vertex cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that a and f are not adjacent in G, otherwise ac and cf are adjacent, which means the distance between ac and ce is two, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ce and cf are adjacent, so are ae and af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Moreover, bf must be a vertex, otherwise there is an edge between ab and af in D1,n,2 = TS2(G) which creates a C3 having {ab, ae, af} as its vertex-set, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ab and ac are adjacent, so are cf and bf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Now, df must be a vertex, 17 otherwise bd and bf are adjacent which contradicts D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ab and bd are adjacent, so are af and df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' From the proof of Lemma 28(a)(ii) c and d are adjacent in G, so df and cf are adjacent, which again contradicts D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' – Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='2: the distance between ce and one of {ac, bd, ae} is exactly two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that bd and ce has no common neighbor, otherwise that neighbor must be labelled as one of {bc, be, dc, de}: the first two can be ignored because ab and ac (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', ab and ae) are adjacent, the last two can be ignored because ab and bd are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Again, since the roles of c and e are equal, we assume without loss of generality that ae and ce has a common neighbor ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since n ≥ 4, ce must have another neighbor which is different from ef, which can be either cg or eg for some vertex g of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' ∗ If it is cg then ag must be a vertex, otherwise cg and ac must be adjacent, which creates a C6 whose vertex-set is {ac, ab, ae, ef, ce, cg}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ce and cg are adjacent, so are ae and ag, which contradicts D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' ∗ If it is eg then ag must be a vertex, otherwise eg and ae must be adjacent, which creates a C4 whose vertex-set is {ae, ef, ce, eg}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since ce and eg are adjacent, so are ag and ac, which contradicts D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Case 2: K1,3 is labelled {ab, ac, ad, ae}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As before, cd, ce, and de must be vertices in D1,n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Without loss of generality, since the roles of c, d, e are equal, we may assume that only ae is adjacent to another vertex of D1,n,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' As shown in the proof of Lemma 28(b), D1,n,2 must also contain vertices cd, ce, de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Let P be the path between ae and cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since the roles of c and d are equal, we can assume without loss of generality that cd is adjacent to a vertex cf in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that if af is not a vertex ac and cf are adjacent contradicting the choice of ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' So af is a vertex and since cd and cf are adjacent so are ad and af, which contradicts D1,n,2 = TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We remark that if we add a vertex g to G in Figure 1 joining it to all vertices except d the corresponding TS2-graph is obtained by adding the edge between bd and dg to TS2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Note that this tree is not in the class Dr,n,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In the next proposition we consider two arbitrary stars whose centers are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proposition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Dr,2,s (1 ≤ r ≤ s) is a TSk-graph if and only if s ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇐) It follows directly from Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' (⇒) Suppose that Dr,2,s (r ≤ s) is obtained from P2 = p1p2 by attaching r leaves u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ur at p1 and s leaves v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , vs at p2 for some s ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We show that this graph is not a TSk-graph for any fixed k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Suppose to the contrary that there exists a graph G such that Dr,2,s ≃ TSk(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=', there exists a bijective mapping f : V (Dr,2,s) → V (TSk(G)) such that uv ∈ E(Dr,2,s) if and only if f(u)f(v) ∈ E(TSk(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Without loss of generality, let f(p2) = I = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , ak}, where I is a size-k independent set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since p2 has s + 1 neighbors, from the pigeonhole principle, it follows that there must be some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k} such that f(u) = I − ai + x and f(v) = I − ai + y, where u, v ∈ N(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Observe that J = (I − ai − aj) + x + y /∈ {f(p2), f(u), f(v)} must be a size-k independent set of G, where j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' , k} − i and therefore there exists z ∈ V (Dr,2,s) − {p2, u, v} such that f(z) = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We consider the following cases: – Neither u nor v is p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In this case, we must have z /∈ N(p2), otherwise it must be adjacent to p2, but then f(z) = J and f(p2) = I must be adjacent in TSk(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' It follows that z ∈ N(p1) − p2 and thus f(p1) must be in {I − ai + x, I − ai + y, I − aj + x, I − aj + y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Since neither u nor v is p1, the first two can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Now, if f(p1) = I − aj + x, the vertices x and aj must be adjacent in G, which contradicts the fact that f(u) ∈ TSk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A similar contradiction can be derived for the case f(p1) = I − aj + y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Thus, f(p1) cannot be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' – u is p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Again, z /∈ N(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Thus, z ∈ N(p1) − p2, which implies that y and aj must be adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' This contradicts f(v) ∈ TSk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Thus, f(z) cannot be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' In both cases, we showed that some contradiction must occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Our proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' 18 5 Conclusions In this paper, we considered two token sliding problems for trees and forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' The two questions studied seem remarkably complicated, even for this simple class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the first question, finding necessary and sufficient conditions on G for TSk(G) to be a forest, we could only get a complete solution for k = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' For the second question, finding necessary and sufficient conditions for a tree or forest to be a token sliding graph, we could get more general results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Nevertheless, as noted in Section 4 several interesting important questions remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' We expect the join and decomposition operations introduced there will be of use for similar questions for more general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Acknowledgments Avis’ research is partially supported by the Japan Society for the Promotion of Science (JSPS) KAK- ENHI Grants JP18H05291, JP20H00579, and JP20H05965 (AFSA) and Hoang’s research by JP20H05964 (AFSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' References [1] David Avis and Duc A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Hoang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' On reconfiguration graphs of independent sets under token sliding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' arXiv preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='16861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' [2] Nicolas Bousquet, Amer E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' Mouawad, Naomi Nishimura, and Sebastian Siebertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' arXiv preprint, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content='10526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} +page_content=' [3] 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AyT4oBgHgl3EQfePhg/content/2301.00317v1.pdf'} diff --git a/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf b/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e609e988a2ae2c1b70dd2477435a9464f7e377f8 --- /dev/null +++ b/DNE1T4oBgHgl3EQfqAUl/content/2301.03337v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:39a48f14a2b2308f53cf4cdb5a02a810b1e06031e9258d89c0267a9f453f6d25 +size 3910529 diff --git a/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf b/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb23ac30202383e84ec696c85119b67ef6dbf0f5 --- /dev/null +++ b/EtAyT4oBgHgl3EQfevi1/content/2301.00328v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:066035c919034b4192353c9691d110ffee94906dac45f1ae5ebfd01d56490c68 +size 566624 diff --git 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+ulukus@umd.edu +Abstract—We consider gossiping in a fully-connected wireless +network consisting of n nodes. The network receives Poisson up- +dates from a source, which generates new information. The nodes +gossip their available information with the neighboring nodes +to maintain network timeliness. In this work, we propose two +gossiping schemes, one semi-distributed and the other one fully- +distributed. In the semi-distributed scheme, the freshest nodes +use pilot signals to interact with the network and gossip with the +full available update rate B. In the fully-distributed scheme, each +node gossips for a fixed amount of time duration with the full +update rate B. Both schemes achieve O(1) age scaling, and the +semi-distributed scheme has the best age performance for any +symmetric randomized gossiping policy. We compare the results +with the recently proposed ASUMAN scheme [1], which also gives +O(1) age performance, but the nodes need to be age-aware. +I. INTRODUCTION +Gossiping is an information sharing mechanism where +nodes transmit their own data to the neighboring nodes ran- +domly. Gossiping does not require any centralized schedul- +ing and is particularly suitable for communication in dense +networks. There are many gossip algorithms in the literature, +e.g., [2]–[4], that focus on maximizing the effectiveness of +information dispersion. Gossip algorithms have been studied +from a timeliness point of view in [5]. The analysis in [5] +uses the version age metric, which is one of the measures of +information freshness in the literature [6]–[19]. +The analysis in [5] shows that for a fully-connected network +of n nodes, if each node gossips with a fixed rate λ, the +average version age of any individual node scales as O(log n) +with the network size. Subsequent works [20]–[26] show that +there can be improvements in the age scaling by introducing +particular network mechanisms, such as clustering, file slicing +and network coding. In this paper, we focus on another aspect +of existing gossip schemes, which is their uniform gossip rate +assignment to all nodes. A main drawback of uniform rate +gossiping is that it allocates the same gossip rate to nodes +with relatively stale and relatively fresh information. This +negatively impacts the timeliness performance of the network. +Our goal in this paper is to efficiently and distributedly allocate +the total network gossip capacity dynamically among the users, +thereby enabling opportunistic gossiping, where fresher nodes +gossip with higher gossip rates. +The first paper to address the inefficiency of uniform +rate gossiping is [1] which proposed the ASUMAN scheme, +which is an opportunistic gossiping scheme that relies on the +assumption that the nodes are age-aware. In ASUMAN, since +the nodes are age-aware, whenever the source updates itself, +all the nodes in the network get synchronized and a new +gossiping frame starts. When a new frame starts, the nodes +stop gossiping and send a small pilot signal after waiting for +a back-off period proportional to their current age. In this way, +the freshest nodes get to start gossiping first as their back-off +period is smallest and the relatively staler nodes do not gossip +after receiving the pilot signal from the freshest nodes. If in +any frame, the number of fresh nodes is more than one, then +that number is estimated from the received pilot signals and +the total update rate B = nλ is equally divided between them. +The analysis in [1] shows that the version age of an individual +node scales as O(1) with the network size n. +Although ASUMAN achieves better age performance, the +system model poses some challenges in real-life implementa- +tions. One such challenge is that when multiple nodes have +the same minimum age, all of them transmit the pilot signal +simultaneously. Thus, multiple short signals overlap over the +air, which leads to incorrect estimation of the minimum- +age nodes, causing interference within the gossiping nodes. +Another downside of ASUMAN is that the nodes have to be +age-aware. This can be achieved if the source sends a signal to +the nodes when it updates itself, adding additional complexity +to the simple gossiping model. +gossiping scheme +age scaling +ASUMAN proposed in [1] +2 λe +λ + 1 +semi-distributed proposed here +2 λe +λ +fully-distributed proposed here +(1 + e) λe +λ +TABLE I +AGE SCALING COMPARISON FOR DIFFERENT GOSSIPING SCHEMES. +In this paper, we propose two new gossiping schemes, one +semi-distributed and the other fully-distributed, that both yield +O(1) performance. These schemes are able to circumvent +the previously mentioned downsides. In the semi-distributed +scheme, each time a node gets updated by the source, it +transmits a pilot signal to the neighboring nodes and starts +gossiping with the maximum capacity until it receives a signal +from some other node. In the fully-distributed scheme, each +time a node gets updated by the source, it gossips for a +fixed duration with the maximum capacity and stops. The +age scaling comparison of these schemes is shown in Table I. +Further, we prove that the semi-distributed gossiping scheme +yields the best age performance among all possible symmetric +gossiping schemes with an upper bound on the instantaneous +maximum gossip rate. For our analysis, we use stochastic + +hybrid system (SHS) formulation [27], similar to [1], [5], to +calculate of mean steady-state version age of the nodes. +II. SYSTEM MODEL +We consider a gossip network consisting of a source labeled +node 0, and a set of nodes labeled N = {1, 2, . . ., n}, as +shown in Fig. 1. The source updates its information with +Poisson arrivals of rate λe, and it sends Poisson updates to +the network with a total rate of λ. For simplicity, we consider +a symmetric network, i.e., each of the nodes receives updates +from the source with a rate λ +n. In the timely gossiping papers +in the literature [5], [20]–[26], it is assumed that each node +of the network gossips with a rate of λ; thus, on a fully +connected network where each node is connected to (n − 1) +other nodes, each node i gossips with a node j with a rate +of +λ +n−1. Therefore, the total update capacity of the network is +B = nλ. As in [1], in this paper, we consider allocating this +total update rate B to users dynamically. Once a gossip rate is +assigned to a node, it gossips with its (n − 1) neighbors with +equal rates in the fully connected network. +Thus, the network has an upper bound of B on the instan- +taneous gossiping rate. If at any time, multiple nodes transmit +and the total instantaneous gossip rate exceeds B, there will be +interference, and the gossiped data is lost. Hence, for effective +gossiping, at any time instant, the total instantaneous gossip +rate has to be less than or equal to B. The goal of our work +is to improve the timeliness of such a network. To measure +the timeliness of the ith node, we use version age, denoted +as ∆i(t). This measure counts how many versions the data at +the ith node is lagging, compared to the data available at the +source at time t. Mathematically, we write +∆i(t) = Ns(t) − Ni(t), +(1) +where Ns(t) and Ni(t) are the versions of the data available +at the source and at the ith node, respectively, at time t. +We denote all the ages of nodes at time t as the age vector +∆(t) = [∆1(t), ∆2(t), . . . , ∆n(t)]. When the source updates +itself, all the ages of the nodes increase by 1. If the source +sends an update to a node, its age becomes 0. When node +i sends a gossip update to node j, it stores the data with +the freshest version, i.e., the age of the jth node becomes +ˆ∆j(t) = ∆{i,j}(t) = min{∆i(t), ∆j(t)}. +III. SEMI-DISTRIBUTED GOSSIPING +In this section, we introduce the semi-distributed gossiping +scheme. The motivation for this is to allow the freshest node +of the network to gossip with maximum capacity. Suppose +we denote the kth source-to-ith node update as t(i) +k . In this +scheme, at time t(i) +k , i transmits a small pilot signal to all the +other nodes in the network and starts gossiping with rate B to +the other nodes with equal rate. While gossiping, if i receives +a pilot signal from any other node, it will stop gossiping. We +define the gossiping node at any given time t as M(t). Since, +the probability of two simultaneous Poisson arrivals is 0, i.e., +P(|M(t)| ≥ 2) = 0, here we do not face the problem of +overlapping pilot signals like ASUMAN [1]. +0 +1 +2 +3 +4 +λe +λ +5 +Fig. 1. Source 0 updates itself with rate λe and sends updates to the nodes +N = {1, 2, 3, 4, 5} uniformly with total rate λ, i.e., with rate λ/5 to each +of the nodes. The nodes gossip with each other with total update rate B. +We investigate the mean steady-state age of an individual +node, denoted as, +ai = lim +t→∞ ai(t) = lim +t→∞ E[∆i(t)], +(2) +in particular, how network size n affects ai, in Theorem 1. +Theorem 1 If B = nλ, the average version age of a node ai +in a semi-distributed gossip network scales as O(1). +Proof: We use SHS formulation of [27]. Note that, for any +time t, the gossiping node is the minimum age node in the +network. Let us denote this minimum age as ∆min(t) = +min{∆1(t), ∆2(t), . . . , ∆n(t)}. From [5], we know that +limt→∞ E[∆min(t)] = +λe +λ . Since for any given t, only the +node with the minimum age is gossiping, we can express +the state transition of the system as an SHS with only one +type of transition, i.e., Q = 0. We choose the test function +ψi : +Rn × [0, ∞) → +R, where i ∈ N, as +ψi(∆(t), t) = ∆i(t). +(3) +Now, following [27, Thm. 1], we evaluate the extended gen- +erator function as +E[(Lψi)(∆(t), t)] = +� +(j,ℓ)∈L +λj,ℓ(∆(t), t)E +� +ψi(φj,ℓ(∆(t), t)) +− ψi(∆(t), t) +� +, +(4) +where L denotes all possible state transitions. We define the +reset maps φj,ℓ(∆(t), t) = ˆ∆(t) = [ ˆ∆1(t), ˆ∆2(t), . . . , ˆ∆n(t)] +as follows +ˆ∆i(t) = + + + + + + + +∆i(t) + 1, +if j = 0, ℓ = 0 +0, +if j = 0, ℓ = i +min(∆j(t), ∆ℓ(t)), +if j ∈ N, ℓ = i +∆i(t), +otherwise. +(5) +The update rates λj,ℓ are given as +λj,ℓ(∆(t), t) = + + + +λe, +if j = 0, ℓ = 0 +λ +n, +if j = 0, ℓ = i +B +n−1 +1{j = M(t)}, +otherwise, +(6) + +where +1{·} denotes the indicator function. Now, we can +rewrite (4) as +E[(Lψi)(∆(t), t)] += E +� +λe(∆i(t) + 1 − ∆i(t)) + λ +n(0 − ∆i(t)) ++ +� +j∈N +B +n − 1 +1{j = M(t)} +� +∆{j,i}(t) − ∆i(t) +� � +. (7) +Since the gossiping node is always the minimum age node, +we can write +E[(Lψi)(∆(t), t)] += λe − λ +nai(t) + E +� +� +j=M(t) +B +n − 1(∆min(t) − ∆i(t)) +� += λe − λ +nai(t) + +B +n − 1(amin(t) − ai(t)). +(8) +Now, since the version age is a piece-wise constant function +of time, we obtain +dE[ψi(∆(t), t)] +dt += dE[∆i(t)] +dt += 0, +(9) +for any continuity point t. Hence, the expected value in (8) is +0, by Dynkin’s formula, as given in [27]. Thus, (8) becomes +0 = λe − λ +nai(t) + +B +n − 1(amin(t) − ai(t)). +(10) +Hence, the mean age of an individual node is expressed as +ai(t) = +λe + +B +n−1amin(t) +λ +n + +B +n−1 +. +(11) +To evaluate the steady-state mean age, we take t → ∞ in (11) +which gives +ai = +λe + +B +n−1 +λe +λ +λ +n + +B +n−1 +. +(12) +Finally, to calculate the scaling of the average age, we use +B = nλ, which yields +lim +n→∞ ai = lim +n→∞ +λe +λ +� +1 + +n +n−1 +1 +n + +n +n−1 +� += 2λe +λ , +(13) +concluding the proof. ■ +Next, we show that this semi-distributed scheme gives +the best version age performance for any possible gossip- +ing scheme with a constraint on the instantaneous gossiping +scheme, in Theorem 2. +Theorem 2 For any symmetric network with maximum in- +stantaneous gossip rate of B, the semi-distributed gossiping +scheme yields the minimum average age for the nodes. +Proof: Suppose we use any arbitrary gossiping policy. Since +the total gossip rate is upper bounded by B, we have +� +j,i∈N,j̸=i +λj,i(∆(t), t) ≤ B, +∀t. +(14) +From the symmetry of the network, we can write +E + + +� +j∈N,j̸=i +λj,i(∆(t), t) + + ≤ +B +n − 1. +(15) +Note that the sum in (14) is over all i, j whereas the sum +in (15) is over j only. Now, equating the extended generator +function to 0, yields +λ +nai(t) + E + + +� +j∈N,j̸=i +λj,i(∆(t), t)∆i(t) + + += λe + E + + +� +j∈N,j̸=i +λj,i(∆(t), t)∆{j,i}(t) + + . +(16) +Using the inequality in (15) and by definition the fact that +∆{j,i}(t) ≥ ∆min(t), we can rewrite (16) as +λ +nai(t) + +B +n − 1ai(t) ≥ λe + +B +n − 1amin(t). +(17) +Taking t → ∞ in (17) and using the expression of amin(t), +we obtain +ai ≥ +λe + +B +n−1 +λe +λ +λ +n + +B +n−1 +, +(18) +where the right-hand side of the inequality is the average +age of a node with the proposed semi-distributed policy. This +concludes the proof. ■ +IV. FULLY-DISTRIBUTED GOSSIPING +In this section, we introduce a gossiping policy which is +fully-distributed. In ASUMAN [1], the nodes need to be age- +aware and in the semi-distributed scheme, the nodes need to +implement a pilot-signal based communication in the network. +We improve upon them and formulate a gossiping policy that +does not require age-awareness or pilot-signal transmissions. +In this scheme, whenever node i receives an update from the +source at time t(i) +k , it starts gossiping to all the other nodes +with rate B for a fixed time duration δ, and then it stops, as +shown in Fig. 2. We investigate the age performance of this +scheme in Theorem 3. +Theorem 3 If B = nλ, the average version age of a node in +a fully-distributed gossip network scales as O(1). +Proof: From Fig. 2, we observe that at any given time, if +there is any effective gossiping, only the minimum age node +is responsible for it. This is because, effective gossiping is +possible only if a single node is gossiping and in that case, the +node has to be a minimum age node. Whereas, when multiple +nodes are gossiping with rate B, there will be no effective +gossiping due to interference. Additionally, each update from +the source is a Poisson arrival with rate λ, and gossiping starts +immediately for a time duration of δ. Hence, we can model +this process as an M/D/∞ queue. Now, from [28], [29], we + +version age +t +t(1) +1 +t(2) +1 +t(1) +2 +t(2) +2 +t(1) +3 +t(2) +3 +t(1) +4 +t(2) +4 +t(1) +5 +t(1) +6 +t(2) +5 +t(2) +6 +t(1) +7 +δ +δ +number of entries in M/D/∞ queue +effective gossiping +interference +1 +2 +1 +2 +t +∆1(t) +∆2(t) +∆min(t) +Fig. 2. Distributed gossiping in a 2 node network. At each t(i) +k , ∆i(t) becomes zero and node i starts gossiping for a δ duration. The corresponding M/D/∞ +queue indicates the number of nodes gossiping simultaneously. Effective gossiping only happens when only one node is gossiping. Presence of multiple +gossiping nodes creates interference, resulting in no net gossip. +know that the stationary distribution for any general M/G/∞ +queue follows the Poisson distribution, +πk = (λ/µ)ke−λ/µ +k! +, +k = 0, 1, 2, . . . +(19) +For this M/D/∞ queue, µ = 1 +δ . Since effective gossip happens +only when there is one entry in the queue, the effective gossip +rate becomes +˜B = π1B = λδe−λδB. +(20) +The rest of the analysis is the same as in Theorem 1. Therefore, +we can directly substitute ˜B instead of B in (12) to obtain the +mean age of the ith node as +ai = +λe + +˜ +B +n−1 +λe +λ +λ +n + +˜ +B +n−1 +. +(21) +Using B = nλ and taking n → ∞ in (21), we get the age +scaling as +lim +n→∞ ai = lim +n→∞ +λe + λδe−λδnλ +n−1 +λe +λ +λ +n + λδe−λδnλ +n−1 +(22) += λe +λ +� +1 + +1 +λδe−λδ +� +, +(23) +which concludes the proof. ■ +Finally, we note that the age expression in (23) for the fully- +distributed gossiping scheme depends on the chosen gossiping +duration δ. Thus, we can improve the age expression in (23) +by choosing an optimal δ that minimizes the mean age. Since +λδe−λδ ≤ 1 +e, the maxima being at δ∗ = 1 +λ, the lower bound +of mean age of distributed gossiping is λe +λ +� +1 + +1 +e−1 +� += (1 + +e) λe +λ . This result matches our intuition, because if δ is too +small, it will not allow sufficient time to gossip. On the other +hand, if δ is too large, there will not be effective gossiping +due to interference from simultaneous gossiping nodes. The +minimum age is achieved when the effective gossiping rate ˜B +is maximized, which is ˜B|δ∗ = B +e . +V. NUMERICAL RESULTS +In this section, we present simulation results for the two +proposed gossiping schemes, and compare them with the +theoretically derived age expressions. We also show the results +for ASUMAN [1] as a benchmark. +In Fig. 3, we present the numerical results for λe +λ = 0.4, +λe +λ += 1 and +λe +λ += 2 in Fig. 3(a), Fig. 3(b) and Fig. 3(c), +respectively, with λ = 1 in all cases. From the figures, +it is evident that all the gossiping schemes result in O(1) +performance and the semi-distributed gossiping scheme yields +the best performance among all. +In Fig. 3(a), where λe +λ = 0.4 < +1 +e−1, ASUMAN gives the +worst age performance among the three schemes. However, +in Fig. 3(b) and Fig. 3(c), i.e., for +λe +λ +> +1 +e−1, ASUMAN +performs worse than the semi-distributed scheme, but is better +than the fully-distributed scheme. This matches our intuition +because, in ASUMAN, we use the information about source +self-updates to allocate gossip rate more efficiently, while in +the fully-distributed scheme, multiple nodes gossiping together +causes interference to lose some portion of the total gossip +rate. This effect of interference becomes more prominent when +the source to network update rate λ is high as compared +to source self-update rate λe. We have chosen δ = +1 +λ = 1 +for the simulation to get the minimum average age for fully- +distributed gossiping. +For +ASUMAN, +the +asymptotic +age +scales +as +limn→∞ λe +λ +� +1+ +n +n−1 (1+ λ +λe ) +1 +n + +n +n−1 +� += 2 λe +λ + 1, while the other two + +0 +100 +200 +300 +400 +500 +600 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +Theorem 1 formula (12) +semi-distributed gossip +Theorem 3 formula (21) +fully-distributed gossip +ASUMAN +(a) λe +λ = 0.4 +0 +100 +200 +300 +400 +500 +600 +1 +1.5 +2 +2.5 +3 +3.5 +4 +Theorem 1 formula (12) +semi-distributed gossip +Theorem 3 formula (21) +fully-distributed gossip +ASUMAN +(b) λe +λ = 1 +0 +100 +200 +300 +400 +500 +600 +2 +3 +4 +5 +6 +7 +8 +Theorem 1 formula (12) +semi-distributed gossip +Theorem 3 formula (21) +fully-distributed gossip +ASUMAN +(c) λe +λ = 2 +Fig. 3. +Average version age of a single node versus the total number of +nodes in the network n for semi-distributed, fully-distributed and ASUMAN +schemes. +schemes obtain 2 λe +λ +and (1 + e) λe +λ , as shown in (13) and +(23) (with optimized δ), respectively, and as listed in Table I. +The numerical simulation results exactly match the derived +formulas. With an increase in the ratio +λe +λ , the average +age increases due to source being updated more frequently +compared to the network for all schemes, as we observe +going from Fig. 3(a) to Fig. 3(b) to Fig. 3(c). +VI. CONCLUSION +We proposed a semi-distributed and a fully-distributed +gossiping scheme for a fully-connected network. The semi- +distributed scheme allows the freshest node to communicate +in the network through pilot signals and to gossip with full +capacity. This scheme archives the lowest possible average +age for any symmetric network, with a constraint on the +instantaneous gossip rate. On the other hand, in the fully- +distributed scheme, the freshest node gossips for a fixed time +duration with full capacity. The effective gossip happens only +a fraction of the total time, when there is no interference +from multiple nodes gossiping. Both of the proposed schemes +yield O(1) age performance. Compared to our previous work +ASUMAN, which also gives O(1) age scaling, this work is an +improvement because here we do not require the nodes to be +age-aware or to transmit pilot signals for channel reservation. +REFERENCES +[1] P. Mitra and S. Ulukus. ASUMAN: Age sense updating multiple access +in networks. In Allerton Conference, September 2022. +[2] Y. Minsky. +Spreading Rumors Cheaply, Quickly, and Reliably. +PhD +thesis, Cornell University, March 2002. +[3] D. Shah. Gossip algorithms. Foundations and Trends in Networking, +3(1):1–125, 2008. +[4] S. Sanghavi, B. Hajek, and L. Massoulie. +Gossiping with multiple +messages. +IEEE Trans. on Information Theory, 53(12):4640–4654, +December 2007. +[5] R. D. Yates. The age of gossip in networks. In IEEE ISIT, July 2021. +[6] S. K. Kaul, M. Gruteser, V. Rai, and J. Kenney. Minimizing age of +information in vehicular networks. In IEEE Infocom, March 2011. +[7] A. Kosta, N. Pappas, and V. Angelakis. Age of information: A new +concept, metric, and tool. In Foundations and Trends in Networking, +volume 12, pages 162–259, November 2017. +[8] Y. Sun, I. Kadota, R. Talak, and E. H. Modiano. Age of information: A +new metric for information freshness. In Age of Information, volume 12, +pages 1–224, December 2019. +[9] R. D. Yates, Y. Sun, D. Brown, S. K. Kaul, E. Modiano, and S. Ulukus. +Age of information: An introduction and survey. IEEE Jour. on Selected +Areas in Communications, 39(5):1183–1210, May 2020. +[10] J. Cho and H. Garcia-Molina. Effective page refresh policies for web +crawlers. ACM Trans. on Database Systems, 28(4):390–426, December +2003. +[11] J. Zhong, R. D. Yates, and E. Soljanin. Two freshness metrics for local +cache refresh. In IEEE ISIT, June 2018. +[12] A. Maatouk, S. Kriouile, M. Assaad, and A. Ephremides. The age of +incorrect information: A new performance metric for status updates. +IEEE/ACM Trans. on Networking, 28(5):2215–2228, October 2020. +[13] M. Bastopcu and S. Ulukus. Who should Google Scholar update more +often? In IEEE Infocom, July 2020. +[14] B. Abolhassani, J. Tadrous, A. Eryilmaz, and E. Yeh. Fresh caching for +dynamic content. In IEEE Infocom, May 2021. +[15] M. Wang, W. Chen, and A. Ephremides. Reconstruction of counting +process in real-time: The freshness of information through queues. In +IEEE ICC, July 2019. +[16] M. Bastopcu and S. Ulukus. Information freshness in cache updating +systems. IEEE Trans. on Wireless Communications, 20(3):1861–1874, +March 2021. + +[17] M. Bastopcu and S. Ulukus. +Maximizing information freshness in +caching systems with limited cache storage capacity. +In Asilomar +Conference, November 2020. +[18] P. Kaswan, M. Bastopcu, and S. Ulukus. Freshness based cache updating +in parallel relay networks. In IEEE ISIT, July 2021. +[19] M. Bastopcu and S. Ulukus. +Timely tracking of infection status of +individuals in a population. In IEEE Infocom, May 2021. +[20] R. D. Yates. Timely gossip. 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Bolch, S. Greiner, H. de Meer, and K. S. Trivedi. +Queueing +Networks and Markov Chains: Modeling and Performance Evaluation +with Computer Science Applications. John Wiley & Sons, 2006. +[29] G. F. Newell. The M/G/∞ queue. SIAM Journal on Applied Mathe- +matics, 14(1):86–88, January 1966. + diff --git a/FdAyT4oBgHgl3EQf4_rs/content/tmp_files/load_file.txt b/FdAyT4oBgHgl3EQf4_rs/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..81d1cc5c8a576922fed818ddc9beb687ae88ed32 --- /dev/null +++ b/FdAyT4oBgHgl3EQf4_rs/content/tmp_files/load_file.txt @@ -0,0 +1,393 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf,len=392 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='00798v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='IT] 2 Jan 2023 Timely Opportunistic Gossiping in Dense Networks Purbesh Mitra Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 pmitra@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='edu ulukus@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='edu Abstract—We consider gossiping in a fully-connected wireless network consisting of n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The network receives Poisson up- dates from a source, which generates new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The nodes gossip their available information with the neighboring nodes to maintain network timeliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this work, we propose two gossiping schemes, one semi-distributed and the other one fully- distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In the semi-distributed scheme, the freshest nodes use pilot signals to interact with the network and gossip with the full available update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In the fully-distributed scheme, each node gossips for a fixed amount of time duration with the full update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Both schemes achieve O(1) age scaling, and the semi-distributed scheme has the best age performance for any symmetric randomized gossiping policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We compare the results with the recently proposed ASUMAN scheme [1], which also gives O(1) age performance, but the nodes need to be age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' INTRODUCTION Gossiping is an information sharing mechanism where nodes transmit their own data to the neighboring nodes ran- domly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Gossiping does not require any centralized schedul- ing and is particularly suitable for communication in dense networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' There are many gossip algorithms in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', [2]–[4], that focus on maximizing the effectiveness of information dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Gossip algorithms have been studied from a timeliness point of view in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The analysis in [5] uses the version age metric, which is one of the measures of information freshness in the literature [6]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The analysis in [5] shows that for a fully-connected network of n nodes, if each node gossips with a fixed rate λ, the average version age of any individual node scales as O(log n) with the network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Subsequent works [20]–[26] show that there can be improvements in the age scaling by introducing particular network mechanisms, such as clustering, file slicing and network coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this paper, we focus on another aspect of existing gossip schemes, which is their uniform gossip rate assignment to all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' A main drawback of uniform rate gossiping is that it allocates the same gossip rate to nodes with relatively stale and relatively fresh information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This negatively impacts the timeliness performance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Our goal in this paper is to efficiently and distributedly allocate the total network gossip capacity dynamically among the users, thereby enabling opportunistic gossiping, where fresher nodes gossip with higher gossip rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The first paper to address the inefficiency of uniform rate gossiping is [1] which proposed the ASUMAN scheme, which is an opportunistic gossiping scheme that relies on the assumption that the nodes are age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In ASUMAN, since the nodes are age-aware, whenever the source updates itself, all the nodes in the network get synchronized and a new gossiping frame starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' When a new frame starts, the nodes stop gossiping and send a small pilot signal after waiting for a back-off period proportional to their current age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this way, the freshest nodes get to start gossiping first as their back-off period is smallest and the relatively staler nodes do not gossip after receiving the pilot signal from the freshest nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' If in any frame, the number of fresh nodes is more than one, then that number is estimated from the received pilot signals and the total update rate B = nλ is equally divided between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The analysis in [1] shows that the version age of an individual node scales as O(1) with the network size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Although ASUMAN achieves better age performance, the system model poses some challenges in real-life implementa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' One such challenge is that when multiple nodes have the same minimum age, all of them transmit the pilot signal simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Thus, multiple short signals overlap over the air, which leads to incorrect estimation of the minimum- age nodes, causing interference within the gossiping nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Another downside of ASUMAN is that the nodes have to be age-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This can be achieved if the source sends a signal to the nodes when it updates itself, adding additional complexity to the simple gossiping model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' gossiping scheme age scaling ASUMAN proposed in [1] 2 λe λ + 1 semi-distributed proposed here 2 λe λ fully-distributed proposed here (1 + e) λe λ TABLE I AGE SCALING COMPARISON FOR DIFFERENT GOSSIPING SCHEMES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this paper, we propose two new gossiping schemes, one semi-distributed and the other fully-distributed, that both yield O(1) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' These schemes are able to circumvent the previously mentioned downsides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In the semi-distributed scheme, each time a node gets updated by the source, it transmits a pilot signal to the neighboring nodes and starts gossiping with the maximum capacity until it receives a signal from some other node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In the fully-distributed scheme, each time a node gets updated by the source, it gossips for a fixed duration with the maximum capacity and stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The age scaling comparison of these schemes is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Further, we prove that the semi-distributed gossiping scheme yields the best age performance among all possible symmetric gossiping schemes with an upper bound on the instantaneous maximum gossip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' For our analysis, we use stochastic hybrid system (SHS) formulation [27], similar to [1], [5], to calculate of mean steady-state version age of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' SYSTEM MODEL We consider a gossip network consisting of a source labeled node 0, and a set of nodes labeled N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', n}, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The source updates its information with Poisson arrivals of rate λe, and it sends Poisson updates to the network with a total rate of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' For simplicity, we consider a symmetric network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', each of the nodes receives updates from the source with a rate λ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In the timely gossiping papers in the literature [5], [20]–[26], it is assumed that each node of the network gossips with a rate of λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' thus, on a fully connected network where each node is connected to (n − 1) other nodes, each node i gossips with a node j with a rate of λ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Therefore, the total update capacity of the network is B = nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' As in [1], in this paper, we consider allocating this total update rate B to users dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Once a gossip rate is assigned to a node, it gossips with its (n − 1) neighbors with equal rates in the fully connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Thus, the network has an upper bound of B on the instan- taneous gossiping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' If at any time, multiple nodes transmit and the total instantaneous gossip rate exceeds B, there will be interference, and the gossiped data is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Hence, for effective gossiping, at any time instant, the total instantaneous gossip rate has to be less than or equal to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The goal of our work is to improve the timeliness of such a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' To measure the timeliness of the ith node, we use version age, denoted as ∆i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This measure counts how many versions the data at the ith node is lagging, compared to the data available at the source at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Mathematically, we write ∆i(t) = Ns(t) − Ni(t), (1) where Ns(t) and Ni(t) are the versions of the data available at the source and at the ith node, respectively, at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We denote all the ages of nodes at time t as the age vector ∆(t) = [∆1(t), ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' , ∆n(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' When the source updates itself, all the ages of the nodes increase by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' If the source sends an update to a node, its age becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' When node i sends a gossip update to node j, it stores the data with the freshest version, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', the age of the jth node becomes ˆ∆j(t) = ∆{i,j}(t) = min{∆i(t), ∆j(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' SEMI-DISTRIBUTED GOSSIPING In this section, we introduce the semi-distributed gossiping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The motivation for this is to allow the freshest node of the network to gossip with maximum capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Suppose we denote the kth source-to-ith node update as t(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this scheme, at time t(i) k , i transmits a small pilot signal to all the other nodes in the network and starts gossiping with rate B to the other nodes with equal rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' While gossiping, if i receives a pilot signal from any other node, it will stop gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We define the gossiping node at any given time t as M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Since, the probability of two simultaneous Poisson arrivals is 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', P(|M(t)| ≥ 2) = 0, here we do not face the problem of overlapping pilot signals like ASUMAN [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 0 1 2 3 4 λe λ 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Source 0 updates itself with rate λe and sends updates to the nodes N = {1, 2, 3, 4, 5} uniformly with total rate λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', with rate λ/5 to each of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The nodes gossip with each other with total update rate B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We investigate the mean steady-state age of an individual node, denoted as, ai = lim t→∞ ai(t) = lim t→∞ E[∆i(t)], (2) in particular, how network size n affects ai, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Theorem 1 If B = nλ, the average version age of a node ai in a semi-distributed gossip network scales as O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Proof: We use SHS formulation of [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Note that, for any time t, the gossiping node is the minimum age node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Let us denote this minimum age as ∆min(t) = min{∆1(t), ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' , ∆n(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' From [5], we know that limt→∞ E[∆min(t)] = λe λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Since for any given t, only the node with the minimum age is gossiping, we can express the state transition of the system as an SHS with only one type of transition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We choose the test function ψi : Rn × [0, ∞) → R, where i ∈ N, as ψi(∆(t), t) = ∆i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (3) Now, following [27, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 1], we evaluate the extended gen- erator function as E[(Lψi)(∆(t), t)] = � (j,ℓ)∈L λj,ℓ(∆(t), t)E � ψi(φj,ℓ(∆(t), t)) − ψi(∆(t), t) � , (4) where L denotes all possible state transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We define the reset maps φj,ℓ(∆(t), t) = ˆ∆(t) = [ ˆ∆1(t), ˆ∆2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' , ˆ∆n(t)] as follows ˆ∆i(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∆i(t) + 1, if j = 0, ℓ = 0 0, if j = 0, ℓ = i min(∆j(t), ∆ℓ(t)), if j ∈ N, ℓ = i ∆i(t), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (5) The update rates λj,ℓ are given as λj,ℓ(∆(t), t) = \uf8f1 \uf8f2 \uf8f3 λe, if j = 0, ℓ = 0 λ n, if j = 0, ℓ = i B n−1 1{j = M(t)}, otherwise, (6) where 1{·} denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Now, we can rewrite (4) as E[(Lψi)(∆(t), t)] = E � λe(∆i(t) + 1 − ∆i(t)) + λ n(0 − ∆i(t)) + � j∈N B n − 1 1{j = M(t)} � ∆{j,i}(t) − ∆i(t) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (7) Since the gossiping node is always the minimum age node, we can write E[(Lψi)(∆(t), t)] = λe − λ nai(t) + E � � j=M(t) B n − 1(∆min(t) − ∆i(t)) � = λe − λ nai(t) + B n − 1(amin(t) − ai(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (8) Now, since the version age is a piece-wise constant function of time, we obtain dE[ψi(∆(t), t)] dt = dE[∆i(t)] dt = 0, (9) for any continuity point t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Hence, the expected value in (8) is 0, by Dynkin’s formula, as given in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Thus, (8) becomes 0 = λe − λ nai(t) + B n − 1(amin(t) − ai(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (10) Hence, the mean age of an individual node is expressed as ai(t) = λe + B n−1amin(t) λ n + B n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (11) To evaluate the steady-state mean age, we take t → ∞ in (11) which gives ai = λe + B n−1 λe λ λ n + B n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (12) Finally, to calculate the scaling of the average age, we use B = nλ, which yields lim n→∞ ai = lim n→∞ λe λ � 1 + n n−1 1 n + n n−1 � = 2λe λ , (13) concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' ■ Next, we show that this semi-distributed scheme gives the best version age performance for any possible gossip- ing scheme with a constraint on the instantaneous gossiping scheme, in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Theorem 2 For any symmetric network with maximum in- stantaneous gossip rate of B, the semi-distributed gossiping scheme yields the minimum average age for the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Proof: Suppose we use any arbitrary gossiping policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Since the total gossip rate is upper bounded by B, we have � j,i∈N,j̸=i λj,i(∆(t), t) ≤ B, ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (14) From the symmetry of the network, we can write E \uf8ee \uf8f0 � j∈N,j̸=i λj,i(∆(t), t) \uf8f9 \uf8fb ≤ B n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (15) Note that the sum in (14) is over all i, j whereas the sum in (15) is over j only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Now, equating the extended generator function to 0, yields λ nai(t) + E \uf8ee \uf8f0 � j∈N,j̸=i λj,i(∆(t), t)∆i(t) \uf8f9 \uf8fb = λe + E \uf8ee \uf8f0 � j∈N,j̸=i λj,i(∆(t), t)∆{j,i}(t) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (16) Using the inequality in (15) and by definition the fact that ∆{j,i}(t) ≥ ∆min(t), we can rewrite (16) as λ nai(t) + B n − 1ai(t) ≥ λe + B n − 1amin(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (17) Taking t → ∞ in (17) and using the expression of amin(t), we obtain ai ≥ λe + B n−1 λe λ λ n + B n−1 , (18) where the right-hand side of the inequality is the average age of a node with the proposed semi-distributed policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' ■ IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' FULLY-DISTRIBUTED GOSSIPING In this section, we introduce a gossiping policy which is fully-distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In ASUMAN [1], the nodes need to be age- aware and in the semi-distributed scheme, the nodes need to implement a pilot-signal based communication in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We improve upon them and formulate a gossiping policy that does not require age-awareness or pilot-signal transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In this scheme, whenever node i receives an update from the source at time t(i) k , it starts gossiping to all the other nodes with rate B for a fixed time duration δ, and then it stops, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We investigate the age performance of this scheme in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Theorem 3 If B = nλ, the average version age of a node in a fully-distributed gossip network scales as O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Proof: From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 2, we observe that at any given time, if there is any effective gossiping, only the minimum age node is responsible for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This is because, effective gossiping is possible only if a single node is gossiping and in that case, the node has to be a minimum age node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Whereas, when multiple nodes are gossiping with rate B, there will be no effective gossiping due to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Additionally, each update from the source is a Poisson arrival with rate λ, and gossiping starts immediately for a time duration of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Hence, we can model this process as an M/D/∞ queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Now, from [28], [29], we version age t t(1) 1 t(2) 1 t(1) 2 t(2) 2 t(1) 3 t(2) 3 t(1) 4 t(2) 4 t(1) 5 t(1) 6 t(2) 5 t(2) 6 t(1) 7 δ δ number of entries in M/D/∞ queue effective gossiping interference 1 2 1 2 t ∆1(t) ∆2(t) ∆min(t) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Distributed gossiping in a 2 node network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' At each t(i) k , ∆i(t) becomes zero and node i starts gossiping for a δ duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The corresponding M/D/∞ queue indicates the number of nodes gossiping simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Effective gossiping only happens when only one node is gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Presence of multiple gossiping nodes creates interference, resulting in no net gossip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' know that the stationary distribution for any general M/G/∞ queue follows the Poisson distribution, πk = (λ/µ)ke−λ/µ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' , k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (19) For this M/D/∞ queue, µ = 1 δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Since effective gossip happens only when there is one entry in the queue, the effective gossip rate becomes ˜B = π1B = λδe−λδB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (20) The rest of the analysis is the same as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Therefore, we can directly substitute ˜B instead of B in (12) to obtain the mean age of the ith node as ai = λe + ˜ B n−1 λe λ λ n + ˜ B n−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' (21) Using B = nλ and taking n → ∞ in (21), we get the age scaling as lim n→∞ ai = lim n→∞ λe + λδe−λδnλ n−1 λe λ λ n + λδe−λδnλ n−1 (22) = λe λ � 1 + 1 λδe−λδ � , (23) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' ■ Finally, we note that the age expression in (23) for the fully- distributed gossiping scheme depends on the chosen gossiping duration δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Thus, we can improve the age expression in (23) by choosing an optimal δ that minimizes the mean age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Since λδe−λδ ≤ 1 e, the maxima being at δ∗ = 1 λ, the lower bound of mean age of distributed gossiping is λe λ � 1 + 1 e−1 � = (1 + e) λe λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This result matches our intuition, because if δ is too small, it will not allow sufficient time to gossip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' On the other hand, if δ is too large, there will not be effective gossiping due to interference from simultaneous gossiping nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The minimum age is achieved when the effective gossiping rate ˜B is maximized, which is ˜B|δ∗ = B e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we present simulation results for the two proposed gossiping schemes, and compare them with the theoretically derived age expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We also show the results for ASUMAN [1] as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3, we present the numerical results for λe λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='4, λe λ = 1 and λe λ = 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(a), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(c), respectively, with λ = 1 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' From the figures, it is evident that all the gossiping schemes result in O(1) performance and the semi-distributed gossiping scheme yields the best performance among all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(a), where λe λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='4 < 1 e−1, ASUMAN gives the worst age performance among the three schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' However, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=', for λe λ > 1 e−1, ASUMAN performs worse than the semi-distributed scheme, but is better than the fully-distributed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This matches our intuition because, in ASUMAN, we use the information about source self-updates to allocate gossip rate more efficiently, while in the fully-distributed scheme, multiple nodes gossiping together causes interference to lose some portion of the total gossip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This effect of interference becomes more prominent when the source to network update rate λ is high as compared to source self-update rate λe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' We have chosen δ = 1 λ = 1 for the simulation to get the minimum average age for fully- distributed gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' For ASUMAN, the asymptotic age scales as limn→∞ λe λ � 1+ n n−1 (1+ λ λe ) 1 n + n n−1 � = 2 λe λ + 1, while the other two 0 100 200 300 400 500 600 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='8 2 Theorem 1 formula (12) semi-distributed gossip Theorem 3 formula (21) fully-distributed gossip ASUMAN (a) λe λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='4 0 100 200 300 400 500 600 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content='5 4 Theorem 1 formula (12) semi-distributed gossip Theorem 3 formula (21) fully-distributed gossip ASUMAN (b) λe λ = 1 0 100 200 300 400 500 600 2 3 4 5 6 7 8 Theorem 1 formula (12) semi-distributed gossip Theorem 3 formula (21) fully-distributed gossip ASUMAN (c) λe λ = 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Average version age of a single node versus the total number of nodes in the network n for semi-distributed, fully-distributed and ASUMAN schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' schemes obtain 2 λe λ and (1 + e) λe λ , as shown in (13) and (23) (with optimized δ), respectively, and as listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The numerical simulation results exactly match the derived formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' With an increase in the ratio λe λ , the average age increases due to source being updated more frequently compared to the network for all schemes, as we observe going from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(a) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(b) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' CONCLUSION We proposed a semi-distributed and a fully-distributed gossiping scheme for a fully-connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The semi- distributed scheme allows the freshest node to communicate in the network through pilot signals and to gossip with full capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' This scheme archives the lowest possible average age for any symmetric network, with a constraint on the instantaneous gossip rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' On the other hand, in the fully- distributed scheme, the freshest node gossips for a fixed time duration with full capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' The effective gossip happens only a fraction of the total time, when there is no interference from multiple nodes gossiping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Both of the proposed schemes yield O(1) age performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' Compared to our previous work ASUMAN, which also gives O(1) age scaling, this work is an improvement because here we do not require the nodes to be age-aware or to transmit pilot signals for channel reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdAyT4oBgHgl3EQf4_rs/content/2301.00798v1.pdf'} +page_content=' REFERENCES [1] P.' metadata={'source': 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b/GNFJT4oBgHgl3EQfDyxI/content/tmp_files/2301.11435v1.pdf.txt @@ -0,0 +1,1472 @@ +Learning Modulo Theories +Matt Fredrikson 1 Kaiji Lu 1 Saranya Vijayakumar 1 Somesh Jha 2 Vijay Ganesh 3 Zifan Wang 1 +Abstract +Recent techniques that integrate solver layers +into Deep Neural Networks (DNNs) have shown +promise in bridging a long-standing gap between +inductive learning and symbolic reasoning tech- +niques. In this paper we present a set of tech- +niques for integrating Satisfiability Modulo Theo- +ries (SMT) solvers into the forward and backward +passes of a deep network layer, called SMTLayer. +Using this approach, one can encode rich domain +knowledge into the network in the form of mathe- +matical formulas. In the forward pass, the solver +uses symbols produced by prior layers, along with +these formulas, to construct inferences; in the +backward pass, the solver informs updates to the +network, driving it towards representations that +are compatible with the solver’s theory. Notably, +the solver need not be differentiable. We imple- +ment SMTLayer as a Pytorch module, and our +empirical results show that it leads to models that +1) require fewer training samples than conven- +tional models, 2) that are robust to certain types +of covariate shift, and 3) that ultimately learn +representations that are consistent with symbolic +knowledge, and thus naturally interpretable. +1. Introduction +A recent class of techniques aims at integrating solver +layer(s) within deep neural networks (DNNs) (Wang et al., +2019a; Pogancic et al., 2020; Huang et al., 2021; Manhaeve +et al., 2018), both during training and inference. A class +of problems which can benefit from such an integration is +one that has both a perceptual and a symbolic sub-problem, +such as “visual” Sudoku (Wang et al., 2019a), or the prob- +lem of determining the shortest path from a picture of a +map (Pogancic et al., 2020). +The most straightforward way to incorporate a solver layer +1Carnegie Mellon University, Pittsburgh, PA, USA 2University +of Wisconsin-Madison, Madison, WI, USA 3University of Water- +loo, Waterloo, ON, Canada. Correspondence to: Matt Fredrikson +. +into an ML model is to learn models with representations +that are compatible with symbols used by the solver. For +example, if one wanted to leverage symbolic domain knowl- +edge to classify images of birds, or diagnose ailments from +CT scans, then one could train a model in a fashion similar +to “concept bottlenecking” (Koh et al., 2020). This requires +detailed labels for supervision, which may be prohibitively +expensive to obtain and keep consistent with a potentially +evolving domain theory. +We present a set of techniques for incorporating a Satisfia- +bility Modulo Theories (SMT) solver into a DNN layer so +that symbolic knowledge can be leveraged to learn such a +compatible representation, without requiring label super- +vision. Our approach is general, and can handle a broad +range of domain knowledge encoded as SMT constraints, +provided that they interface with the surrounding neural net- +work layers over propositional variables. Unlike the most +closely related prior work (Wang et al., 2019a), our approach +does not approximate the solver’s behavior by formulating +a differentiable relaxation. Rather, we extract information +from the solver as it works on a set of constraints, that is +geared towards checking the correctness of the output of the +model that precedes the solver, and use that information to +construct updates to the model during training (Section 4.2). +We present two different approaches for this, one based +on unsatisfiable cores, and another based on weighted +MaxSMT (Section 4.2). There are several advantages to +this approach. Aside from the mild interface constraints +mentioned earlier (i.e., solver and neural layers interface +with each other via boolean variables), our approach does +not place any restrictions on the theory solver embedded in +the layer, such as linearity (Pogancic et al., 2020) or even +decidability—if the solver is capable of efficiently discharg- +ing the relevant constraints, then the layer can operate as +intended. Because there is no need to provide a differen- +tiable relaxation for each theory or solver technique that one +may want to incorporate, we can leverage the continuous +and unabated progress being made in solver technology. +We implement our approach as a PyTorch (Paszke et al., +2019) layer, using the Z3 (De Moura & Bjørner, 2008) SMT +solver as the solver layer to solve SMT and MaxSMT con- +straints. On three applications involving vision and natural +language: visual arithmetic, algebraic equation solving, and +arXiv:2301.11435v1 [cs.LG] 26 Jan 2023 + +Learning Modulo Theories +a so-called natural language “liar’s puzzle,” we demonstrate +that our implementation can be incorporated into DNN ar- +chitectures to solve problems more effectively than conven- +tional DNNs (Section 5). In particular, our results show that +the data needed to train a DNN with symbolic knowledge +may be much simpler than may be necessary otherwise, and +that while doing so is more expensive computationally, of- +ten times the more efficient (i.e., not involving MaxSMT) +algorithms perform well in practice. +Our contributions are as follows: +1. We present SMTLayer, a framework for incorporating +an SMT solver into a DNN, as a layer that leverages +symbolic knowledge during training and inference. +2. We prototype our approach in Pytorch1, and show that +it can be applied to solve a range of problems that +incorporate symbolic knowledge. +3. Our empirical evaluation, over four diverse applica- +tions, shows that models using SMTLayer require sig- +nificantly less training data, can be trained more ef- +ficiently, and are more robust than those based on +closely-related prior work (Wang et al., 2019a; Huang +et al., 2021). +Section 3 provides background on ERM and the first- +order theories used in our framework. Section 4 describes +SMTLayer, Section 5 gives our empirical evaluation, and +Section 6 concludes the paper. +2. Related Work +Combining logical solvers and deep models can be diffi- +cult because logic has discrete structure while the most +successful way to construct neural networks today requires +differentiability (Riegel et al., 2020). +Combinatorial Solver Layers. +Vlastelica et al. (2019) in- +tegrate a blackbox, non-differentiable combinatorial solver +on top of a deep network. To propagate the gradient through +the solver on the backward pass, they linearly interpolate +the loss w.r.t the solver’s input and define the gradient of the +solver as the slopes of the line segments. CSL solves a set +of problems where the solver’s objective must be linear w.r.t +its input, e.g. finding the shortest path and travel salesman +problem (TSP). Further, the authors assume that the only +labels available are the outputs of solvers, e.g. the minimum +cost in TSP, and hence their tool has to discover the label for +the output of the network itself. These requirements limit +the choices one has for the solver layer. +1We plan to release our implementation as an open source +library upon publication of this paper +Neural Logic Programming. +While SATNet integrates +a logic-based solver on top of a network, DeepProbLog +takes the opposite approach, extending the capability of a +probabilistic logic programming language with neural pred- +icates Manhaeve et al. (2018). In the context of our work, +the logic program can be viewed as a “solver layer” that +explicitly encodes symbolic knowledge. Scallop (Huang +et al., 2021) extends DeepProbLog to scale without sacri- +ficing accuracy compared to DeepProbLog. Similarly to +DeepProbLog, each possible result of the sum of two digits +in MNIST is given a probability, in the form of a weighted +Boolean formula. They prune unlikely clauses of the for- +mula, represented by proofs, only keeping the top-k most +likely. Likelihood is computed using weighted model count- +ing (Huang et al., 2021; Chavira & Darwiche, 2008). These +techniques are well-suited to problems that benefit from +probabilistic Datalog, but have inherent limitations: they +cannot handle quantifiers, general negation, and the range +of supported first-order theories is more restrictive. +SATNet. +Wang et al. (2019b) present SATNet, a network +architecture with a differentiable approximate MAXSAT +solver layer. Their approximation is based on a coordinate +descent approach to solving the semidefinite program (SDP) +relaxation of the MAXSAT problem. SATNet does not as- +sume that the logical structure of the problem is given, and +instead attempts to learn it. By placing the MAXSAT solver +layer on top of a convolution network to learn represen- +tations from images, SATNet directly solve problems like +Visual Sudoku, for which neural networks alone are not well +suited (Wang et al., 2019b). +Differentiable Logic. +Another recent direction has ex- +plored differentiable logics (Fischer et al., 2019; Varnai & +Dimarogonas, 2020; van Krieken et al., 2022). These ap- +proaches provide ways of integrating symbolic knowledge +into training, by making logical formulas differentiable, and +therefore amenable to optimization when included in a loss +function. This line of work does not explicitly aim to make +use of symbolic information during inference. In contrast, +the information that our approach extracts from the solver +during training is used to condition the model towards a rep- +resentation that will allow it to communicate effectively with +the solver during inference. Additionally, we do not require +the logical formulas, or the solver, to be differentiable. +3. Background +Let X denote a domain of features, Y a domain of labels, +and D a distribution over X × Y. Formally, D is a proba- +bility measure on a space given by a σ-algebra over subsets +of ℘(X × Y). The goal of a learning algorithm A is to +find a function h : X → Y that, for (x, y) ∼ D, can be +used to predict y when given x. To do this, A is given a set + +Learning Modulo Theories +of training examples S = (x1, y1), . . . , (xm, ym) sampled +i.i.d. from D, and uses some criterion to select h from a +hypothesis class H of functions. We refer to h as the model +learned by A on S. When the learning algorithm A is clear +from the context, we will write hS to denote the model pro- +duced from the given sample. Throughout this paper, we +will generally assume that the loss is either the 0-1 loss ℓ01 +or binary cross-entropy ℓbce. +A theory T consists of a signature Σ of constant, predicate, +and function symbols, as well as a set of axioms over Σ. +Formulas in a theory are composed of elements of Σ, vari- +ables, and logical symbols such as quantifiers and Boolean +operations. We use the term decision procedure to refer to +an algorithm that is given an open T-formula, and returns +true if it is satisfiable, and false otherwise. Additionally, it +may return an assignment to all of the variables that demon- +strates satisfiability, or if the formula is not satisfiable, then +it may return an unsatisfiable core, which is a subset of +clauses taken from the formula’s representation in conjunc- +tive normal form that remains unsatisfiable. Loosely, we +also refer to such an algorithm as a “solver”, but this term is +more general, and could also refer to an algorithm that iden- +tifies the maximal set of clauses, possibly weighted by some +user-defined values, that are satisfiable when conjoined. +4. Constructing SMTLayer +In this section, we present SMTLayer, a set of algorithms +for computing the forward and backward passes of a layer +whose behavior is defined by a set of user-defined SMT +constraints. SMTLayer does not have trainable parame- +ters, and its functionality is wholly defined by a set of +SMT constraints φ that are provided by the model de- +signer. SMTLayer can be used in modern deep-learning +frameworks as a drop-in replacement for more conven- +tional neural network layers, e.g., dense, convolutional, and +LSTM (Hochreiter & Schmidhuber, 1997) are prominent +examples of widely-used layers. +Section 4.1 provides a high-level overview of our approach, +Section 4.2 describes them in detail, and Section 4.3 begins +an analysis of this setting that we hope future work will +continue developing. +4.1. Overview +We envision SMTLayer being used primarily at the top of +a DNN taking inputs from a stack of conventional DNN +layers that convert raw input features into ground terms for +the constraints φ(z0, . . . , zp−1, y0, . . . , yq−1) embedded in +SMTLayer, and producing outputs that are consistent with +φ and the given ground terms. Figure 1 shows an illustrative +example, with the previously-studied problem of MNIST +addition (Manhaeve et al., 2018; Huang et al., 2021). +Algorithm 1 Fφ +max(z) +MaxSMT-based forward pass of SMTLayer +Inputs: z ∈ Rp layer input +φ(z0, . . . , zp−1, y0, . . . , yq−1) T-formula +Output: y ∈ Rq +1 begin +2 +zb ← [z[i] > 0 : i = 0 . . . p − 1] +3 +C ← � +i∈I softmax(|z|)[i] +4 +y ← arg maxyb maxI C·1(φ∧� +i∈I yi = yb[i]∧zi = zb[i]) +5 +return y +6 end +During the forward pass the outputs of the previous layer +are mapped to designated free variables z0, . . . , zp−1. The +layer then checks the satisfiability of φ, a formula in an +appropriate combination of first-order theories, after substi- +tuting these ground terms for the zi, and the output of the +layer consists of the solver’s model for y0, . . . , yq−1. These +outputs are converted from Boolean to floating-point values +by mapping false to -1 and true to 1. At the moment, the +only restriction on φ that our layer requires is that z and y +be vectors of Booleans, so that they can be appropriately +mapped to continuous values; any other symbols appear- +ing in φ can come from arbitrary domains (e.g. strings) +supported by the underlying SMT solver. +In the backward pass, the layer receives the gradient of +its output with respect to the function whose derivative +is being computed, which we will assume is the binary +cross-entropy loss ℓ(y, y⋆). Unless stated otherwise, we +will assume this loss for the remainder of the section. This +gradient is used, along with the inputs and outputs of the +corresponding forward pass, to first compute an amended +output ˆy which corresponds to an output that would have +yielded a smaller loss. Because the outputs are Boolean, +it is always possible to determine the ground truth output +y⋆ from this information. Using ˆy, the layer determines +which of components of its inputs are inconsistent with +φ and ˆy, and provides the corresponding gradients to the +previous layer. Section 4.2 details the manner in which these +gradients are computed. +4.2. SMTLayer, forward and backward +We now present the details of the forward and backward +passes of SMTLayer. There are two algorithms for each +pass, Fφ +max and Fφ +smt are forward passes, and Bφ +max, Bφ +core +are backward passes. Fφ +max and Bφ +max both make use of +MaxSMT solvers, whereas Fφ +smt and Bφ +core rely on satisfia- +bility solvers (SMT). Despite the symmetry in which type +of solver each algorithm uses, they are all compatible with +each other. That is, Fφ +smt can be used with either Bφ +max or +Bφ +core, and the same for Fφ +max. + +Learning Modulo Theories +Features X +Symbolic Domain Z +0010000111 +Neural +Network +φ( +z1∥ . . . ∥z10, +y +) ≡ +a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧ +a + b = y +Prediction Logic φ +Labels Y +{01011} +Satisfying +Assignments +Figure 1. MNIST Addition example. +Algorithm 2 Fφ +smt(z) +SMT-based forward pass of SMTLayer +Inputs: z ∈ Rp layer input +φ(z0, . . . , zp−1, y0, . . . , yq−1) T-formula +Output: y ∈ Rq +1 begin +2 +zb ← [z[i] > 0 : i = 0 . . . p − 1] +3 +ˆφ ← φ(zb[0], . . . , zb[p − 1]) +4 +if ˆφ is satisfiable then +5 +yb[0], . . . , yb[q − 1] ← solve(ˆφ, y0, . . . , yq−1) +6 +y ← [yb[i] > 0 +: i = 0 . . . q − 1] +7 +else +8 +y ← 0 +9 +end +10 +return y +11 end +Forward pass. +Algorithms 1 and 2 illustrate Fφ +max and +Fφ +smt, the methods for computing the forward pass based +on weighted MaxSMT and SMT, respectively. Both of the +algorithms are parameterized by a user-provided first-order +formula φ, and take a single vector-valued input consisting +of unscaled floating-point values (logits). These values are +cast to Boolean constants by taking their sign on line 2 +of both algorithms, so that they can be equated with the +corresponding free variables z0, . . . , zp−1. +The key difference between Fφ +max and Fφ +smt is the way in +which they handle inputs that are inconsistent with φ when +interpreted as Booleans. Fφ +smt addresses this by provid- +ing an output that is also inconsistent with φ, i.e. a vector +of zeroes, effectively signaling that the network below it +did not provide consistent inputs. Alternatively, we can +interpret the values provided by the network as Booleans +enriched with “confidence” values. Although we expect +inputs to SMTLayer to be unscaled floating-point values, +Algorithm 1 scales them to a formal probability distribution +via the softmax function (line 3) for use as weights to find +the weighted MaxSMT solution of φ. With this approach, +Algorithm 3 Bφ +max(z, y, ∂yℓ(y, y⋆)) +MaxSMT-based backward pass of SMTLayer +Inputs: z ∈ Rp input of forward pass +y ∈ Rq output of forward pass +∂yℓ(y, y⋆) gradient with respect to output +φ(z0, . . . , zp−1, y0, . . . , yq−1) a T-formula +Output: ∂zℓ(y, y⋆) ∈ Rp approximate gradient of ℓ +1 begin +2 +Gz ← ∂zℓ(z, sign(z)) +3 +ˆy = sign(y) − 2 · sign (∂yℓ(y, y⋆)) +4 +if sign(y) ̸= sign(ˆy) then +5 +zb ← [z[i] > 0 : i = 0 . . . p − 1] +6 +ˆyb ← [ˆy[i] > 0 : i = 0 . . . q − 1] +7 +φy ← � +0≤i 0 : i = 0 . . . p − 1] +6 +ˆyb ← [ˆy[i] > 0 : i = 0 . . . q − 1] +7 +φz, φy ← � +0≤i

0 and ∀S in the support of Dn, +Pr +(x,y)∼D[(x, y) ∈ S] = +Pr +(x,·)∼D[(x, g(⟨x⟩f,φ)) ∈ S] +where ⟨x⟩f,φ = {y : φ(f(x), y) is satisfied}. + +Learning Modulo Theories +In (2), f is called the grounding function and φ is called the +prediction logic. +Intuitively, a learning problem defined in terms of a distribu- +tion D and hypothesis class H is decomposable if members +of H can be decomposed into functions that are responsible +for grounding and prediction, and D can be expressed in +terms of a grounding function and a first-order formula φ. +There are a few important things to note. First, there is no re- +quirement that the grounding function f be a member of Hf. +While this may be realized at times, we should not assume +that the data is actually generated, or otherwise described +with perfect fidelity, by a function in the class that one learns +over. In fact, we do not assume that f is efficiently com- +putable, as it may correspond to a natural process, or an +aspect of data generation that is not understood well enough +to make such computational claims. +Second, for a given x, there may be more than one satis- +fying assignment for y to φ(f(x), y). The function g in +(2) accounts for this, requiring only that when solutions +to φ(f(x), y) are sampled by g, the result is distributed +identically to D. This paper will focus on the case where +satisfying assignments for y are unique, as these are more in +line with ”classic” ERM classification problems. We leave +exploration of the more general setting to future work. +We note that if the grounding function is known, can be +computed efficiently, and φ is efficiently solvable, then the +learning problem effectively has a closed-form solution. +Rather, we assume that only φ and perhaps g are known, +and a sample of D is given. The remaining challenge is +to identify a grounding hypothesis hf ∈ Hf for which the +construction in (2) is an effective solution to the end-to-end +learning problem posed by D, H. This stands in contrast to +traditional ERM, in which a good solution h ∈ H must ei- +ther solve both grounding and prediction, or find a “shortcut” +that manages to predict D as well as the decomposition. +Convergence. +Regarding the backward passes, Theo- +rem 2 below demonstrates that when φ satisfies certain +conditions, and the companion hypothesis class Hf satisfies +conditions that are sufficient to guarantee convergence with +SGD, then training with Fφ +smt and Bφ +max will converge to +the optimal solution in the number of iterations. The proof +of this theorem is based on the observation that when the +conditions on φ are met, then training with Bφ +max obtains the +same solution that would be obtained if the labels of φ were +available for supervised learning. Thus, the conditions on +Hf are sufficient to ensure the stated convergence, as stated +in a well-known result outlined in Chapter 14 of (Shalev- +Shwartz & Ben-David, 2014). +It is also worth noting that Theorem 2 does not necessar- +ily hold if Bφ +core is used instead of Bφ +max. The reason is +that there may be many unsatisfiable cores that are locally +minimal in cardinality, and gradients are set only for in- +puts that appear in the computed core. These gradients will +not match those of the loss on a grounding sample, so the +training dynamics are likely to be different. We believe that +training with Bφ +core may have more in common with block +coordinate descent than gradient descent, and save a more +detailed exploration of the topic for future work. +Theorem 2. Let D, H be a T-decomposable problem with +grounding function f and prediction logic φ where: +1. Z and Y are Cartesian products of Booleans. +2. For any (x, y) ∼ D and y′ ̸= y, φ(f(x), y′) is T- +equivalent to false and there is exactly one z such that +φ(z, y) is T-equivalent to true. +3. Hf is a convex set and for all hf ∈ Hf, ∥hf∥ ≤ B, +and the loss ℓ(hf(·), z) is M-Lipschitz and convex in +x for any fixed z. +Then for any ϵ > 0, selecting hf by minimizing either +LS(Fφ +smt(hf(·))) with τ ≥ M 2B2/ϵ2 iterations of stochastic +gradient descent, with gradients provided by Bφ +max, and +learning rate η = +� +B2/M 2τ yields a grounding hypothesis +ˆhf ∈ Hf that satisfies: E[LD( ˆhf)] ≤ minhf ∈Hf LD(hf)+ +ϵ. The randomness in this expectation is taken over the +choices of the SGD algorithm. +5. Experimental Evaluation +In this section we present an empirical evaluation of +SMTLayer on four learning problems that can be decom- +posed into perceptual and symbolic subtasks. Our results +demonstrate the following primary findings. 1) SMTLayer +is effective: on every benchmark, it provides superior re- +sults over “conventional” learning that takes place without +encoded symbolic knowledge. 2) SMTLayer has distinct +advantages over prior approaches. Compared with SAT- +Net (Wang et al., 2019a), it requires significantly less train- +ing data to converge, and in all cases yields a more accurate +model; compared with Scallop (Huang et al., 2021), it is +less computationally expensive, requires less training data, +and it is more expressive in terms of the knowledge that +it can encode. 3) Models trained with SMTLayer may be +more robust to certain types of covariate shift that occur +relative to the symbolic component of the problem; when +SMTLayer succeeds at learning a compatible representation, +then it will continue to produce correct inferences provided +the perceptual component remains stationary. +5.1. Datasets +Additional details on the datasets and corresponding ar- +chitectures used in our evaluation can be found in Ap- + +Learning Modulo Theories +pendix A.2, and specific hyperparameters used when train- +ing on each dataset are in Appendix A.3. +MNIST Addition. +The MNIST addition problem is illus- +trated in Figure 1, and is similar to the benchmark described +by Huang et al. (2021). For training, we use “MNIST +p%” +to denote a training set of size 60,000 that contains p% of +the possible pairs of digits. So p = 100 indicates all pos- +sible pairs of digits are used, and for p = 10, we only use +pairs of the same digit. We use p = 10, 25, 50, 75 and 100 +in our experiments. In all cases, we use the same test set +consisting of instances from all possible pairs of digits. +Visual Algebra. +The task is to solve for the variable x in +a graphical depiction of the equation ax + b = c, where +a, b and c are randomly-chosen numbers, and each symbol +is depicted visually using EMNIST (Cohen et al., 2017a) +and HASY graphics (Thoma, 2017b). Similar to MNIST +addition, the training sample selects a and b uniformly from +pairs of the same digit, and x uniformly from the odd num- +bers between 0 and 9. The test sample was generated by +sampling a, b uniformly from all pairs of digits, and x from +all numbers 0 to 9. +Liar’s Puzzle. +The liar’s puzzle is comprised of three +sentences spoken by three distinct agents: Alice, Bob, and +Charlie. One of the agents is “guilty” of an unspecified +offense, and in each sentence, the corresponding agent either +states that one of the other parties is either guilty or innocent. +For example, “Alice says that Bob is innocent.” It is assumed +that two of the agents are honest, and the guilty party is not. +The solution to the problem is an identification of the guilty +party. A formal characterization of the underlying logic +is given in Appendix A.2. We note that the logic has non- +stratified occurrences of negation, so it cannot be encoded +with Scallop. We select a training sample that does not +fully specify the logic, so conventional training should be +insufficient to identify a good model. +Visual Sudoku. +This task is to complete a 9 × 9 Sudoku +board where each entry is an MNIST digit. We use the +dataset from the SATNet evaluation (Wang et al., 2019a), +and examine three configurations obtained by sampling 10%, +50%, and 100% of the original training set. Although there +are examples of Sudoku solvers implemented as logic pro- +grams, we were not able to implement one in Scallop with- +out violating stratified negation. When calculating accuracy, +we check that the entire Sudoku board is correct. +5.2. Setup +We implemented a prototype of our approach using Py- +torch (Paszke et al., 2019) and Z3 (De Moura & Bjørner, +2008), which will be made available in open-source when +this paper is published. +When training models with +SMTLayer, we use SGD with Nesterov momentum at rate +0.9 and gradient clipping rate 0.1. Before training a model +with SMTLayer (or a comparison technique, unless stated +otherwise), we first pre-train the neural network by replac- +ing SMTLayer with a dense network containing one hidden +layer of 512 neurons. This can potentially limit the num- +ber of training updates needed at lower layers, but will not +result in a model with a representation that is compatible +with symbolic knowledge, so further training is needed. The +models labeled “conventional” in our evaluation have the +same architecture as the one used for pre-training. Results +were averaged over five runs of training. +Our evaluation was performed on a machine with an Intel +i9 1050K CPU, 64GB memory, and a GeForce RTX 3080 +accelerator running Ubuntu 20.04.4, with CUDA 11.1.0 and +cuDNN 8.0.4. We developed and tested our prototype with +Pytorch version 1.7.0a0+7036e91 and Z3 4.8.14, and the +results in our evaluation use these versions as well. +5.3. Results +Overall performance. +In terms of accuracy, Table 1 +shows that SMTLayer outperforms both the conventional +network and prior work in terms of accuracy, training +time, or both, on all configurations. While training with +SMTLayer (or any of the above approaches) is more ex- +pensive than conventional, SMTLayer is consistently faster +than Scallop (nearly 4× in the case of visual algebra). The +per-epoch time to train the SATNet models is less expensive +than SMTLayer, but this is not always conclusive. In the +case of visual sudoku, the 10% SMTLayer model achieved +superior error rates in 15 epochs, compared with 100 epochs +for the 100% SATNet model; this means that the SMTLayer +model took less than one-tenth the amount of time to train. +It is also worth noting that although Theorem 2 suggests +that Algorithm 3 might have learning advantages over Algo- +rithm 4, we found this not to be the case on these datasets. +All of the results in Table 1 were trained with Algorithm 4, +and test inference was done using Algorithm 1. +Training sample size. +Because SMTLayer encodes ex- +plicit knowledge that is essential to correct inference on +these datasets, our approach is able to perform well in data- +impoverished settings where the training sample is insuffi- +cient to fully specify the symbolic component of the learning +task. This is readily apparent across the results in Table 1: in +the MNIST addition and first visual algebra configuration, +SMTLayer yields a model that performs nearly perfectly +despite not being given a sufficient sample in most cases. +Because SATNet must learn the symbolic component, it is +at a disadvantage, and in these settings performs similarly +to a conventional model. In theory, Scallop should be able + +Learning Modulo Theories +Conventional +w/ SMTLayer +w/ SATNet +w/ Scallop +configuration +test +epoch +test +epoch +test +epoch +test +epoch +acc. (%) +time (sec.) +acc.(%) +time (sec.) +acc.(%) +time (sec.) +acc. (%) +time (sec.) +MNIST+ 10% +10.0 +7.1 +98.1 +75.4 +10.0 +31.0 +33.7 +96.3 +MNIST+ 25% +32.5 +7.1 +98.3 +74.8 +34.2 +30.9 +65.8 +96.4 +MNIST+ 50% +51.5 +7.0 +98.6 +75.8 +54.8 +32.8 +98.4 +96.5 +MNIST+ 75% +76.1 +7.0 +98.5 +75.0 +78.4 +31.9 +93.5 +96.4 +MNIST+ 100% +98.3 +7.1 +98.5 +75.8 +96.7 +33.5 +98.6 +96.6 +Vis. Alg. #1 +24.1 +13.2 +98.2 +168.2 +19.6 +80.1 +18.7 +602.8 +Vis. Alg. #2 +25.4 +11.2 +25.4 +127.2 +18.6 +52.5 +21.3 +636.1 +Liar’s Puzzle +54.2 +3.1 +86.1 +28.7 +84.6 +3.0 +— +— +Vis. Sudoku 10% +0.0 +6.3 +66.0 +135.7 +0.0 +9.9 +— +— +Vis. Sudoku 50% +0.0 +28.3 +73.1 +608.1 +0.0 +45.4 +— +— +Vis. Sudoku 100% +0.0 +26.7 +79.1 +1199.0 +63.2 +86.5 +— +— +Table 1. Results after training and inference with SMTLayer versus a conventional architecture. We use the publicly-available imple- +mentations of SATNet (Wang et al., 2019a) and Scallop (Huang et al., 2021), with hyperparameters matching those in their code. All +SMTLayer test accuracies were measured with the MaxSMT forward pass. Epoch times are averaged over all epochs on which the model +was trained. Cells marked — denote that the problem is not compatible with the approach. +to perform as well as SMTLayer, as it also encodes explicit +knowledge. However, it is unable to learn a useful model +for either visual algebra configuration, and does not learn +the correct representation for MNIST addition until it is +exposed to half of the possible digit pairs during training. +SMTLayer does particularly well on the visual Sudoku +dataset introduced by Wang et al.. When trained on just +10% of the original sample, it learns a function that exceeds +the performance of the SATNet model by a healthy margin, +which continues to grow as it is exposed to more training +data. On the other hand, we found that SATNet failed to +converge with less than the full original training sample. +Robustness +& +interpretability. +The +reason +that +SMTLayer is able to perform well, and often near the +optimum, in configurations that other approaches perform +poorly on, is that it learns a representation that is consistent +with the symbolic knowledge encoded in the SMTLayer. +For example, the constraints that we use for MNIST +addition, visual algebra, and visual sudoku all encode digits +as bitvectors. In order to make a correct inference, the +neural network must learn to encode MNIST digits in their +correct bitvector representation. If learning succeeds at +this, then there are two positive outcomes that follow. First, +the model’s representation will be inherently interpretable, +because it will coincide with the provided symbolic domain +knowledge, which is also (presumably) interpretable. +Second, the resulting model is naturally robust to covariate +shift that does not affect the distribution of perceptual data +that the network translates into theory symbols, but that +does affect the statistics of their composition. +This type of shift is on display in the MNIST 10% and visual +algebra experiments, where at training time, the model only +sees pairs of same-numbered digits, and at test time it is +exposed to a substantially different distribution of digit pairs +or formulas. We verified this by examining the representa- +tions learned by SMTLayer and Scallop on MNIST Addi- +tion 10%; it is unreasonable to expect SATNet to learn an +interpretable representation, as it is not provided with an in- +terpretable theory during training. As expected, SMTLayer +produces the correct representation at the rate of accuracy of +a typical MNIST model (≈ 99%), whereas Scallop’s digit +representation was correct roughly 50% of the time. How- +ever, architecture plays a role in this robustness, as shown in +the SMTLayer results for the second visual algebra configu- +ration. Because the network is shown the full instance, and +not the individual digits, it learns the training bias. Despite +having access to the symbolic formulas in SMTLayer, it +cannot disentangle the perceptual symbols from their covari- +ance. Understanding this issue is an important direction for +future work. +6. Conclusion +Our approach for integrating logical theories into deep learn- +ing, SMTLayer, provides a pragmatic solution to the prob- +lem of incorporating symbolic knowledge into learning for +training and inference, which we demonstrate on several +problems involving both perceptual tasks—vision and natu- +ral language—and logical reasoning. 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SAT/SMT by example, 2020. Available at +https://sat-smt.codes/ (January, 2023). + +Learning Modulo Theories +A. Appendix +A.1. Proofs +Theorem 2. Let D, H be a T-decomposable problem with grounding function f and prediction logic φ where: +1. Z and Y are Cartesian products of Booleans. +2. For any (x, y) ∼ D and y′ ̸= y, φ(f(x), y′) is T-equivalent to false and there is exactly one z such that φ(z, y) is +T-equivalent to true. +3. Hf is a convex set and for all hf ∈ Hf, ∥hf∥ ≤ B, and the loss ℓ(hf(·), z) is M-Lipschitz and convex in x for any +fixed z. +Then for any ϵ > 0, selecting hf by minimizing either LS(Fφ +smt(hf(·))) with τ ≥ M 2B2/ϵ2 iterations of stochastic gradient +descent, with gradients provided by Bφ +max, and learning rate η = +� +B2/M 2τ yields a grounding hypothesis ˆhf ∈ Hf that +satisfies: E[LD( ˆhf)] ≤ minhf ∈Hf LD(hf) + ϵ. The randomness in this expectation is taken over the choices of the SGD +algorithm. +Proof. To prove this result, we introduce the notion of a grounding sample. +Definition 4 (Grounding sample). Let D, H be a T-decomposable problem with grounding function f. The grounding +sample Sf for S ∼ D is given by [(xi, f(xi)) : (xi, yi) ∈ S], i.e., tuples that consist of the first element of each instance in +S and its image under f. +Now observe that the conditions stated in assumption (3) are sufficient to yield the result if instead of optimizing +LS(Fφ +smt(hf(·))), we were given the grounding sample Sf and minimized LSf (hf) (see (Shalev-Shwartz & Ben-David, +2014), Theorem 14.8). The result follows as stated then because of assumptions (1) and (2), which imply that the update +vectors provided by Bφ +max are the gradients of LSf (hf). +To understand why, observe that the sign of ˆy computed on line 3 of both algorithms must be equal to that of y⋆. This +follows from two facts: +1. At any coordinate i where y[i] ̸= y⋆[i], sign(∂yℓ(y, y⋆))[i] = sign(y)[i]. +2. At any coordinate i where y[i] = y⋆[i], sign(∂yℓ(y, y⋆))[i] = −1 · sign(y)[i]. +Now there are two cases to consider. +Case 1: sign(y) = sign(ˆy). +In this case the algorithm returns ∂zℓ(z, sign(z)). Because solutions to φ(·, y) are unique, +sign(z) is the correct grounding, i.e., z = f(x) for the original features x. +Case 2: sign(y) ̸= sign(ˆy). +Because of assumption (2), the set of indices computed on line 8 will contain all of the +coordinates at which z matches the correct value z⋆ = f(x). Note that at these coordinates, the vector returned by the +algorithm matches the gradient of ℓ(z, z⋆), which is ∂z[i]ℓ(z[i], sign(z[i])). In the remaining coordinates, the vector will +contain ∂z[i]ℓ(z[i], 1 − sign(z[i])), which also matches the gradient of ℓ(z, z⋆). The result follows. +A.2. Dataset details +Two of the three problems that we examine are based on the MNIST handwritten digit dataset (Deng, 2012), which consists of +60,000 28x28 gray-scale images of handwritten numerals for training and 10,000 instances for testing. The digits on the left +of Figure 1 are examples of instances from this data. To generate data for the visual algebra problem, we additionally drew +from EMNIST (Cohen et al., 2017b), which extends MNIST with handwritten letters, and HASY (Thoma, 2017a), which +contains handwritten symbols with similar characteristics to MNIST. For the liar’s puzzle, inspired by examples (Yurichev, +2020) which formulate similar examples as SMT constraints, we constructed examples using a set of common phrases that +we devised ourselves, and did not otherwise draw from publicly-available data. + +Learning Modulo Theories +Below, we describe the way in which we used these data sources to construct training and test samples, and the neural +network architectures that we used with SMTLayer to solve them. +MNIST Addition. +The MNIST addition problem is described in Example 1. In each instance, two MNIST digits are +presented as features, and the task of the model is to provide their sum represented as a bitvector. The architecture that +we use consists of four convolutional layers with kernels of size 3, depths in the order 64, 64, 128, 128, and a stride of +width 2 on the first layer, and two dense layers of width 256 and 4. This network is applied to each digit, and the results are +concatenated to obtain a vector of size 8 that is passed to an instance of SMTLayer with SMT constraints from Example 1, +which ultimately produces a vector of width 5 that represents the bitvector sum of the digits. +We generated five training samples starting with one containing only pairs of the same digit, i.e. (0, 0), (1, 1), . . .. We then +added progressively more from the full set of possible pairs, using 25%, 50%, and 100%, and trained on batches of 128 +across all datasets. Note that although we change the number of digit pairs that appear between samples, we always map +these pairs to random MNIST images to obtain 60,000 training instances. This is to ensure that the training sample contains +a sufficient sample of MNIST images to be able to perform well on the test data. In all cases, we use the same test set +consisting of instances from all possible pairs of digits. The purpose of this is to demonstrate that the conventional network +will not generalize until it has seen the full distribution, whereas the model with SMTLayer should be able to generalize +after seeing many fewer examples. +Visual Algebra. +The visual algebra problem is described in Example 2 and Example 3. Recall that features depict +handwritten depictions of linear equations of the form ax + b = c. Values for a, x and b are randomly drawn from the +range 1-9 to ensure that solutions are unique. Then the corresponding value of c is decomposed into c = 10 · c1 + c2, +and MNIST digits are selected at random to represent a, b, c1, c2. A random EMNIST alphabet character is drawn for the +variable, and random multiplication, addition, and equality symbols are drawn from HASY. A minor note is that HASY does +not contain the standard equality symbol “=”, so we instead use “ .=”. These images are then concatenated horizontally in +the appropriate order. +We evaluate two architectures for this problem. The first uses the same neural network that was used for MNIST addition, +and an instance of SMTLayer with the constraints given in Example 2. It assumes that the four numeric digits in the problem +have already been extracted, e.g. by a separate vision routine that can recognize digits from letters and arithmetic symbols, +and are provided directly to the model. The four inputs are given separately to the neural network, which produces four +4-bit bitvectors that are concatenated and passed to SMTLayer, which produces a 4-bit bitvector result. We refer to this as +configuration #1 in our results. +The second uses an architecture which takes the entire image containing the problem as a whole, and produces a 16-bit +bitvector that is passed directly to SMTLayer. This architecture uses a similar stack of convolutional layers, but has a larger +initial dense layer containing 26,112 neurons, as it is given a larger image. The difference in the convolutional stack is at the +second layer, which also has a stride of width 2, to reduce the size of the feature map and mitigate the need for an even larger +dense connection. The difference between these architectures relates to one of the challenges of this problem. Much of the +information contained in the features is irrelevant to the solution, e.g., it is irrelevant which letter is chosen for the variable, +or what the arithmetic operators look like, so this architecture must also learn to disregard these parts of the instance. We +refer to this as configuration #2 in our results and in Example 3. +We generated a training sample by selecting a and b uniformly from pairs of the same digit, i.e. (0, 0), (1, 1), . . ., and +sampling x uniformly from the odd numbers between 0 and 9. The test sample was generated by sampling a, b uniformly +from all pairs of digits, and x from all numbers 0 to 9. We then map these values to random MNIST, EMNIST, and HASY +images to obtain 60,000 samples. The intention is to study a problem wherein the model is not shown all possible problems +(modulo representation as digits), or all of the solutions. This is more challenging than MNIST addition for two reasons: +for a given solution, there are many more compatible ground terms, and the model does not see examples of some of +the solutions it must provide for the test set. Thus, in order for SMTLayer to succeed, it must use the provided symbolic +knowledge to approximate the correct grounding function, despite these deficiencies in the data. +Liar’s Puzzle. +The liar’s puzzle is comprised of three sentences spoken by three distinct agents: Alice, Bob, and Charlie. +One of the agents is “guilty” of an unspecified offense, and in each sentence, the corresponding agent either states that one +of the other parties is either guilty or innocent. It is assumed that two of the agents are honest, and the guilty party is not. +The solution to the problem is an identification of the guilty party. An example is described in Example 4 + +Learning Modulo Theories +Features +Symbolic Domain Z +0010000111 +Grounding +Function f +φ( +z1∥ . . . ∥z10, +y +) ≡ +a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧ +c = 1z5>0∥ . . . ∥1z10>0 +∧ +ax + b = c +Prediction Logic φ +Labels Y +{00101} +Satisfying +Assignments +Figure 2. Visual Algebra configuration 1 example. +We synthesized a dataset for the liar’s puzzle based on a limited set of utterances about who speaks in each sentence, +which agent is the subject, and whether the subject is guilty or innocent. There are five ways of denoting the speaker: “* +says”, “* says that”, “* said”, “* said that”, and a colon “* :” separating the speaker’s name from the rest of the sentence. +There are five ways of uttering either innocence or guilt: “* did it/did not do it”, “* is guilty/innocent”, “* is definitely +guilty/innocent”, and “* definitely did it/did not do it”, “* is the criminal/is a good person”. We generated all of the +combinations of subject, speaker, and proclaimed innocence or guilt, and took the product with all possible combinations of +these utterances. The result is a dataset of 375,000 instances, each containing three natural language sentences. +The +prediction +logic +for +this +problem +assumes +a +set +of +ground +predicates +speaker(agent, sentence), +subject(agent, sentence), accuses(sentence), and guilty(agent). +For example, if the first sentence was “Alice +says that Bob is innocent”, the ground predicates would be speaker(alice, 1), subject(bob, 1), and ¬guilty(bob). Then the +prediction logic is shown in Equation 1. +|{a : guilty(a)}| = 1 +∧ +∀a.|{s : speaker(a, s)}| = 1 +∧ +∀s.|{a : subject(a, s)}| = 1 +∧ +∀s, a1, a2.speaker(a1, s) ∧ subject(a2, s) → +(¬guilty(a1) ↔ accuses(s) ↔ guilty(a2)) +(1) +In our implementation, agents and sentence identifiers are encoded unary as 3-bit bitvectors. The quantifiers are removed +by substituting for each ground term or sentence identifier, and the cardinality constraints are expanded into propositional +formulas. The architecture that we adopt is a two-layer bidirectional long short-term model (LSTM) with 512 dimensions at +each layer, and a 300-dimension trainable embedding layer initialized from GloVe-6B (Pennington et al., 2014). The hidden +units of the last LSTM layer were connected to 2-layer dense network containing 128 followed by 7 neurons. This network +is applied to each sentence in the input, and the concatenated results are passed to the SMTLayer, which solves the formula +in Equation 1 to produce a unary encoding of the guilty party. +To evaluate solver layers on this problem, we selected training and test samples by first subsampling half of the full 375,000 +available instances. We then selected half of the speaker, subject, accuses predicate configurations for all three sentences +appearing in this subsample to appear in the training sample, and the other half to appear in the test sample. To further limit +the amount of information in the training sample, we randomly selected one ordering for each predicate configuration to +remain for training. There were 9,400 resulting training instances, and 28,062 test instances. Restricting the training set as +described ensures that the model is trained on a limited subset of possible sentence configurations, and one that is logically +disjoint from those that appear in the test sample. Because there is not enough information in the training sample to learn +Equation 1, we expect only the model with SMTLayer to succeed, but to do so it must approximate the grounding function +well from a limited sample. +A.3. Hyperparameters +The four problems that our evaluation studies vary considerably in size and complexity, and the models used to train them +require different considerations. This section details these differences. Table 2 relates the optimizers and epochs for each + +4X+Learning Modulo Theories +Features +Symbolic Domain Z +0100010000100100 +Grounding +Function f +φ( +z1∥ . . . ∥z10, +y +) ≡ +a = 1z1>0∥ . . . ∥1z5>0 ∧ b = 1z5>0∥ . . . ∥1z10>0 ∧ +ax + b = c +Prediction Logic φ +Labels Y +{00101} +Satisfying +Assignments +Figure 3. Visual Algebra configuration 2 example. +Input +Alice said that Bob +did not do it. +Bob: Alice is +definitely innocent +Charlie: Alice did it +Symbolic Domain Z +100010001010100100110 +Grounding +Function f +|{a : guilty(a)}| = 1 +∧ +∀a.|{s : speaker(a, s)}| = 1 +∧ +∀s.|{a : subject(a, s)}| = 1 +∧ +∀s, a1, a2.speaker(a1, s) ∧ subject(a2, s) → +(¬guilty(a1) ↔ accuses(s) ↔ guilty(a2)) +Prediction Logic φ +Labels (Liar) Y +{2} +Satisfying +Assignments +Figure 4. Liar’s Puzzle example. +dataset and solver layer configuration. For SATNet and Scallop, we use the optimization settings described in their respective +papers, and found in their public implementations. For SATNet models, the MaxSAT clause parameters were trained at a +rate of 2e-3, and the convnet at rate 1e-5. When SGD(1.0) is stated, we used a warmup period spanning the first epoch, and +cosine annealing for the remainder of training. +MNIST Addition. +The MNIST addition problem is the easiest of the problems that we study, at least in its full (100%) +configuration. We find that for the 100% configuration, all of the solver layers converge to a nearly optimal solution with +three epochs of supervised pre-training, and five epochs of subsequent training with the solver layer attached. For the +subsample configurations, all of the solver layers converge to a stable, although in many cases suboptimal, solution within +these parameters as well. For SMTLayer, we clipped gradients for all parameters at 0.1, and did not clip gradients for the +other solver layer models. For all configurations, we used batches of size 128. +Visual Algebra. +Although visual algebra is a more difficult learning problem than MNIST addition, as evidenced by the +results in Table 1, we find that the same parameters allow all of the configurations studied in our evaluation to converge. After +3/5 epochs of training, the models stabilize, and in some cases, further training yields an overfit model. For SMTLayer, we +clipped gradients for all parameters at 0.1, and did not clip gradients for the other solver layer models. For all configurations, +we used batches of size 64. +Liar’s Puzzle. +The Liar’s Puzzle is the only problem to use a recurrent model, and we found that it required more epochs +of pre-training to reduce the variance of the final model with the solver layer. Additionally, the SGD optimizer used +with SMTLayer on other datasets caused the model to converge at local minima. We found that pre-training at a higher +learning rate, and using Adam with a default learning rate, let to the best results. Additionally, we did not clip gradients +for the SMTLayer model. We used the same parameters, but the normal SATNet optimizer, for the SATNet model. For all +configurations, we used batches of size 32. + +4x+4=ayLearning Modulo Theories +Conventional +w/ SMTLayer +w/ SATNet +w/ Scallop +optimizer +epochs +optimizer +epochs +optimizer +epochs +optimizer +epochs +MNIST+ 10% +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +MNIST+ 25% +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +MNIST+ 50% +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +MNIST+ 75% +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +MNIST+ 100% +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +Vis. Alg. #1 +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +Vis. Alg. #2 +SGD(1.0) +0/5 +SGD(1.0) +3/5 +Adam(2e-3, 1e-5) +3/5 +Adam(1e-3) +3/5 +Liar’s Puzzle +Adam(2e-3) +0/15 +Adam(1e-3) +15/5 +Adam(2e-3, 1e-5) +15/5 +— +— +Vis. Sudoku 10% +SGD(1.0) +0/100 +SGD(1.0) +30/15 +Adam(2e-3, 1e-5) +30/100 +— +— +Vis. Sudoku 50% +SGD(1.0) +0/100 +SGD(1.0) +30/5 +Adam(2e-3, 1e-5) +30/100 +— +— +Vis. Sudoku 100% +SGD(1.0) +0/100 +SGD(1.0) +30/5 +Adam(2e-3, 1e-5) +0/100 +— +— +Table 2. Training hyperparameters. Numbers in parentheses after the optimizer denote the learning rate; when there are multiple numbers, +different learning rates were applied to different parameter groups, as detailed in the text. Each epoch pair corresponds to the pre-training +and training phases, i.e., 3/5 denotes three epochs of pre-training and five subsequent training epochs with the solver layer. +Visual Sudoku. +Visual Sudoku is the most challenging problem that we studied, for all solver layers as well as the +conventional model. We did not use supervised pre-training, as the supervision in this problem leaks the correct labels +directly to the model, bypassing the solver layer and the need for its updates (Chang et al., 2020). We instead used the +unsupervised pre-training method described in (Topan et al., 2021), and found that 30 epochs of unsupervised pre-training +was sufficient to yield consistent and quick convergence with the solver layer. For the most data-scarce configuration (10%), +15 epochs of training with the solver layer were needed to converge, and for the others five epochs were sufficient. For the +SMTLayer model, we used batches of size 1 after pretraining (batch size 64 during pre-training), primarily due to the fact +that SMTLayer returns only the indices masked as non-hint elements on the Sudoku board. Because each instance has a +different number of hint elements, this would lead to ragged tensors during training, which Pytorch does not support. +When assessing SATNet on Visual Sudoku, we were unable to converge to a useful model on any except the full (100%) +configuration, as discussed in Section 5.2. Using the authors’ public implementation and training script, we attempted the +10% and 50% configurations with and without unsupervised pre-training, with and without the measures taken in (Chang +et al., 2020) to prevent label leakage, and with batches of size 40 (as used in the original paper) as well as 1, to no avail. For +the 100% configuration, we were able to reproduce a useful model; we use the accuracy reported in the original paper in +Table 1 for consistency, as the average that we obtained did not differ significantly from this. + diff --git a/GNFJT4oBgHgl3EQfDyxI/content/tmp_files/load_file.txt b/GNFJT4oBgHgl3EQfDyxI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96a0f3c54be98cf1f949f8f4cdb387cef94ce1a3 --- /dev/null +++ b/GNFJT4oBgHgl3EQfDyxI/content/tmp_files/load_file.txt @@ -0,0 +1,985 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf,len=984 +page_content='Learning Modulo Theories Matt Fredrikson 1 Kaiji Lu 1 Saranya Vijayakumar 1 Somesh Jha 2 Vijay Ganesh 3 Zifan Wang 1 Abstract Recent techniques that integrate solver layers into Deep Neural Networks (DNNs) have shown promise in bridging a long-standing gap between inductive learning and symbolic reasoning tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' In this paper we present a set of tech- niques for integrating Satisfiability Modulo Theo- ries (SMT) solvers into the forward and backward passes of a deep network layer, called SMTLayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Using this approach, one can encode rich domain knowledge into the network in the form of mathe- matical formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' In the forward pass, the solver uses symbols produced by prior layers, along with these formulas, to construct inferences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' in the backward pass, the solver informs updates to the network, driving it towards representations that are compatible with the solver’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Notably, the solver need not be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' We imple- ment SMTLayer as a Pytorch module, and our empirical results show that it leads to models that 1) require fewer training samples than conven- tional models, 2) that are robust to certain types of covariate shift, and 3) that ultimately learn representations that are consistent with symbolic knowledge, and thus naturally interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Introduction A recent class of techniques aims at integrating solver layer(s) within deep neural networks (DNNs) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Pogancic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Manhaeve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2018), both during training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' A class of problems which can benefit from such an integration is one that has both a perceptual and a symbolic sub-problem, such as “visual” Sudoku (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2019a), or the prob- lem of determining the shortest path from a picture of a map (Pogancic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' The most straightforward way to incorporate a solver layer 1Carnegie Mellon University, Pittsburgh, PA, USA 2University of Wisconsin-Madison, Madison, WI, USA 3University of Water- loo, Waterloo, ON, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNFJT4oBgHgl3EQfDyxI/content/2301.11435v1.pdf'} +page_content=' Correspondence to: Matt Fredrikson ht +Xt +Input gate +Output gateit=o(atU+ht-iW) +x, input +ft=o(atUf +ht-1Wf) +f. forget gate +i， +inputgate +Ot = o(atUo + ht-1Wo) +cellupdate +c, cell state +Ct =tanh (ctU9+ht-iW9) +o,outputgate +Ct = o(ft* Ct-1 + it*Ct) +h,output +G sigmoidactivationfunction +ht = tanh(Ct) * Ot +tanh Tanh activationfunctionIn the specification of this Prophet model, there are several +places where we can alter the model to apply your experience +and external expert knowledge without needing to understand +the underlying statistics.[4] +For the Prophet model with general theory: +We use a mathematical decomposable time series model[9] +and this model has these components: trend, seasonality, and +holidays. They are combined into the following equation: +Series Model Formula. +Here g(t) is the trend function that models non-periodic +changes in the value of the time series, s(t) represents periodic +changes (for example, weekly and yearly seasonality), and h(t) +represents the effects of the seasons, The error term represents +any idiosyncratic changes that do not fit the model. +In PROPHET we incorporate trend changes into the growth +model by explicitly determining the change points where the +growth rate is allowed to change. Suppose there are exchange +points at moments j,j= 1,..., S. We give N a vector of adjust- +ments, where only the rate change occurs at moment j. The +rate at any time is then the base rate k, plus all adjustments +up to the point of This is best represented by the vectors as +follows: +Adjustments represented. +When rate k is adjusted, the parameter set must also be +adjusted to connect the endpoints of the segments. The correct +fit at the shift point is easily calculated as: +That would be the Adjust at shift point. +The logistic piece of the growth model then looks as +follows: +The Fourier series is also applied in Prophet to provide +a flexible model of periodic effects[10]. Let P be the regular +period that we expect the time series to have (for example, P= +365:25 for annual data or P= 7 for weekly data, when we scale +our indices of time variables). We can approximate arbitrary +uniform seasonal effects with this definition: +Fourier Definition +Sometimes we can’t just randomly split the data accord- +ingly. PROPHET develops simulated historical forecasts by +producing K forecasts at various cut-off points in history, +chosen such that the horizons are within the historical record +and the total error can be evaluated. +This procedure is based on the classic ’rolling source’ fore- +cast evaluation procedures[11], but uses only a small sequence +of target dates instead of forecasting by historical date) is that +it saves on computation and provides less correlated precision +measurements. +C. Work Sequence +• +We apply an EDA (Exploratory Data Analysis) it is +necessary to clean the data and adapt it for the job. +• +Normalization of the data, we adapt the data to follow +a supervised sequence model according to a time +series. +• +Let’s divide our dataset into proof and validation tests. +• +Coding and implementation of the models according +to the Prophet or LSTM case. +• +Adjustment of the parameters and extraction of pre- +dictions with their respective evaluation metrics. +IV. +RESULTS +The results using PROPHET and LSTM are shown in this +section together with the general comparison of the mentioned +models. +We will also appreciate the comparisons and metrics de- +veloped according to the models applied to the study variable: +PESO. +Result according to the validation of the LSTM model: +Fig. 3. +LSTM VALIDATION +3 | P a g e + +PESO +600000 +LSTM_Predictions +500000 +400000 +300000 +20000 +100000 +0 +00:45:38 +00:45:40 +00:45:42 +00:45:44 +00:45:46 +00:45:48 +Fechay(t) = g(t) + s(t) + h(t) + Et1, +if t≥sj, +a;(t) +0, +otherwise.=(s-m-)(1 +1-)C(t)s(t)-(an cos()+bsin())Result according to the validation of the PROPHET model: +Fig. 4. +PROPHET VALIDATION +Comparison to the validation of the PROPHET and LSTM +model: +Fig. 5. +LSTM AND PROPHET COMPARISON +MSE(mean squared error) and the RMSE(root mean +squared error) of both models: +We can appreciate each model according to its evaluation. +Fig. 6. +RMSE Y MSE +V. +DISCUSSION AND CONCLUSIONS +In keeping with the theory, all machine learning algorithms +are unique, which is the root cause of why the prediction +results are different algorithms on the same data set differ. +LSTM was proposed for being an improvement of the RNNs +and Prophet for its versatility in data with less presence of +seasonality. +Given what was taken into consideration for LSTM, the +applied model was with the minimum parameters and show +results with much better conditioning than was expected, +remembering that in different studies the superiority of LSTM +is detailed over the basic algorithms that apply Recurrent +Neural Networks. Furthermore, it is noted from the theory +that the number of training times, known as the ’epoch’ in +learning[12], did not take into account the expected effect on +the performance of the trained forecast model and exhibited +mostly random behavior. How intuitive The developed model +based on LSTM incorporates additional ’gates’ in order to store +longer sequences of input data. +One of the main questions when developing and analyzing is +whether the gates incorporated in the LSTM architecture would +give a good prediction and if additional data training would +be needed to further improve the prediction[13] and in this +case, we can deduce that the quantity of data was acceptable +but the predictions were affected by the not very well defined +seasonality of the dataset. A more effective solution would be +to add exact dates and continuous seasonalities. +For the Prophet model, we should have some more +intuitive results according to its theory, the application of +the Fourier series could develop more precision. The model +is expected to obtain a reasonable forecast on disordered +data without too much manual effort, unlike LSTM, which +has more hyperparameters. Prophet proved to be resistant +to outliers, missing data, and drastic changes in its time +series, the intention to fit the timeline is noted. Compared +to other classical forecasting methods, Prophet should be +fast and easy to apply to time series, which is what it +was designed for in the first place; however, it could be +considerably less accurate[14] and in this case, we confirm +this appreciation by highlighting once again its intuitive factor. +The Prophet procedure should include more parameters for +users to modify and adjust the forecasts in a more effective +way. Also, a hybrid model could improve significantly. +According to the RMSE results of the import predictions, +we can conclude that the LSTM model presents a significantly +better performance and reliability with respect to the Prophet +model, however, as we deduced previously, the seasonality +of the dataset was an important key in the variation of the +development of the models and their predictions. Therefore, +increasing the size of the dataset and adapting an exact timeline +for our dataset of vegetable imports from Peru should be a +priority, in this way, we would undoubtedly obtain results +with better relevance and reliability and, of course, the field of +application with the use of machine learning techniques would +be widely used and its results would be of a strongly necessary +relevance. +REFERENCES +[1] +A. J. V. Perez, N. H. T. Vazquez, and C. M. V. Sol´orzano, “Aprendizaje +autom´atico en la industria 4.0 (machine learning)”Bolet´ın No, vol. 91, +no. 1o, 2022 +[2] +J. Sarshar, S. S. Moosapour, and M. Joorabian, “Multi-objective energy +management of a micro-grid considering uncertainty in wind power +forecasting,” Energy, vol. 139, pp. 680–693, 2017. [Online]. Available: +https://www.sciencedirect.com/science/article/pii/S0360544217313221 +[3] +H. Abbasimehr, M. Shabani, and M. Yousefi, “An optimized model +using lstm network for demand forecasting,” Computers & industrial +engineering, vol. 143, p. 106435, 2020. +[4] +S. J. Taylor and B. Letham, “Forecasting at scale,” The American Sta- +tistician, vol. 72, no. 1, pp. 37–45, 2018. +4 | P a g e + +1e6 +1.5 +10 +PESO +0.5 +0.0 +0.5 +00:45:40 +00:45:45 +Fecha1e6 +1.5 +10 +0.5 +0.0 +0.5 +00:45:40 +00:45:45Models +RMsE Errors +MsE Errors +0 +LSTM +678271.532637 +4.600523e+11 +1 +Prophet +688513.667236 +4.740511e+11[5] +SENASA. https://www.datosabiertos.gob.pe/dataset/dataset/importaci´on- +de-productos-vegetales-en-senasa-para-el-2021-2022-ministerio-de- +desarrollo. +[6] +J. L. Elman, “Finding structure in time,” Cognitive science, vol. 14, no. +2, pp. 179–211, 1990. +[7] +M. C. Sorkun, ¨O. D. Incel, and C. Paoli, “Time series forecasting on +multivariate solar radiation data using deep learning (lstm),” Turkish +Journal of Electrical Engineering and Computer Sciences, vol. 28, no. 1, +pp. 211–223, 2020. +[8] +W. Liu, Y. Liu, T. Zhang, Y. Han, X. Zhou, Y. Xie, and S. Yoo, “Use of +physics to improve solar forecast: Part ii, machine learning and model +interpretability,” Solar Energy, vol. 244, pp. 362–378, 2022. +[9] +A. C. Harvey and S. Peters, “Estimation procedures for structural time +series models,” Journal of forecasting, vol. 9, no. 2, pp. 89–108, 1990. +[10] +A. C. Harvey and N. Shephard, “10 structural time series models,” 1993. +[11] +L. J. Tashman, “Out-of-sample tests of forecasting accuracy: an analy- +sis and review,” International journal of forecasting, vol. 16, no. 4, pp. +437–450, 2000. +[12] +S. Siami-Namini, N. Tavakoli, and A. S. Namin, “A comparison of arima +and lstm in forecasting time series,” in 2018 17th IEEE inter- national +conference on machine learning and applications (ICMLA). IEEE, 2018. +[13] +Siami-Namini, Sima, Neda Tavakoli, and Akbar Siami Namin. ”The +performance of LSTM and BiLSTM in forecasting time series.” 2019 +IEEE International Conference on Big Data (Big Data). IEEE, 2019.. +[14] +L. Menculini, A. Marini, M. Proietti, A. Garinei, A. Bozza, C. Moretti, +and M. Marconi, “Comparing prophet and deep learning to arima in +forecasting wholesale food prices,” Forecasting, vol. 3, no. 3, pp. 644– +662, 2021. +5 | P a g e + diff --git a/JNE2T4oBgHgl3EQfAAax/content/tmp_files/load_file.txt b/JNE2T4oBgHgl3EQfAAax/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbfa5fe0488404d4e8597495288e26f5940a1d86 --- /dev/null +++ b/JNE2T4oBgHgl3EQfAAax/content/tmp_files/load_file.txt @@ -0,0 +1,275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf,len=274 +page_content='MACHINE LEARNING APPLIED TO PERUVIAN VEGETABLES IMPORTS Ticona-Salluca Hugo Faculty of Statistic and Computer Engineering, Universidad Nacional del Altiplano de Puno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Box 291 Puno - Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Email: hticonas@est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='pe Torres-Cruz Fred Faculty of Statistic and Computer Engineering, Universidad Nacional del Altiplano de Puno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Box 291 Puno - Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Email: ftorres@.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='pe Tumi-Figueroa Ernesto Nayer Faculty of Statistic and Computer Engineering, Universidad Nacional del Altiplano de Puno, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Box 291 Puno - Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Email: nayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='tumi@unap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='pe Abstract—The current research work is being developed as a training and evaluation object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' the performance of a predictive model to apply it to the imports of vegetable products into Peru using artificial intelligence algorithms, specifying for this study the Machine Learning models: LSTM and PROPHET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The forecast is made with data from the monthly record of imports of vegetable products(in kilograms) from Peru, collected from the years 2021 to 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' As part of applying the training methodology for automatic learning algorithms, the exploration and construction of an appropriate dataset according to the parameters of a Time Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Subsequently, the model with better performance will be selected, evaluating the precision of the predicted values so that they account for sufficient reliability to consider it a useful resource in the forecast of imports in Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Keywords—Machine Learning: forecasting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' im- ports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='artificial intelligence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='dataset I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' INTRODUCTION The Machine Learning methodology is one of the most valuable technologies in the Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='0, and advances in Ma- chine Learning have provided significant benefits in strategic decision-making in organizations[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' In recent years, predictive models using Machine Learning algorithms have been imple- mented in real environments of different organizations, and as in this particular case that concerns us, it also operates in the agriculture industry and optimal results are expected from these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Likewise, Machine Learning technology has a high poten- tial to optimize your processes and, consequently, make correct and anticipated decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' For this reason, Learning Algorithms should also be considered as the next innovative direction to significantly improve the prediction of these use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Therefore, this research will seek to demonstrate and encourage the use of Machine Learning applications in the field of agribusiness in Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Through the present work, the examination and modeling of real data will be developed in order to forecast future values with the least possible prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' In this sense, this applied research work covers the design, training, validation, and evaluation of 2 predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The research models to be explored are LSTM and Prophet models, these being adequate and recommended to develop applied foresting, on the one hand, LSTM is rec- ognized for its hyperparameters which have the ability to capture non-linear patterns using the grid search method[3] and PROPHET for their modular regression with intuitive and adjustable interpretation parameters applied on Time Series[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' DATASET According to the purpose of the study, the official records of the export volume (in kilograms) of vegetables in Peru were used, collected from 2021 to 2022 in the National Open Data Platform,[5], officially uploaded by the Ministry of Agrarian Development and Irrigation of Peru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Table 1 Description of the variables in the Vegetable Import Dataset in Peru 2021-2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Based on this dataset, it is proposed to make a reliable prediction, and according to the models to be developed, we will therefore need the continuous quantitative variable, PESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Also according to the observed data, we find it necessary to adapt a proper timeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' therefore, we must join the variables A ˜NO and MES to foresee a more accurate seasonality, spec- ifying the variables to forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Subsequently, to get a more accurate overview of our data regarding products, we define IDs for each product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' From the dataset, we highlight 3P: Soja, cake with 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='1% in 1st position, in second position 5P: Apple, fresh fruit with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='3% and in third position 27P: Soja, grain with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The data ranges from 2021-05-01 to 2022-06-01, with 34,364 observations on imports of a total of 848 plant products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 1 | P a g e arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='03587v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='LG] 8 Jan 2023 VAR SCALA DESCRIPTION 1 ANO CUANTITATIVA DISCRETA Anodelaimportacion 2 MES CUANTITATIVA DISCRETA Mes de la importacion 3 SEDE CUALITATIVA NOMINAL Sede donde se registra de la Importacion 4 PRODUCTO CUANTITATIVA DISCRETA Productoimportado 7 PESO CUANTITATIVA CONTINUA Pesoenkilogramosdelproductc 8 TIPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='ENVASE CUALITATIVA NOMINAL Envasecontenedordelaimportacion PAIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='ORIGEN CUALITATIVA NOMINAL Origendelaimportacion del productoFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Percentage description of each vegetable product imported in Peru III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' METHODS In this section, we will describe the models used for the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' LSTM MODEL The LSTM name refers to the analogy that a standard RNN has both ’long-term memory’ and ’short-term memory’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The connection weights and biases in the network change once per training episode, analogous to how physiological changes in synaptic strengths store long-term memories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' activation patterns in the network change once per time step, analogous to how moment-to-moment changes in electrical activation patterns in the general brain store short-term memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The LSTM architecture aims to provide a short-term memory for RNNs that can last for thousands of time steps and continue to be reliable, hence ’short-term long-term memory’[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' LSTM is considered a special type of recurrent neural network (RNN), developed to solve the potential problem of descending gradient found in traditional RNN training, and is able to learn both short-term and long-term dependencies[7] and is constructed of four main components: an entry gate, an exit gate, memory cell and a forget gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Input Gate: controls the sending addition of information to the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' In other words, the gateway will consider what information needs to be added to the cell state to ensure that important information is added and that there is no redundant information or noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Memory Cell: controls the value that might be deleted or updated, and contains a value that might need to be kept as additional information for many other time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Output Gate: controls the selection of useful learning information from the current state of the cell as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Forget Gate: controls the removal of information that is no longer needed for LTSM to learn things or, less importantly, the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' This helps to optimize the performance of the LSTM proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Also, The LSTM model follows the sequence: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Decide to discard the cell state information by a Sigmoid layer called ”forget gate”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Decide new information to store in the cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The Sigmoid layer called the ”gateway layer” decides which values will be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' the tan coat creates a vector of new candidate values that could be added to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Update the old cell state to the new cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Decide the filtered output based on the state of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' MODEL LSTM Considered for a normal LSTM model is the Tanh property, which is a non-linear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' It regulates the values that flow through the network, keeping the values between - 1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' To avoid information fading, a function is needed whose second derivative can survive longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' There might be a case where some values become huge, causing the values to be irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' And of course an LSTM principal property: The Sigmoid that belongs to the family of nonlinear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Unlike Tanh, the Sigmoid holds values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' It helps the network update or forget data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' If the multiplication results in 0, the information is considered forgotten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Similarly, the information is kept if the value is relevant[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' This will assist the network in determining what data can be lost and what data should be kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' PROPHET MODEL PROPHET is a procedure for forecasting time series data based on an additive model in which nonlinear trends are adjusted for annual, weekly, and daily seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' It works best with time series that have strong seasonal effects and multiple seasons of historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 2 | P a g e 3P 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='1% 6 iF 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='3% 5P 145P 43P 10P 239P 56P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='2% 86P 11p 106P 31P 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='5% 104P 1P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='8% 42P 27P 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='2% 29P 64P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='8% 59P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='4% 77P 61P 23P 8P 91P 17P 84P 6P 18P 34p 57P 12P 44P 89P 22p 9P 28P 41P 70P 69P 26P 16P 50PForget gate ht Memory cell Ct Ct-1 文 Ct x tanh a tanh Hidden state ht ht-1 - X >ht Xt Input gate Output gateit=o(atU+ht-iW) x, input ft=o(atUf +ht-1Wf) f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' forget gate i， inputgate Ot = o(atUo + ht-1Wo) cellupdate c, cell state Ct =tanh (ctU9+ht-iW9) o,outputgate Ct = o(ft* Ct-1 + it*Ct) h,output G sigmoidactivationfunction ht = tanh(Ct) * Ot tanh Tanh activationfunctionIn the specification of this Prophet model, there are several places where we can alter the model to apply your experience and external expert knowledge without needing to understand the underlying statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' [4] For the Prophet model with general theory: We use a mathematical decomposable time series model[9] and this model has these components: trend, seasonality, and holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' They are combined into the following equation: Series Model Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Here g(t) is the trend function that models non-periodic changes in the value of the time series, s(t) represents periodic changes (for example, weekly and yearly seasonality), and h(t) represents the effects of the seasons, The error term represents any idiosyncratic changes that do not fit the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' In PROPHET we incorporate trend changes into the growth model by explicitly determining the change points where the growth rate is allowed to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Suppose there are exchange points at moments j,j= 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' We give N a vector of adjust- ments, where only the rate change occurs at moment j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The rate at any time is then the base rate k, plus all adjustments up to the point of This is best represented by the vectors as follows: Adjustments represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' When rate k is adjusted, the parameter set must also be adjusted to connect the endpoints of the segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The correct fit at the shift point is easily calculated as: That would be the Adjust at shift point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The logistic piece of the growth model then looks as follows: The Fourier series is also applied in Prophet to provide a flexible model of periodic effects[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Let P be the regular period that we expect the time series to have (for example, P= 365:25 for annual data or P= 7 for weekly data, when we scale our indices of time variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' We can approximate arbitrary uniform seasonal effects with this definition: Fourier Definition Sometimes we can’t just randomly split the data accord- ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' PROPHET develops simulated historical forecasts by producing K forecasts at various cut-off points in history, chosen such that the horizons are within the historical record and the total error can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' This procedure is based on the classic ’rolling source’ fore- cast evaluation procedures[11], but uses only a small sequence of target dates instead of forecasting by historical date) is that it saves on computation and provides less correlated precision measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Work Sequence We apply an EDA (Exploratory Data Analysis) it is necessary to clean the data and adapt it for the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Normalization of the data, we adapt the data to follow a supervised sequence model according to a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Let’s divide our dataset into proof and validation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Coding and implementation of the models according to the Prophet or LSTM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Adjustment of the parameters and extraction of pre- dictions with their respective evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' RESULTS The results using PROPHET and LSTM are shown in this section together with the general comparison of the mentioned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' We will also appreciate the comparisons and metrics de- veloped according to the models applied to the study variable: PESO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Result according to the validation of the LSTM model: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' LSTM VALIDATION 3 | P a g e PESO 600000 LSTM_Predictions 500000 400000 300000 20000 100000 0 00:45:38 00:45:40 00:45:42 00:45:44 00:45:46 00:45:48 Fechay(t) = g(t) + s(t) + h(t) + Et1, if t≥sj, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='(t) 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content='=(s-m-)(1 1-)C(t)s(t)-(an cos()+bsin())Result according to the validation of the PROPHET model: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' PROPHET VALIDATION Comparison to the validation of the PROPHET and LSTM model: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' LSTM AND PROPHET COMPARISON MSE(mean squared error) and the RMSE(root mean squared error) of both models: We can appreciate each model according to its evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' RMSE Y MSE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' DISCUSSION AND CONCLUSIONS In keeping with the theory, all machine learning algorithms are unique, which is the root cause of why the prediction results are different algorithms on the same data set differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' LSTM was proposed for being an improvement of the RNNs and Prophet for its versatility in data with less presence of seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Given what was taken into consideration for LSTM, the applied model was with the minimum parameters and show results with much better conditioning than was expected, remembering that in different studies the superiority of LSTM is detailed over the basic algorithms that apply Recurrent Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Furthermore, it is noted from the theory that the number of training times, known as the ’epoch’ in learning[12], did not take into account the expected effect on the performance of the trained forecast model and exhibited mostly random behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' How intuitive The developed model based on LSTM incorporates additional ’gates’ in order to store longer sequences of input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' One of the main questions when developing and analyzing is whether the gates incorporated in the LSTM architecture would give a good prediction and if additional data training would be needed to further improve the prediction[13] and in this case, we can deduce that the quantity of data was acceptable but the predictions were affected by the not very well defined seasonality of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' A more effective solution would be to add exact dates and continuous seasonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' For the Prophet model, we should have some more intuitive results according to its theory, the application of the Fourier series could develop more precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The model is expected to obtain a reasonable forecast on disordered data without too much manual effort, unlike LSTM, which has more hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Prophet proved to be resistant to outliers, missing data, and drastic changes in its time series, the intention to fit the timeline is noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Compared to other classical forecasting methods, Prophet should be fast and easy to apply to time series, which is what it was designed for in the first place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' however, it could be considerably less accurate[14] and in this case, we confirm this appreciation by highlighting once again its intuitive factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' The Prophet procedure should include more parameters for users to modify and adjust the forecasts in a more effective way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Also, a hybrid model could improve significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' According to the RMSE results of the import predictions, we can conclude that the LSTM model presents a significantly better performance and reliability with respect to the Prophet model, however, as we deduced previously, the seasonality of the dataset was an important key in the variation of the development of the models and their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Therefore, increasing the size of the dataset and adapting an exact timeline for our dataset of vegetable imports from Peru should be a priority, in this way, we would undoubtedly obtain results with better relevance and reliability and, of course, the field of application with the use of machine learning techniques would be widely used and its results would be of a strongly necessary relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE2T4oBgHgl3EQfAAax/content/2301.03587v1.pdf'} +page_content=' Perez, N.' metadata={'source': 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Using Google Scholar, we gathered data about publication +citations for 5,574 tenure-track faculty from 185 U.S. universities. For each faculty, we extracted +their t10 index, defined as the number of citations received by their 10th highest cited paper. For each +department, we calculated four quality metrics: median t10 (m10), the geometric mean of t10 (g10), +and the number of well-cited faculty with t10 above 40% (c40) and 60% (c60) of the national average. +We fitted a linear regression model using those four measures to match the 2022 U.S. News ranking +scores of CS doctoral programs. The resulting model provides Scholar Ranking 2023, which can be +found at https://chi.temple.edu/csranking_scholar. +1 +Introduction +A previous version of the Scholar ranking [1] was published in the spring of 2017, based on citation data collected +during the fall of 2016. This previous effort demonstrated that it is possible to learn a simple formula from citation +measures that has a high correlation with peer assessment scores of CS doctoral programs published by the U.S. News +(USN). +A few years have passed since the last publication of the Scholar ranking, and a new U.S. News ranking came out +in 20221. Given the fact that the data on which the last ranking was performed is a few years old, we felt that it would +be helpful to conduct another round of data collection and validate our method with the recent U.S. News ranking. The +first objective is to refine the data collection method and collect a new set of high-quality faculty citation data. The +second objective is to use the 2022 U.S. News ranking to validate the method proposed in the first version of the scholar +ranking and observe changes in the ranking. The third objective is to analyze the trends in aggregated metrics used to +perform the ranking given the data sets, with the first collected during the fall of 2016 and the other during the fall of +2021. +1https://www.usnews.com/best-graduate-schools/top-science-schools/computer-science-rankings +arXiv:2301.03140v1 [cs.DL] 9 Jan 2023 + +A PREPRINT - JANUARY 10, 2023 +2 +Data collection +In this section, we explain the data collection process which took place from September 2021 to December 2021. +The data collection team consisted of two CS graduate students and a CS professor. +2.1 +U.S. News (USN) Data +USN is well-known for producing several rankings. We gathered the scores from the most recent ranking of CS +doctoral programs, Best Computer Science Schools, which was published in 2022. USN collected the names of those to +be surveyed for the science doctoral surveys in the summer of 2021. We retained the scores from the 2013 version of +Best Computer Science Schools from previous data collection. +USN ranks programs using scores generated from surveys sent to academic professionals2. Only survey responses +from fall 2021 and early 2022 were used to compute the scores. The surveys asked respondents to rate each program +from 1 to 5, with one being marginal and five being outstanding. Respondents could skip programs and select "don’t +know" if they were unfamiliar with them. Each program’s score is the average of its survey ratings if it has at least ten +ratings. Programs with less than ten ratings are not scored. Unlike the scores reported in USN 2017, where the program +is ranked if it has a score of at least 2.0, USN 2022 published and ranked the scores of programs that are lower than +2.0. USN does not provide raw survey data or information about potential sources of bias in responses. USN does not +attempt to fill in missing values. +2.2 +Computer Science Faculty List Data +We collected the data on 5,574 tenure-track CS professors from 185 departments ranked by USN. We identified +2,011 faculty on our 2022 list but not on our 2017 list, including 1,750 new professors and 261 professors who joined +another department. In contrast, 4,728 professors were collected in 2017 from 173 departments. The number of CS +professors included in our list increased by 17.89%. +We consider a professor part of a department if the professor is listed on the website’s faculty list. We found each +website by performing Google searches on the school’s name followed by "computer science" or "cs." In most cases, +lists of faculty and their appointments were on pages labeled "Directory," "People," or "Faculty." Some pages did not +specify appointments. In these cases, we found a professor’s appointment by performing a Google search on their name +and exploring their website or profile page. +We only consider tenure-track professors, which would have the rank of assistant, associate, or full professor. +We excluded professors who have the titles "Clinical", "Courtesy", "Adjunct", "Research", "Teaching", "Emeritus", +"Visiting", or other additional labels that indicate that the professor was not a tenure-track professor. +For universities with CS departments, we added all professors because they were in a department for CS professors +only. In some universities, the computer science faculty are part of joint departments called "Electrical Engineering +and Computer Sciences" or "Computer Science and Engineering." Some universities have colleges or departments +of computing or informatics, which contain faculty in CS, library science, information sciences, or management +information systems. These departments made it harder to distinguish who was a CS professor. We determined that +professors with research interests and publications in CS topics will be CS professors. We looked at the publications or +research interests on their department profile page or website. CS topics include artificial intelligence, machine learning, +data science, human-computer interaction, bioinformatics, cybersecurity, and others. Some cases we do not consider +within CS are sensor networks, hardware, genomics, signals, and others. +There were some unique cases in choosing the departments. New York University has the Department of +Computer Science and Department of Computer Science and Engineering within two separate colleges. We only +considered the department within the Courant Institute of Mathematical Sciences. Case Western University has the +Department of Computer and Data Sciences and Department of Electrical, Computer and Systems Engineering. We +only included professors within The Department of Computer and Data Sciences because the faculty of the other +department generally had research interests that we excluded. Rochester Institute of Technology has a big college of +Computing and Information Sciences, which consists of several departments, such as Computer Science, Computing +and Information Sciences, Software Engineering, etc. We only included faculty from the department of Computer +Science since they listed their faculty in separate departments. +Another issue was affiliated professors and professors who have non-primary appointments. We did not include +affiliated faculty if they were in a separate section from the main faculty or labeled as having a joint or secondary +2https://www.usnews.com/education/best-graduate-schools/articles/science-schools-methodology +2 + +A PREPRINT - JANUARY 10, 2023 +appointment. In cases when affiliated, joint, or secondary appointed professors were mixed in with primary faculty, we +added all professors who were on the list because the department listed them as its professors. +Outside these departments, some professors have made significant contributions in CS venues but have primary +appointments within engineering, biology, statistics, business, or other departments. We did not include non-CS +professors because we needed a time-effective and unbiased method of finding these professors. +In 2017, we found that 23.6% (1114/4728) of CS faculty are assistant professors, and that percentage has changed +to 29.2% (1630/5574) in 2022. Since assistant professors are starting to establish their publications, we treat them +differently from associate and full professors. We refer to associate and full professors as senior faculty, and our +collection of Google Scholar data focused on senior faculty. +2.3 +Google Scholar Data +After determining what professors are in each department, we identified each professor’s Google Scholar page +to collect their citation measures3. Despite some limitations of automated web crawling [2][3], the quality of Google +Scholar data is comparable to the data coming from the subscription-based services for journal publications such as +Web of Science [4]. A Google Scholar page lists the individual’s publications with each respective number of citations +and overall aggregate citation measures. A profile’s aggregate citation measures include the h-index and i10-index +(i10). The h-index [5] is defined as the highest number x for which the individual has x number of papers with at least x +number of citations. The i10 is the total number of papers with above ten citations by an individual. +We found each page by searching Google Scholar for the professor’s name and university. Some professors have +common names, and multiple people appeared in the search result. We ensured that the professor’s rank, university, and +research topics aligned with the page. Sometimes, the Google Scholar page would list a university where the professor +was a previous student or faculty member. We confirmed that it was the correct page by looking at past affiliations +through their websites or department website pages. +We found about 89.8% (5,005/5,574) of professors’ Google Scholar pages. About 85.7% (3,379/3,944) of senior +faculty have a page. We determined that using the data of professors who have Google Scholar pages is biased because +they tend to have higher citation measures. To prevent bias from affecting the results, we decided to collect citation +measures for professors who did not have a page. We introduced the t10 index, a citation measure that would be easy to +collect for professors with or without a Google Scholar page. This measure is explained in the next section. +2.4 +t10 Index +The t10 index (t10) is defined as the number of citations of one’s 10th most cited paper. Identifying this index is +more convenient and less prone to error than the h-index when performing a manual search. The t10 is obtained by +identifying an individual’s ten most cited papers. In contrast, the h-index is obtained by identifying the top x number +of papers that have at least x citations. Because assistant professors are starting to build their publication records, we +decided to focus on collecting the t10 of the senior faculty. +We obtained the t10 of the associate and full professors with a Google Scholar page using a web scraping program +that takes the 10th highest paper on each page. This is a simple process because a Google Scholar page lists the +individual’s highest-cited articles in descending order. We collected them manually for professors who do not have a +Google Scholar page. To save time, we took the t10 that were gathered in 2017 and matched them to each professor +who did not have a t10. +To manually gather the t10, we searched Google Scholar for the professor’s name. The search engine typically +retrieves publications with the name in the author list by descending the number of citations. We looked for the 10th +highest cited paper from the search results with the professor’s name in the author list. For professors with common +names, the search results would show publications from multiple people. We checked each publication to ensure the +author was the correct professor. +We identified the t10 of 5,553 of the 5,574 CS faculty (99.6% coverage) and for 3,932 of the 3,944 senior CS +faculty (99.7% coverage) by manually searching Google Scholar. However, when a faculty has a common name or a +faculty name listed on the people pages does not precisely match the name listed in their papers, obtaining t10-index +can be too time-consuming. To save time, the curators were instructed to abort the extraction if it took more than 5 +minutes. As a result, we did not collect t10 for 21 of the 5,574 CS faculty (0.4%). Since a faculty’s name should not +have an influence on their citation count, the resulting sample of faculty with known t10 can be treated as an unbiased +sample of the senior CS faculty. +3https://scholar.google.com/ +3 + +A PREPRINT - JANUARY 10, 2023 +Furthermore, among 3,416 faculty in both the 2017 and 2022 data, we found that 2,520 (73.8% coverage) have an +h-index and t10. 459 of them have added google scholar profiles since 2017, and none of them removed their google +scholar profile. 926 of them were promoted during the past few years. Among 1,750 new faculty in our 2022 data, +1,636 have a google scholar profile, where 205 are full professors, 196 are associate professors, and 1,235 are assistant +professors. 1,741 of these new faculty have t10, where 252 are full professors, 219 are associate professors, and 1,270 +are assistant professors. +3 +Methods +3.1 +Program Strength Measures +We propose two approaches for using individual citation measures to calculate the strength of a program, averaged +and cumulative citation measures, the same as those used in 2017. +3.2 +Averaged citation measures +The first method we use to measure the strength of a program is by averaging citations of its faculty members [6]. +We use three different averaging schemes. The first averaging scheme is the median of t10 values of senior CS faculty, +which we denote as m10. The second is the geometric mean of (1+t10) values of senior CS faculty, which we denote +as g10. The third is the average percentile of the senior faculty’s t10, which we denote as p10. We exclude assistant +professors from the averaged measures because their citations are typically smaller, and their inclusion would hurt +departments with more assistant professors. +3.3 +Cumulative citation measures +The second method to measure the strength of a program is to count the number of highly cited faculty in a +program. We define a t10 threshold to determine which professors are highly cited. We introduce cN, which denotes the +number of faculty whose t10 is higher than N% of senior faculty, with 0 < N < 100. For example, c40 is the count of +professors within a department with a t10 higher than 40% of senior faculty. We include all faculty to calculate cN. +3.4 +Regression Models +We use regression models that combine the averaged and cumulative citation measures into an aggregated score. +The regression models that we consider are of the type in formula 1: +si = β0 + β1ai + β2ci, +(1) +si is the predicted USN CS score, ai is an aggregated citation measure, and ci is a cumulative citation measure of +the i-th program. The regression parameters are β0, β1, and β2. Instead of learning the intercept parameter β0, we set it +to β0 = 1 by default. The primary justification is that a program with ai = 0 and si = 0 does not have active research +faculty; based on the peer assessment instructions by USN, this program would have a score of 1 ("marginal"). The +secondary justification is that the resulting regression models would be simpler because they only require fitting two +regression parameters, β1, and β2. We train one regression model for each combination of the three averaged and the +ranges of cumulative measures. We average the individual regression models to create an aggregate score. +4 +Results +4.1 +Department size +Fig.1 compares the number of assistant, associate, and full professors between our newly collected data and that +from our 2017 paper. The percentage of assistant professors increased during the past four years, which indicates that +more young professors are joining CS academia. The reason might be that recently popular areas, such as machine +learning or deep learning, are attracting more young professors to contribute to research. +Fig.2 shows the distribution of department sizes, defined as the number of tenure-track faculty in each of the 185 +CS programs. The median faculty size is 23, the mode is 20, the minimum is 3, and the maximum is 170 (Carnegie +Mellon University). We also show the scatter plot between the department size in 2017 and 2022 in Fig.3. The colors +represent the USN CS scores, and it can be seen there is an overall increase in department sizes. The department +4 + +A PREPRINT - JANUARY 10, 2023 +Figure 1: Trend of faculty size +Figure 2: The distribution of the U.S. CS department size +size in 2022 has a similar distribution to that in 2017, since there is a nearly linear relationship between the two. The +correlation coefficient of the department sizes and the USN CS scores of the 185 programs is 0.755, which is higher +than our previous finding (0.676), indicating that larger departments are more likely to be ranked higher. +4.2 +Distribution of the Citation Indices +As shown in Fig.4, the median value of the h-index, i10-index, and t10 index all increased compared to our 2017 +result. Particularly, the t10 index has a larger rise compared to the h-index and i10-index, indicating that it is a more +sensitive citation metric. +In Fig.5, we show the histogram of the t10 for the 3,944 senior faculty. We observe a similar distribution pattern +compared to the 2017 result. Since the t10 distribution resembles a lognormal distribution, the histogram of t10 is +shown in a log scale. We observe a bump at low values, representing the 70 senior faculty with a t10 of 0, meaning they +have less than ten cited papers listed in Google Scholar. The median of t10 is 114, and the percentiles of t10 are shown +in Table 1. Overall, the t10 values increased compared to our results from 2017. +5 + +60 +old data +new data +49.6% +50 +46.8% +40 +Percentage +29.2% +30 +26.9% +24.0% +23.6% +20 +10 +0 +Full professor +Associate professor +Assistant professor50 +45 +40 +35 +30 +uno +25 +20 +15 +10 +5 +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +Faculty SizeA PREPRINT - JANUARY 10, 2023 +Figure 3: Department size in 2016 and 2022 +Figure 4: Trend of citation measurements (median value) +The correlation coefficient between logarithms of h-index and t10 for the 3,379 senior faculty with both indices is +0.943, which is close to our 2017 result. The sufficiently high correlation concludes that the t10 is a good proxy for the +h-index. +Table 1: Percentiles of t10 +Percentile +t10 +10% +21 +20% +40 +30% +60 +40% +83 +50% +115 +60% +154 +Percentile +t10 +70% +212 +80% +301 +90% +493 +95% +751 +98% +1247 +99% +1843 +6 + +Score. +175 +150 +(2022) +125 +100 +3 +size +Faculty +75 +2 +50 +25 +1 +F 0 +20 +-0 +0 +40 +60 +80 +100 +120 +140 +Faculty size (2016)80 +old data +75 +new data +70 +62 +60 +indices +50 +48 +le +44 +40 +30 +27 +25 +20 +10 +0 +h-index +il0-index +tl0-indexA PREPRINT - JANUARY 10, 2023 +Figure 5: Histogram of t10 of associate and full CS professors +Figure 6: Number of tenured CS faculty with (blue) and without (orange) Google scholar profile as a function of the t10 +percentile +4.3 +Scholar profile bias +While the median of t10 for the 3,932 senior CS faculty is 114, it increases to 131 among the 3,379 professors +who have a Google scholar profile and drops to 33 among the 553 without a profile. In Fig. 6 we show a stacked bar +plot of the numbers of faculty with and without Google scholar profiles as a function of their t10 percentile. These +results are consistent with the observations in our 2017 paper, which indicate that CS faculty who have Google scholar +profiles are a biased sample of the entire CS faculty and validate our effort to gather the t10 values and use them in our +study instead of the h-index. +4.4 +Scholar model +According to the above analysis, it can be observed our 2022 data shows a similar pattern and trend compared to +our 2017 data. Hence, we decided to keep the same linear regression scholar model used in 2017 and directly applied it +to our 2022 data, which is shown in formula 2: +s = 1 + 0.058 +√ +m10 + 0.059 +� +g10 + 0.121 +√ +c40 + 0.127 +√ +c60 +(2) +7 + +600 +500 +400 +Count +300 +200 +100 +0 +0.5 +1.5 +2 +2.5 +3 +3.5 +4 +log10(t10)450 +400 +350 +300 +Count +250 +200 +150 +100 +50 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +- +Percentile RangeA PREPRINT - JANUARY 10, 2023 +Figure 7: Comparison of scholar model scores and USN CS scores of CS graduate programs. +In Fig 7, we show a scatter plot of the USN CS scores and scholar model scores for the 185 CS programs. A +closer look at the scatter plot reveals that two groups of CS programs can be distinguished with respect to the correlation +between the USN CS scores and scholar model scores. The first group contains 73 programs that were scored 2.8 +and higher by the USN. The correlation between the USN CS scores and joint model scores in this group is 0.914. +The second group contains 112 programs with USN scores between 0 and 2.7. The correlation between the USN CS +scores and scholar model scores in this low-scoring group is 0.673, which is much higher than what we observed in +2017 (0.360) but still lower than those 73 programs. We hypothesize that the CS programs whose USN CS scores are +between 0 and 2.8 are not sufficiently well-known among the peers to provide objective and reliable peer assessment at +the national level. +4.5 +Comparison Study +We first study the newly added professor list to compare our new ranking result with the previous one conducted +in 2017. Among those 1,750 new professors, it is observed that 1,518 (86.7%) of them have a t10-index lower than +the average t10-index of all faculty from the department they join in. 223 of them are higher, and 9 of them have a +missing t10-index. Furthermore, among those 261 professors who transferred to another department, 114 have joined +a higher-ranking department than their previous department, and 147 have joined a lower-ranking department. We +also compared their t10-index in 2017 with the previous department average before they moved. It is an interesting +finding that only 66 have a t10-index higher than the department average, and 176 have a t10-index lower or equal to +the department average. 19 of them have a missing t10-index. Based on these observations, it can be inferred that new +professors are primarily young, with a lower t10-index. It is also true that young professors at the starting stage of their +careers are more likely to transfer to another university, and most of them tend to join a higher-ranking university. The +story might be they built a stronger profile during the past 5 years and then joined a higher-ranking department. +4.5.1 +2017 ranking vs. 2022 ranking +To better understand how the ranking has changed during the past five years, we calculate the ranking difference +between the results we produced in 2017 and 2022 using the same scholar model created in 2017. Fig. 8 shows the +box plot of the absolute difference between our old (2017) and new (2022) scholar ranking results. It can be seen that +the placements of top universities above rank 90 are more stable, while lower-ranked universities tend to have more +significant variations. However, we observe some outliers. For example, Northeastern University jumped from rank 40 +to 21 because this department may have recruited many new professors since 2017. +Another observation is that larger departments are not necessarily ranked higher. For example, Stanford University +and Princeton University both have relatively small departments. Still, they are within the top 10 departments in our +ranking, indicating that other factors, such as faculty citations, significantly influence the ranking result. +8 + +5.5 +5 +00 +O +000 +4.5 +8 +o +4 +900 +o +8 +o +8 +o +08 +Model +08 +80 +3 +000 +. +2.5 +08 +009 +00 +00 +8000 +08 +1.59 +8000 +8 +o +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +U.S.News ScoreA PREPRINT - JANUARY 10, 2023 +Figure 8: Absolute value of ranking difference between our 2017 and 2022 scholar ranking results +Figure 9: Absolute value of ranking difference between USN 2017 and USN 2022 results +Fig. 9 shows the absolute difference between the USN 2017 and USN 2022 ranking. It can be observed that the +ranking difference has a similar pattern compared to what has been shown in Fig. 8. This indicates that our method +using the t-10 index of faculty to rank the program produces similar ranking results compared to USN. +4.5.2 +Scholar ranking vs. USN +The new ranking results and a comparison to the USN scores are shown in the Appendix. To justify that it +is appropriate to apply the linear regression model obtained from the 2017 data to the 2022 data, we calculated the +correlations between our scholar ranking result and the USN ranking in 2017 and 2022, respectively. The results are +shown in Table 2. +Table 2: Correlation between USN ranking and our scholar ranking using the 2017 regression model +R2 +Pearson +Spearman +USN 2017 vs. scholar ranking 2017 +0.8731 +0.9357 +0.8978 +USN 2022 vs. scholar ranking 2022 +0.8734 +0.9390 +0.9126 +Despite the fact that we used the same regression formula for both 2017 and 2022 data, our scholar ranking results +show a high correlation with both USN 2017 and USN 2022 rankings. This result confirms that our practice of using +the 2017 ranking formula on our 2022 data is justifiable. +To further investigate the relationship between our scholar ranking and the USN ranking, we calculated the +difference between our new ranking and the USN 2022 ranking. Fig. 10 shows the boxplot of the absolute value of +difference. We separate universities into six groups based on their new scholar rank. It can be observed that our ranking +9 + +50 +40 +Rank difference +30 +20 +10 +0 +Rank 1-30 +Rank 31-60 +Rank 61-90 +Rank 91-120 +University ranking (2022)50 +40 +0 +Rank difference +30 +0 +20 +10 +0 +Rank 1-30 +Rank 31-60 +Rank 61-90 +Rank91-120 +University ranking (2022)A PREPRINT - JANUARY 10, 2023 +Figure 10: Absolute value of ranking difference between our 2022 scholar ranking and USN 2022 ranking +Figure 11: Histogram plot of ranking score increase between 2022 and 2017 +model is more likely to match with the USN ranking for higher-ranking departments. For instance, all departments that +rank 1-30 show a rank difference of less than 10. +We also calculate the ranking score increase in 2022 compared to 2017 for both our scholar model and USN and +show the histogram plots in Fig. 11. It can be seen that most departments have a score increase in both the scholar +model and USN, and all of them are within the range between -0.4 and 0.8. There are a few extreme cases. For example, +the ranking score of Northeastern University increased by 0.8; where the reason may be the department recruited many +new faculty during the past few years. UNC Chapel Hill has a score decrease of 0.3. Their department size remains the +same (32), but the m10, g10, c40, and c60 values drop in 2022. +4.5.3 +Scholar ranking vs. CSRankings +Table 3: Correlation between CSRankings, USN 2022 and our scholar ranking +R2 +Pearson +Spearman +USN vs. scholar ranking +0.8734 +0.9390 +0.9126 +USN vs. CSRankings +0.8391 +0.9160 +0.9157 +scholar ranking vs. CSRankings +0.8375 +0.9151 +0.8965 +USN vs. average model +0.8462 +0.9199 +0.9305 +To further investigate the relationship between our ranking and some widely-used CS ranking results, we compared +the CSRankings result with our ranking. Unlike USN ranking, CSRankings4 relies on publications in top-tier computer +science conferences, as reported by DBLP, a computer science bibliography website 5. To study the relationship between +CSRankings and our scholar ranking, we collected the current CSRankings result and calculated its correlations with +4http://csrankings.org +5https://dblp.org/ +10 + +50 +40 +Rank difference +30 +20 +10 +0 +Rank1-30 +Rank 31-60 +Rank 61-90 +Rank 91-120 +Universityranking(2022)Histogram of ranking score increase in 2022 compared to 2017 +Scholarranking +USN ranking +35 +30 +25 +20 +Count +15 +10 +5 +0 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +Ranking score increase +Ranking score increaseA PREPRINT - JANUARY 10, 2023 +Figure 12: Absolute value of ranking difference between CSRankings and USN 2022 +Figure 13: Absolute value of ranking difference between CSRankings and our scholar ranking +USN and our scholar ranking. Since the CSRanking score is based on a different scale, we applied log transformation +to its score before computing the correlations. The result is shown in Table 3. +It can be seen from the Table that our scholar ranking result has a higher correlation with the USN ranking, which +indicates that it is better aligned with the USN ranking compared to CSRankings. This indication is further justified +in Fig. 10 and Fig. 12, where we show the ranking difference between 1) our scholar ranking results with USN and +2) CSRankings with USN. It can be seen that CSRankings shows an overall more significant ranking difference with +USN compared to our scholar model ranking, especially for the top 60 departments. An interesting finding is that +CSRankings is better aligned with USN than our scholar ranking for those lower-ranking departments (>90). The +ranking difference between CSRankings and our scholar model ranking is shown in Fig. 13. The result shows that the +difference increases with a larger variance as the ranking of the department decreases. This indicates that the ranking +of top departments is more stable despite which model is being used, whereas lower-ranking departments are more +sensitive to the ranking method. We also build an average model by computing the average score using the scholar +model score and CSRankings score (shown in the last row of Table), which yields a higher spearman correlation with +the USN ranking. +References +[1] Slobodan Vucetic, Ashis Kumar Chanda, Shanshan Zhang, Tian Bai, and Aniruddha Maiti. Faculty citation measures +are highly correlated with peer assessment of computer science doctoral programs. ArXiv, abs/1708.05435, 2017. +[2] Emilio Delgado López-Cózar, Nicolás Robinson-García, and Daniel Torres-Salinas. The Google scholar experiment: +How to index false papers and manipulate bibliometric indicators. Journal of the Association for Information +Science & Technology, 65(3):446–454, March 2014. +[3] Péter Jacsó. Deflated, inflated and phantom citation counts. Online Inf. Rev., 30:297–309, 2006. +11 + +50 +40 +30 +0 +0 +20 +8 +10 +0 +Rank 1-30 +Rank 31-60 +Rank 61-90 +Rank 91-120 +Universityranking(2022)50 +40 +0 +30 +20 +0 +Rank +0 +10 +0 +Rank 1-30 +Rank 31-60 +Rank 61-90 +Rank91-120 +University ranking (2022)A PREPRINT - JANUARY 10, 2023 +[4] Judit Bar-Ilan. Which h-index?—a comparison of wos, scopus and google scholar. Scientometrics, 74:257–271, 02 +2008. +[5] J. E. Hirsch. An index to quantify an individual’s scientific research output. Proceedings of the National Academy +of Sciences, 102(46):16569–16572, 2005. +[6] Themis Lazaridis. Ranking university departments using the mean h-index. Scientometrics, 82:211–216, 2009. +12 + +A PREPRINT - JANUARY 10, 2023 +Table 4: List of 185 U.S. CS graduate programs: Ranking by our scholar model (Rank), University name (University), +Number of tenured faculty with t10 score (Size), median t10 score of all the faculty (M10), geometric mean of t10 score +of all faculty (G10), number of highly cited faculty based on c40 (C40) and c60 (C60), U.S. News CS score (USN), +Scholar score (Scholar) +Rank +University +Size +M10 +G10 +C40 +C60 +USN +Scholar +USN 2016 +Scholar 2016 +1 +Carnegie Mellon University +170 +280 +262 +121 +84 +4.9 +5 +5 +5 +1 +Cornell University +102 +327 +315 +75 +58 +4.6 +5 +4.5 +4.4 +1 +Massachusetts Institute of Technology +110 +304 +302 +92 +74 +5 +5 +5 +5 +1 +Stanford University +64 +707 +706 +60 +53 +4.9 +5 +5 +5 +1 +University of California Berkeley +82 +421 +461 +68 +60 +4.9 +5 +5 +5 +6 +University of Washington +73 +345 +295 +58 +47 +4.6 +4.9 +4.5 +4.3 +7 +University of California San Diego +63 +341 +330 +49 +39 +4.3 +4.8 +4 +4.2 +8 +Georgia Institute of Technology +105 +203 +198 +75 +53 +4.6 +4.6 +4.3 +4.3 +8 +Princeton University +48 +360 +345 +36 +30 +4.5 +4.6 +4.4 +4.1 +8 +University of Illinois Urbana Champaign +76 +265 +248 +56 +43 +4.7 +4.6 +4.6 +4.1 +11 +University of California Los Angeles +44 +317 +316 +35 +31 +4.3 +4.5 +4.1 +4.2 +11 +University of Pennsylvania +67 +256 +238 +49 +39 +4.1 +4.5 +3.8 +3.7 +13 +Columbia University +52 +249 +288 +42 +34 +4.3 +4.4 +4 +4.1 +13 +New York University +44 +288 +254 +38 +29 +3.5 +4.4 +3.4 +4 +13 +University of Michigan Ann Arbor +67 +251 +254 +47 +35 +4.3 +4.4 +4.1 +4.1 +16 +Harvard University +34 +277 +286 +28 +22 +4.2 +4.2 +3.9 +3.7 +17 +Duke University +56 +216 +195 +37 +26 +3.9 +4.1 +3.6 +3.6 +17 +Johns Hopkins University +32 +258 +309 +23 +17 +4 +4.1 +3.5 +4 +17 +University of Maryland College Park +56 +219 +186 +37 +29 +4.1 +4.1 +4 +4 +17 +University of Wisconsin Madison +45 +285 +225 +28 +20 +4.1 +4.1 +4.2 +3.9 +21 +Northeastern University +77 +165 +180 +45 +30 +3.6 +4 +2.7 +3.2 +21 +University of Southern California +40 +272 +238 +26 +19 +3.9 +4 +3.7 +3.9 +21 +University of Texas Austin +53 +182 +184 +38 +25 +4.5 +4 +4.3 +3.7 +24 +Brown University +31 +224 +233 +22 +17 +3.8 +3.9 +3.7 +3.5 +24 +University of Massachusetts Amherst +59 +191 +187 +33 +22 +3.9 +3.9 +3.6 +3.7 +26 +University of Chicago +48 +174 +208 +29 +17 +3.7 +3.8 +3.3 +3.5 +26 +Yale University +26 +243 +231 +19 +14 +4 +3.8 +3.7 +4 +28 +California Institute of Technology +21 +240 +240 +16 +12 +4.3 +3.7 +4.2 +3.7 +28 +Pennsylvania State University University Park +44 +200 +176 +25 +17 +3.6 +3.7 +3.4 +3.4 +28 +Rice University +28 +213 +204 +18 +13 +3.7 +3.7 +3.7 +3.3 +28 +University of California Santa Barbara +37 +183 +194 +25 +15 +3.7 +3.7 +3.3 +3.6 +28 +University of Minnesota Twin Cities +46 +145 +186 +30 +15 +3.6 +3.7 +3.4 +3.4 +33 +Purdue University West Lafayette +72 +137 +133 +31 +17 +4 +3.6 +3.7 +3.3 +33 +Stony Brook University SUNY +45 +152 +137 +27 +17 +3.4 +3.6 +3.1 +3.3 +33 +University of California Davis +37 +178 +162 +23 +18 +3.4 +3.6 +3.3 +3.5 +33 +University of California Irvine +50 +152 +132 +28 +19 +3.7 +3.6 +3.4 +3.4 +33 +University of Virginia +40 +193 +179 +20 +14 +3.7 +3.6 +3.4 +3.1 +38 +University of California Santa Cruz +39 +185 +153 +20 +14 +3.2 +3.5 +2.8 +3.5 +39 +Boston University +33 +141 +167 +18 +11 +3.3 +3.4 +3 +3.2 +39 +Michigan State University +39 +151 +159 +20 +12 +3 +3.4 +2.8 +3 +13 + +A PREPRINT - JANUARY 10, 2023 +39 +Northwestern University +44 +122 +123 +26 +12 +3.7 +3.4 +3.3 +3.1 +39 +Rutgers University +39 +151 +152 +18 +10 +3.5 +3.4 +3.3 +3.3 +39 +University of Arizona +22 +197 +168 +14 +9 +3.2 +3.4 +3.1 +3.2 +39 +University of California Riverside +34 +153 +133 +22 +12 +3 +3.4 +2.8 +3.3 +45 +Arizona State University +54 +92 +122 +21 +16 +3.2 +3.3 +3 +2.9 +45 +University of Colorado Boulder +51 +127 +100 +24 +12 +3.5 +3.3 +3.1 +3 +45 +University of Rochester +18 +186 +160 +10 +7 +3.1 +3.3 +2.9 +3 +45 +University of Utah +54 +117 +121 +22 +12 +3.4 +3.3 +3.1 +3 +45 +Vanderbilt University +26 +164 +155 +13 +10 +3.1 +3.3 +2.8 +2.9 +45 +Washington University in St Louis +28 +135 +150 +17 +10 +3.4 +3.3 +3.1 +2.9 +51 +University of North Carolina Chapel Hill +32 +139 +122 +16 +10 +3.8 +3.2 +3.6 +3.5 +51 +University of Notre Dame +27 +126 +144 +17 +7 +3.2 +3.2 +2.7 +2.7 +53 +Colorado State University +22 +115 +136 +14 +7 +2.5 +3.1 +2.4 +3 +53 +George Mason University +47 +100 +99 +21 +10 +2.9 +3.1 +2.5 +2.9 +53 +Indiana University Bloomington +36 +103 +99 +17 +9 +3.1 +3.1 +2.9 +3 +53 +Ohio State University +44 +99 +84 +22 +11 +3.6 +3.1 +3.3 +3.1 +53 +Texas AM University College Station +48 +94 +90 +22 +9 +3.5 +3.1 +3.1 +2.9 +53 +University of Central Florida +37 +89 +100 +18 +10 +2.8 +3.1 +2.2 +2.6 +53 +University of Tennessee Knoxville +29 +125 +114 +13 +9 +2.5 +3.1 +2.4 +3 +53 +University of Texas Dallas +52 +88 +76 +24 +15 +2.9 +3.1 +2.4 +2.9 +53 +Virginia Tech +56 +93 +99 +24 +9 +3.5 +3.1 +3.1 +3 +62 +College of William and Mary +21 +136 +140 +9 +6 +2.8 +3 +2.4 +2.8 +62 +Lehigh University +20 +148 +129 +9 +5 +2.5 +3 +2.1 +2.7 +62 +Temple University +23 +130 +103 +12 +7 +2.4 +3 +2 +2.6 +62 +University at Buffalo SUNY +39 +96 +90 +17 +9 +2.9 +3 +2.6 +3 +62 +University of Florida +47 +82 +79 +18 +11 +3.4 +3 +3 +2.7 +67 +North Carolina State University +43 +82 +83 +16 +9 +3.2 +2.9 +3 +2.9 +67 +Rensselaer Polytechnic Institute +19 +89 +134 +9 +6 +3.1 +2.9 +2.9 +2.8 +67 +University of Maryland Baltimore County +29 +94 +101 +12 +6 +2.7 +2.9 +2.4 +2.7 +67 +University of Pittsburgh +23 +104 +104 +11 +6 +3.1 +2.9 +2.9 +2.8 +71 +CUNY Graduate School and University Center +99 +37 +43 +25 +13 +2.1 +2.8 +2.3 +2.6 +71 +Dartmouth College +20 +97 +110 +8 +3 +3.2 +2.8 +3.1 +2.7 +71 +Oregon State University +49 +84 +69 +17 +4 +2.9 +2.8 +2.5 +2.3 +71 +University of Houston +24 +95 +88 +13 +4 +2.2 +2.8 +2.1 +2.4 +71 +University of Illinois Chicago +39 +85 +100 +13 +5 +3 +2.8 +2.7 +2.7 +71 +University of Texas Arlington +37 +85 +69 +13 +6 +2.5 +2.8 +2.2 +2.7 +71 +Wayne State University +22 +109 +89 +11 +3 +2.3 +2.8 +2 +2.4 +78 +New Jersey Institute of Technology +30 +70 +70 +11 +6 +2.5 +2.7 +2.2 +2.4 +78 +Portland State University +19 +94 +71 +7 +6 +2 +2.7 +0 +2.7 +78 +Tufts University +21 +81 +95 +8 +3 +2.8 +2.7 +2.4 +2.4 +78 +University of Memphis +18 +88 +99 +7 +5 +1.8 +2.7 +0 +2.4 +78 +University of New Mexico +16 +93 +89 +7 +3 +2.4 +2.7 +2.3 +2.4 +78 +University of California Merced +20 +91 +106 +7 +4 +2.2 +2.7 +84 +Binghamton University SUNY +27 +76 +56 +11 +4 +2.4 +2.6 +2 +2.2 +14 + +A PREPRINT - JANUARY 10, 2023 +84 +Drexel University +15 +94 +80 +6 +2 +2.7 +2.6 +2.2 +2.4 +84 +Illinois Institute of Technology +23 +74 +65 +9 +4 +2.4 +2.6 +2.1 +2.5 +84 +University of Connecticut +28 +102 +78 +9 +1 +2.7 +2.6 +2.3 +2.3 +84 +University of Georgia +24 +81 +65 +8 +4 +2.5 +2.6 +2.2 +2.2 +84 +University of Massachusetts Lowell +26 +90 +47 +7 +5 +2.1 +2.6 +0 +2.1 +84 +University of Missouri +29 +62 +65 +11 +4 +2.3 +2.6 +2.1 +2.4 +84 +University of Nebraska Lincoln +31 +78 +68 +10 +3 +2.6 +2.6 +2.4 +2.6 +84 +University of Oregon +20 +97 +93 +7 +2 +2.7 +2.6 +2.6 +2.2 +84 +University of South Florida +26 +60 +77 +7 +6 +2.3 +2.6 +2.1 +2.5 +84 +Washington State University +22 +75 +85 +8 +3 +2.7 +2.6 +2.4 +2 +84 +West Virginia University +11 +117 +62 +6 +4 +2 +2.6 +2 +2.3 +84 +Worcester Polytechnic Institute +32 +69 +76 +9 +4 +2.5 +2.6 +2.2 +2.4 +97 +Case Western Reserve University +16 +91 +68 +7 +2 +2.9 +2.5 +2.4 +2.4 +97 +Florida Atlantic University +27 +53 +55 +11 +4 +1.9 +2.5 +0 +2.1 +97 +Florida International University +34 +60 +42 +12 +4 +2.1 +2.5 +0 +2.2 +97 +Georgia State University +28 +59 +65 +9 +3 +2.1 +2.5 +2 +2.3 +97 +Iowa State University +29 +70 +63 +8 +3 +2.9 +2.5 +2.6 +2.2 +97 +Syracuse University +25 +86 +48 +9 +2 +2.8 +2.5 +2.5 +2.2 +97 +University of Delaware +31 +73 +54 +6 +4 +2.6 +2.5 +2.4 +2.5 +97 +University of Iowa +21 +79 +82 +6 +2 +2.8 +2.5 +2.6 +2.3 +97 +University of Massachusetts Boston +17 +81 +82 +6 +1 +2.2 +2.5 +0 +2 +97 +University of South Carolina +28 +65 +73 +10 +1 +2.3 +2.5 +2.1 +2.2 +97 +University of Vermont +11 +113 +95 +2 +2 +1.9 +2.5 +108 +Brandeis University +18 +67 +36 +7 +4 +2.3 +2.4 +2.3 +2.3 +108 +Brigham Young University +36 +66 +32 +8 +3 +2.4 +2.4 +2.2 +2.3 +108 +Clemson University +38 +62 +51 +8 +3 +2.6 +2.4 +2.3 +2.2 +108 +Florida State University +24 +74 +62 +8 +1 +2.6 +2.4 +2.3 +2.1 +108 +Kansas State University +16 +66 +61 +6 +2 +2.3 +2.4 +2.2 +1.9 +108 +University of Alabama Birmingham +9 +94 +80 +3 +1 +2 +2.4 +0 +2 +108 +University of North Texas +33 +61 +63 +8 +2 +1.9 +2.4 +0 +1.8 +115 +George Washington University +12 +67 +59 +3 +2 +2.7 +2.3 +2.3 +2.1 +115 +Louisiana State University +18 +54 +46 +5 +2 +2.1 +2.3 +2.1 +2.2 +115 +University of Nevada Reno +18 +61 +64 +3 +1 +1.8 +2.3 +0 +1.9 +115 +University of North Carolina Charlotte +19 +53 +55 +4 +2 +2.4 +2.3 +2.1 +1.9 +115 +University of Oklahoma +24 +51 +54 +4 +2 +2.2 +2.3 +2 +1.9 +115 +Utah State University +17 +65 +70 +4 +1 +2 +2.3 +0 +1.8 +115 +Virginia Commonwealth University +23 +58 +50 +7 +1 +2.1 +2.3 +0 +2 +115 +Emory University +14 +64 +45 +3 +3 +2.7 +2.3 +123 +Colorado School of Mines +15 +70 +61 +4 +0 +2.6 +2.2 +2.1 +2 +123 +Missouri University of Science Technology +14 +46 +57 +3 +1 +2.1 +2.2 +2 +2.2 +123 +University of Arkansas Fayetteville +22 +59 +51 +4 +1 +1.9 +2.2 +0 +2 +123 +University of Hawaii Manoa +19 +29 +39 +7 +3 +2 +2.2 +0 +2.2 +123 +University of Kansas +19 +60 +56 +2 +1 +2.5 +2.2 +2.3 +2.3 +123 +University of Texas San Antonio +20 +54 +52 +3 +1 +2.1 +2.2 +0 +2.3 +15 + +A PREPRINT - JANUARY 10, 2023 +129 +Mississippi State University +16 +48 +38 +2 +1 +1.9 +2.1 +0 +1.6 +129 +Naval Postgraduate School +20 +45 +39 +2 +1 +0 +2.1 +2.4 +2 +129 +Old Dominion University +21 +34 +46 +4 +1 +2 +2.1 +0 +2 +129 +Oregon Health and Science University +7 +66 +76 +1 +0 +1.8 +2.1 +2.2 +1.8 +129 +Stevens Institute of Technology +20 +54 +53 +3 +0 +2.6 +2.1 +2.1 +2.1 +129 +University of Missouri Kansas City +16 +61 +52 +3 +0 +1.8 +2.1 +0 +1.6 +129 +University of Texas El Paso +19 +34 +40 +4 +1 +1.6 +2.1 +0 +1.9 +129 +University of Tulsa +15 +37 +53 +3 +1 +1.8 +2.1 +0 +2 +129 +Western Michigan University +10 +51 +34 +2 +1 +1.5 +2.1 +0 +1.6 +138 +Auburn University +22 +39 +29 +2 +1 +2.5 +2 +2.2 +1.7 +138 +Claremont Graduate University +5 +66 +43 +2 +0 +1.6 +2 +0 +1.9 +138 +DePaul University +51 +24 +22 +4 +2 +1.9 +2 +0 +2 +138 +Kent State University +20 +39 +27 +3 +1 +1.9 +2 +0 +1.7 +138 +New Mexico State University +14 +59 +50 +2 +0 +2 +2 +0 +1.9 +138 +Texas Tech University +17 +41 +34 +1 +1 +2.1 +2 +0 +1.7 +138 +University at Albany SUNY +14 +62 +47 +1 +0 +2.2 +2 +2.1 +2.2 +138 +University of Alabama +16 +24 +29 +4 +2 +2.3 +2 +0 +2 +138 +University of Colorado Colorado Springs +15 +32 +39 +2 +2 +2 +2 +0 +1.9 +138 +University of Denver +9 +56 +51 +1 +0 +1.9 +2 +0 +1.8 +138 +University of Kentucky +19 +59 +32 +3 +0 +2.3 +2 +2.2 +2.2 +138 +University of Louisville +18 +32 +25 +4 +2 +1.8 +2 +0 +1.8 +138 +Ohio University +16 +49 +33 +3 +0 +1.9 +2 +151 +University of Louisiana Lafayette +22 +30 +24 +2 +1 +1.9 +1.9 +0 +1.8 +151 +University of Wisconsin Milwaukee +12 +57 +44 +1 +0 +2 +1.9 +0 +1.8 +153 +Florida Institute of Technology +27 +19 +18 +1 +1 +1.8 +1.8 +0 +1.7 +153 +Oakland University +23 +16 +25 +2 +1 +1.5 +1.8 +0 +1.9 +153 +Oklahoma State University +10 +18 +24 +2 +1 +2 +1.8 +0 +1.4 +153 +Southern Methodist University +8 +49 +20 +2 +0 +2.1 +1.8 +2 +1.6 +153 +University of Cincinnati +20 +33 +29 +2 +0 +2.2 +1.8 +2 +1.8 +153 +University of Maine +9 +33 +31 +2 +0 +1.8 +1.8 +0 +2 +153 +University of Mississippi +7 +33 +32 +2 +0 +1.9 +1.8 +0 +1.6 +153 +Clarkson University +10 +37 +31 +2 +0 +1.7 +1.8 +161 +Montana State University +10 +30 +34 +0 +0 +1.7 +1.7 +0 +1.6 +161 +North Dakota State University +12 +38 +32 +0 +0 +1.5 +1.7 +0 +1.5 +161 +University of Idaho +14 +33 +31 +0 +0 +1.8 +1.7 +0 +1.5 +161 +University of New Orleans +13 +30 +22 +1 +0 +1.5 +1.7 +0 +1.7 +161 +Rochester Institute of Technology +24 +26 +20 +2 +0 +2.7 +1.7 +161 +San Diego State University +13 +41 +15 +1 +0 +1.7 +1.7 +167 +Michigan Technological University +22 +22 +28 +0 +0 +2.1 +1.6 +0 +1.5 +167 +New Mexico Institute of Mining and Technology +7 +24 +25 +0 +0 +0 +1.6 +0 +1.4 +167 +Nova Southeastern University +17 +14 +12 +3 +0 +1.3 +1.6 +0 +1.4 +167 +Towson University +31 +15 +13 +1 +0 +1.6 +1.6 +0 +1.5 +167 +University of Southern Mississippi +12 +10 +13 +1 +1 +1.4 +1.6 +0 +1.5 +167 +University of Wyoming +7 +30 +25 +0 +0 +1.6 +1.6 +0 +1.6 +16 + +A PREPRINT - JANUARY 10, 2023 +167 +Southern Illinois University +12 +27 +28 +0 +0 +1.6 +1.6 +167 +University of New Hampshire +9 +24 +23 +0 +0 +2 +1.6 +167 +University of Puerto Rico Mayaguez +9 +12 +14 +2 +0 +1.4 +1.6 +167 +University of Rhode Island +10 +17 +14 +1 +0 +2 +1.6 +177 +Air Force Institute of Technology +6 +17 +18 +0 +0 +1.8 +1.5 +0 +1.5 +177 +Louisiana Tech University +6 +25 +15 +0 +0 +1.6 +1.5 +0 +1.3 +177 +University of Alabama Huntsville +13 +21 +23 +0 +0 +1.8 +1.5 +0 +1.7 +177 +University of Colorado Denver +14 +13 +11 +1 +0 +1.9 +1.5 +0 +1.4 +181 +University of Nebraska Omaha +16 +15 +11 +0 +0 +1.7 +1.4 +0 +1.4 +182 +University of Arkansas Little Rock +6 +10 +7 +0 +0 +1.7 +1.3 +0 +1.7 +183 +Bowie State University +12 +2 +5 +0 +0 +1.4 +1.2 +184 +Indiana State University +6 +0 +2 +0 +0 +1.8 +1.1 +0 +1.1 +184 +LIU Post +3 +0 +1 +0 +0 +1.3 +1.1 +0 +1.1 +17 + diff --git a/KdE1T4oBgHgl3EQfYgTD/content/tmp_files/load_file.txt b/KdE1T4oBgHgl3EQfYgTD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8f749ea8f5040336369154ac18c8290fd92e67ff --- /dev/null +++ b/KdE1T4oBgHgl3EQfYgTD/content/tmp_files/load_file.txt @@ -0,0 +1,981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf,len=980 +page_content='SCHOLAR RANKING 2023: RANKING OF COMPUTER SCIENCE DEPARTMENTS BASED ON FACULTY CITATIONS Sai Shi Department of Computer & Information Science Temple University Philadelphia, PA 19122 sai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='shi@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='edu Aniruddha Maiti Department of Computer & Information Science Temple University Philadelphia, PA 19122 aniruddha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='maiti@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='edu Ashis Kumar Chanda Department of Computer & Information Science Temple University Philadelphia, PA 19122 ashis@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='edu Slobodan Vucetic Department of Computer & Information Science Temple University Philadelphia, PA 19122 vucetic@temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='edu January 10, 2023 ABSTRACT Scholar Ranking 2023 is the second edition of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Computer Science (CS) departments ranking based on faculty citation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Using Google Scholar, we gathered data about publication citations for 5,574 tenure-track faculty from 185 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For each faculty, we extracted their t10 index, defined as the number of citations received by their 10th highest cited paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For each department, we calculated four quality metrics: median t10 (m10), the geometric mean of t10 (g10), and the number of well-cited faculty with t10 above 40% (c40) and 60% (c60) of the national average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We fitted a linear regression model using those four measures to match the 2022 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News ranking scores of CS doctoral programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The resulting model provides Scholar Ranking 2023, which can be found at https://chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='temple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='edu/csranking_scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 1 Introduction A previous version of the Scholar ranking [1] was published in the spring of 2017, based on citation data collected during the fall of 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This previous effort demonstrated that it is possible to learn a simple formula from citation measures that has a high correlation with peer assessment scores of CS doctoral programs published by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News (USN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' A few years have passed since the last publication of the Scholar ranking, and a new U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News ranking came out in 20221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Given the fact that the data on which the last ranking was performed is a few years old, we felt that it would be helpful to conduct another round of data collection and validate our method with the recent U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The first objective is to refine the data collection method and collect a new set of high-quality faculty citation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The second objective is to use the 2022 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News ranking to validate the method proposed in the first version of the scholar ranking and observe changes in the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The third objective is to analyze the trends in aggregated metrics used to perform the ranking given the data sets, with the first collected during the fall of 2016 and the other during the fall of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='usnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='com/best-graduate-schools/top-science-schools/computer-science-rankings arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='03140v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='DL] 9 Jan 2023 A PREPRINT - JANUARY 10, 2023 2 Data collection In this section, we explain the data collection process which took place from September 2021 to December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The data collection team consisted of two CS graduate students and a CS professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News (USN) Data USN is well-known for producing several rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We gathered the scores from the most recent ranking of CS doctoral programs, Best Computer Science Schools, which was published in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' USN collected the names of those to be surveyed for the science doctoral surveys in the summer of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We retained the scores from the 2013 version of Best Computer Science Schools from previous data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' USN ranks programs using scores generated from surveys sent to academic professionals2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Only survey responses from fall 2021 and early 2022 were used to compute the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The surveys asked respondents to rate each program from 1 to 5, with one being marginal and five being outstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Respondents could skip programs and select "don’t know" if they were unfamiliar with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Each program’s score is the average of its survey ratings if it has at least ten ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Programs with less than ten ratings are not scored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Unlike the scores reported in USN 2017, where the program is ranked if it has a score of at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0, USN 2022 published and ranked the scores of programs that are lower than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' USN does not provide raw survey data or information about potential sources of bias in responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' USN does not attempt to fill in missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 Computer Science Faculty List Data We collected the data on 5,574 tenure-track CS professors from 185 departments ranked by USN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We identified 2,011 faculty on our 2022 list but not on our 2017 list, including 1,750 new professors and 261 professors who joined another department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In contrast, 4,728 professors were collected in 2017 from 173 departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The number of CS professors included in our list increased by 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We consider a professor part of a department if the professor is listed on the website’s faculty list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We found each website by performing Google searches on the school’s name followed by "computer science" or "cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='" In most cases, lists of faculty and their appointments were on pages labeled "Directory," "People," or "Faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='" Some pages did not specify appointments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In these cases, we found a professor’s appointment by performing a Google search on their name and exploring their website or profile page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We only consider tenure-track professors, which would have the rank of assistant, associate, or full professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We excluded professors who have the titles "Clinical", "Courtesy", "Adjunct", "Research", "Teaching", "Emeritus", "Visiting", or other additional labels that indicate that the professor was not a tenure-track professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For universities with CS departments, we added all professors because they were in a department for CS professors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In some universities, the computer science faculty are part of joint departments called "Electrical Engineering and Computer Sciences" or "Computer Science and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='" Some universities have colleges or departments of computing or informatics, which contain faculty in CS, library science, information sciences, or management information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' These departments made it harder to distinguish who was a CS professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We determined that professors with research interests and publications in CS topics will be CS professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We looked at the publications or research interests on their department profile page or website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CS topics include artificial intelligence, machine learning, data science, human-computer interaction, bioinformatics, cybersecurity, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Some cases we do not consider within CS are sensor networks, hardware, genomics, signals, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' There were some unique cases in choosing the departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' New York University has the Department of Computer Science and Department of Computer Science and Engineering within two separate colleges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We only considered the department within the Courant Institute of Mathematical Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Case Western University has the Department of Computer and Data Sciences and Department of Electrical, Computer and Systems Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We only included professors within The Department of Computer and Data Sciences because the faculty of the other department generally had research interests that we excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Rochester Institute of Technology has a big college of Computing and Information Sciences, which consists of several departments, such as Computer Science, Computing and Information Sciences, Software Engineering, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We only included faculty from the department of Computer Science since they listed their faculty in separate departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Another issue was affiliated professors and professors who have non-primary appointments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We did not include affiliated faculty if they were in a separate section from the main faculty or labeled as having a joint or secondary 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='usnews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='com/education/best-graduate-schools/articles/science-schools-methodology 2 A PREPRINT - JANUARY 10, 2023 appointment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In cases when affiliated, joint, or secondary appointed professors were mixed in with primary faculty, we added all professors who were on the list because the department listed them as its professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Outside these departments, some professors have made significant contributions in CS venues but have primary appointments within engineering, biology, statistics, business, or other departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We did not include non-CS professors because we needed a time-effective and unbiased method of finding these professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In 2017, we found that 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6% (1114/4728) of CS faculty are assistant professors, and that percentage has changed to 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2% (1630/5574) in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Since assistant professors are starting to establish their publications, we treat them differently from associate and full professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We refer to associate and full professors as senior faculty, and our collection of Google Scholar data focused on senior faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 Google Scholar Data After determining what professors are in each department, we identified each professor’s Google Scholar page to collect their citation measures3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Despite some limitations of automated web crawling [2][3], the quality of Google Scholar data is comparable to the data coming from the subscription-based services for journal publications such as Web of Science [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' A Google Scholar page lists the individual’s publications with each respective number of citations and overall aggregate citation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' A profile’s aggregate citation measures include the h-index and i10-index (i10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The h-index [5] is defined as the highest number x for which the individual has x number of papers with at least x number of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The i10 is the total number of papers with above ten citations by an individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We found each page by searching Google Scholar for the professor’s name and university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Some professors have common names, and multiple people appeared in the search result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We ensured that the professor’s rank, university, and research topics aligned with the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Sometimes, the Google Scholar page would list a university where the professor was a previous student or faculty member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We confirmed that it was the correct page by looking at past affiliations through their websites or department website pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We found about 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8% (5,005/5,574) of professors’ Google Scholar pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' About 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7% (3,379/3,944) of senior faculty have a page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We determined that using the data of professors who have Google Scholar pages is biased because they tend to have higher citation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To prevent bias from affecting the results, we decided to collect citation measures for professors who did not have a page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We introduced the t10 index, a citation measure that would be easy to collect for professors with or without a Google Scholar page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This measure is explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 t10 Index The t10 index (t10) is defined as the number of citations of one’s 10th most cited paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Identifying this index is more convenient and less prone to error than the h-index when performing a manual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The t10 is obtained by identifying an individual’s ten most cited papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In contrast, the h-index is obtained by identifying the top x number of papers that have at least x citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Because assistant professors are starting to build their publication records, we decided to focus on collecting the t10 of the senior faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We obtained the t10 of the associate and full professors with a Google Scholar page using a web scraping program that takes the 10th highest paper on each page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This is a simple process because a Google Scholar page lists the individual’s highest-cited articles in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We collected them manually for professors who do not have a Google Scholar page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To save time, we took the t10 that were gathered in 2017 and matched them to each professor who did not have a t10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To manually gather the t10, we searched Google Scholar for the professor’s name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The search engine typically retrieves publications with the name in the author list by descending the number of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We looked for the 10th highest cited paper from the search results with the professor’s name in the author list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For professors with common names, the search results would show publications from multiple people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We checked each publication to ensure the author was the correct professor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We identified the t10 of 5,553 of the 5,574 CS faculty (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6% coverage) and for 3,932 of the 3,944 senior CS faculty (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7% coverage) by manually searching Google Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' However, when a faculty has a common name or a faculty name listed on the people pages does not precisely match the name listed in their papers, obtaining t10-index can be too time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To save time, the curators were instructed to abort the extraction if it took more than 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' As a result, we did not collect t10 for 21 of the 5,574 CS faculty (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Since a faculty’s name should not have an influence on their citation count, the resulting sample of faculty with known t10 can be treated as an unbiased sample of the senior CS faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 3https://scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='com/ 3 A PREPRINT - JANUARY 10, 2023 Furthermore, among 3,416 faculty in both the 2017 and 2022 data, we found that 2,520 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8% coverage) have an h-index and t10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 459 of them have added google scholar profiles since 2017, and none of them removed their google scholar profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 926 of them were promoted during the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Among 1,750 new faculty in our 2022 data, 1,636 have a google scholar profile, where 205 are full professors, 196 are associate professors, and 1,235 are assistant professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 1,741 of these new faculty have t10, where 252 are full professors, 219 are associate professors, and 1,270 are assistant professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 3 Methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 Program Strength Measures We propose two approaches for using individual citation measures to calculate the strength of a program, averaged and cumulative citation measures, the same as those used in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 Averaged citation measures The first method we use to measure the strength of a program is by averaging citations of its faculty members [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We use three different averaging schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The first averaging scheme is the median of t10 values of senior CS faculty, which we denote as m10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The second is the geometric mean of (1+t10) values of senior CS faculty, which we denote as g10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The third is the average percentile of the senior faculty’s t10, which we denote as p10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We exclude assistant professors from the averaged measures because their citations are typically smaller, and their inclusion would hurt departments with more assistant professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 Cumulative citation measures The second method to measure the strength of a program is to count the number of highly cited faculty in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We define a t10 threshold to determine which professors are highly cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We introduce cN, which denotes the number of faculty whose t10 is higher than N% of senior faculty, with 0 < N < 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For example, c40 is the count of professors within a department with a t10 higher than 40% of senior faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We include all faculty to calculate cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 Regression Models We use regression models that combine the averaged and cumulative citation measures into an aggregated score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The regression models that we consider are of the type in formula 1: si = β0 + β1ai + β2ci, (1) si is the predicted USN CS score, ai is an aggregated citation measure, and ci is a cumulative citation measure of the i-th program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The regression parameters are β0, β1, and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Instead of learning the intercept parameter β0, we set it to β0 = 1 by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The primary justification is that a program with ai = 0 and si = 0 does not have active research faculty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' based on the peer assessment instructions by USN, this program would have a score of 1 ("marginal").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The secondary justification is that the resulting regression models would be simpler because they only require fitting two regression parameters, β1, and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We train one regression model for each combination of the three averaged and the ranges of cumulative measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We average the individual regression models to create an aggregate score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 Department size Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 compares the number of assistant, associate, and full professors between our newly collected data and that from our 2017 paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The percentage of assistant professors increased during the past four years, which indicates that more young professors are joining CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The reason might be that recently popular areas, such as machine learning or deep learning, are attracting more young professors to contribute to research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 shows the distribution of department sizes, defined as the number of tenure-track faculty in each of the 185 CS programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The median faculty size is 23, the mode is 20, the minimum is 3, and the maximum is 170 (Carnegie Mellon University).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We also show the scatter plot between the department size in 2017 and 2022 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The colors represent the USN CS scores, and it can be seen there is an overall increase in department sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The department 4 A PREPRINT - JANUARY 10, 2023 Figure 1: Trend of faculty size Figure 2: The distribution of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CS department size size in 2022 has a similar distribution to that in 2017, since there is a nearly linear relationship between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The correlation coefficient of the department sizes and the USN CS scores of the 185 programs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='755, which is higher than our previous finding (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='676), indicating that larger departments are more likely to be ranked higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 Distribution of the Citation Indices As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4, the median value of the h-index, i10-index, and t10 index all increased compared to our 2017 result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Particularly, the t10 index has a larger rise compared to the h-index and i10-index, indicating that it is a more sensitive citation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5, we show the histogram of the t10 for the 3,944 senior faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We observe a similar distribution pattern compared to the 2017 result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Since the t10 distribution resembles a lognormal distribution, the histogram of t10 is shown in a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We observe a bump at low values, representing the 70 senior faculty with a t10 of 0, meaning they have less than ten cited papers listed in Google Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The median of t10 is 114, and the percentiles of t10 are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Overall, the t10 values increased compared to our results from 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 5 60 old data new data 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6% 50 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8% 40 Percentage 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2% 30 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9% 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0% 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6% 20 10 0 Full professor Associate professor Assistant professor50 45 40 35 30 uno 25 20 15 10 5 0 0 20 40 60 80 100 120 140 160 Faculty SizeA PREPRINT - JANUARY 10, 2023 Figure 3: Department size in 2016 and 2022 Figure 4: Trend of citation measurements (median value) The correlation coefficient between logarithms of h-index and t10 for the 3,379 senior faculty with both indices is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='943, which is close to our 2017 result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The sufficiently high correlation concludes that the t10 is a good proxy for the h-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Table 1: Percentiles of t10 Percentile t10 10% 21 20% 40 30% 60 40% 83 50% 115 60% 154 Percentile t10 70% 212 80% 301 90% 493 95% 751 98% 1247 99% 1843 6 Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 175 150 (2022) 125 100 3 size Faculty 75 2 50 25 1 F 0 20 0 0 40 60 80 100 120 140 Faculty size (2016)80 old data 75 new data 70 62 60 indices 50 48 le 44 40 30 27 25 20 10 0 h-index il0-index tl0-indexA PREPRINT - JANUARY 10, 2023 Figure 5: Histogram of t10 of associate and full CS professors Figure 6: Number of tenured CS faculty with (blue) and without (orange) Google scholar profile as a function of the t10 percentile 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 Scholar profile bias While the median of t10 for the 3,932 senior CS faculty is 114, it increases to 131 among the 3,379 professors who have a Google scholar profile and drops to 33 among the 553 without a profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 6 we show a stacked bar plot of the numbers of faculty with and without Google scholar profiles as a function of their t10 percentile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' These results are consistent with the observations in our 2017 paper, which indicate that CS faculty who have Google scholar profiles are a biased sample of the entire CS faculty and validate our effort to gather the t10 values and use them in our study instead of the h-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 Scholar model According to the above analysis, it can be observed our 2022 data shows a similar pattern and trend compared to our 2017 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Hence, we decided to keep the same linear regression scholar model used in 2017 and directly applied it to our 2022 data, which is shown in formula 2: s = 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='058 √ m10 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='059 � g10 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='121 √ c40 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='127 √ c60 (2) 7 600 500 400 Count 300 200 100 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4 log10(t10)450 400 350 300 Count 250 200 150 100 50 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 Percentile RangeA PREPRINT - JANUARY 10, 2023 Figure 7: Comparison of scholar model scores and USN CS scores of CS graduate programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' In Fig 7, we show a scatter plot of the USN CS scores and scholar model scores for the 185 CS programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' A closer look at the scatter plot reveals that two groups of CS programs can be distinguished with respect to the correlation between the USN CS scores and scholar model scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The first group contains 73 programs that were scored 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 and higher by the USN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The correlation between the USN CS scores and joint model scores in this group is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The second group contains 112 programs with USN scores between 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The correlation between the USN CS scores and scholar model scores in this low-scoring group is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='673, which is much higher than what we observed in 2017 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='360) but still lower than those 73 programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We hypothesize that the CS programs whose USN CS scores are between 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 are not sufficiently well-known among the peers to provide objective and reliable peer assessment at the national level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 Comparison Study We first study the newly added professor list to compare our new ranking result with the previous one conducted in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Among those 1,750 new professors, it is observed that 1,518 (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7%) of them have a t10-index lower than the average t10-index of all faculty from the department they join in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 223 of them are higher, and 9 of them have a missing t10-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Furthermore, among those 261 professors who transferred to another department, 114 have joined a higher-ranking department than their previous department, and 147 have joined a lower-ranking department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We also compared their t10-index in 2017 with the previous department average before they moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It is an interesting finding that only 66 have a t10-index higher than the department average, and 176 have a t10-index lower or equal to the department average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 19 of them have a missing t10-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Based on these observations, it can be inferred that new professors are primarily young, with a lower t10-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It is also true that young professors at the starting stage of their careers are more likely to transfer to another university, and most of them tend to join a higher-ranking university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The story might be they built a stronger profile during the past 5 years and then joined a higher-ranking department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 2017 ranking vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2022 ranking To better understand how the ranking has changed during the past five years, we calculate the ranking difference between the results we produced in 2017 and 2022 using the same scholar model created in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 8 shows the box plot of the absolute difference between our old (2017) and new (2022) scholar ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be seen that the placements of top universities above rank 90 are more stable, while lower-ranked universities tend to have more significant variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' However, we observe some outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For example, Northeastern University jumped from rank 40 to 21 because this department may have recruited many new professors since 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Another observation is that larger departments are not necessarily ranked higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For example, Stanford University and Princeton University both have relatively small departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Still, they are within the top 10 departments in our ranking, indicating that other factors, such as faculty citations, significantly influence the ranking result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 5 00 O 000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 8 o 4 900 o 8 o 8 o 08 Model 08 80 3 000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 08 009 00 00 8000 08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='59 8000 8 o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='News ScoreA PREPRINT - JANUARY 10, 2023 Figure 8: Absolute value of ranking difference between our 2017 and 2022 scholar ranking results Figure 9: Absolute value of ranking difference between USN 2017 and USN 2022 results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 9 shows the absolute difference between the USN 2017 and USN 2022 ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be observed that the ranking difference has a similar pattern compared to what has been shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This indicates that our method using the t-10 index of faculty to rank the program produces similar ranking results compared to USN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 Scholar ranking vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' USN The new ranking results and a comparison to the USN scores are shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To justify that it is appropriate to apply the linear regression model obtained from the 2017 data to the 2022 data, we calculated the correlations between our scholar ranking result and the USN ranking in 2017 and 2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Table 2: Correlation between USN ranking and our scholar ranking using the 2017 regression model R2 Pearson Spearman USN 2017 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' scholar ranking 2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8978 USN 2022 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' scholar ranking 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9126 Despite the fact that we used the same regression formula for both 2017 and 2022 data, our scholar ranking results show a high correlation with both USN 2017 and USN 2022 rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This result confirms that our practice of using the 2017 ranking formula on our 2022 data is justifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To further investigate the relationship between our scholar ranking and the USN ranking, we calculated the difference between our new ranking and the USN 2022 ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 10 shows the boxplot of the absolute value of difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We separate universities into six groups based on their new scholar rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be observed that our ranking 9 50 40 Rank difference 30 20 10 0 Rank 1-30 Rank 31-60 Rank 61-90 Rank 91-120 University ranking (2022)50 40 0 Rank difference 30 0 20 10 0 Rank 1-30 Rank 31-60 Rank 61-90 Rank91-120 University ranking (2022)A PREPRINT - JANUARY 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 2023 Figure 10: Absolute value of ranking difference between our 2022 scholar ranking and USN 2022 ranking Figure 11: Histogram plot of ranking score increase between 2022 and 2017 model is more likely to match with the USN ranking for higher-ranking departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For instance, all departments that rank 1-30 show a rank difference of less than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We also calculate the ranking score increase in 2022 compared to 2017 for both our scholar model and USN and show the histogram plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be seen that most departments have a score increase in both the scholar model and USN, and all of them are within the range between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' There are a few extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' For example, the ranking score of Northeastern University increased by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' where the reason may be the department recruited many new faculty during the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' UNC Chapel Hill has a score decrease of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Their department size remains the same (32), but the m10, g10, c40, and c60 values drop in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 Scholar ranking vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CSRankings Table 3: Correlation between CSRankings, USN 2022 and our scholar ranking R2 Pearson Spearman USN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' scholar ranking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9126 USN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CSRankings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9157 scholar ranking vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CSRankings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8965 USN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' average model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9305 To further investigate the relationship between our ranking and some widely-used CS ranking results, we compared the CSRankings result with our ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Unlike USN ranking, CSRankings4 relies on publications in top-tier computer science conferences, as reported by DBLP, a computer science bibliography website 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' To study the relationship between CSRankings and our scholar ranking, we collected the current CSRankings result and calculated its correlations with 4http://csrankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='org 5https://dblp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='org/ 10 50 40 Rank difference 30 20 10 0 Rank1-30 Rank 31-60 Rank 61-90 Rank 91-120 Universityranking(2022)Histogram of ranking score increase in 2022 compared to 2017 Scholarranking USN ranking 35 30 25 20 Count 15 10 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 Ranking score increase Ranking score increaseA PREPRINT - JANUARY 10, 2023 Figure 12: Absolute value of ranking difference between CSRankings and USN 2022 Figure 13: Absolute value of ranking difference between CSRankings and our scholar ranking USN and our scholar ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Since the CSRanking score is based on a different scale, we applied log transformation to its score before computing the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The result is shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be seen from the Table that our scholar ranking result has a higher correlation with the USN ranking, which indicates that it is better aligned with the USN ranking compared to CSRankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This indication is further justified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 10 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 12, where we show the ranking difference between 1) our scholar ranking results with USN and 2) CSRankings with USN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' It can be seen that CSRankings shows an overall more significant ranking difference with USN compared to our scholar model ranking, especially for the top 60 departments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' An interesting finding is that CSRankings is better aligned with USN than our scholar ranking for those lower-ranking departments (>90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The ranking difference between CSRankings and our scholar model ranking is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The result shows that the difference increases with a larger variance as the ranking of the department decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' This indicates that the ranking of top departments is more stable despite which model is being used, whereas lower-ranking departments are more sensitive to the ranking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' We also build an average model by computing the average score using the scholar model score and CSRankings score (shown in the last row of Table), which yields a higher spearman correlation with the USN ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' References [1] Slobodan Vucetic, Ashis Kumar Chanda, Shanshan Zhang, Tian Bai, and Aniruddha Maiti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Faculty citation measures are highly correlated with peer assessment of computer science doctoral programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' ArXiv, abs/1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='05435, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' [2] Emilio Delgado López-Cózar, Nicolás Robinson-García, and Daniel Torres-Salinas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' The Google scholar experiment: How to index false papers and manipulate bibliometric indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Journal of the Association for Information Science & Technology, 65(3):446–454, March 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' [3] Péter Jacsó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Deflated, inflated and phantom citation counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Online Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=', 30:297–309, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 11 50 40 30 0 0 20 8 10 0 Rank 1-30 Rank 31-60 Rank 61-90 Rank 91-120 Universityranking(2022)50 40 0 30 20 0 Rank 0 10 0 Rank 1-30 Rank 31-60 Rank 61-90 Rank91-120 University ranking (2022)A PREPRINT - JANUARY 10, 2023 [4] Judit Bar-Ilan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Which h-index?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='—a comparison of wos, scopus and google scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Scientometrics, 74:257–271, 02 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Hirsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' An index to quantify an individual’s scientific research output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences, 102(46):16569–16572, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' [6] Themis Lazaridis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Ranking university departments using the mean h-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' Scientometrics, 82:211–216, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' 12 A PREPRINT - JANUARY 10, 2023 Table 4: List of 185 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' CS graduate programs: Ranking by our scholar model (Rank), University name (University), Number of tenured faculty with t10 score (Size), median t10 score of all the faculty (M10), geometric mean of t10 score of all faculty (G10), number of highly cited faculty based on c40 (C40) and c60 (C60), U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content=' News CS score (USN), Scholar score (Scholar) Rank University Size M10 G10 C40 C60 USN Scholar USN 2016 Scholar 2016 1 Carnegie Mellon University 170 280 262 121 84 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 5 5 5 1 Cornell University 102 327 315 75 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 1 Massachusetts Institute of Technology 110 304 302 92 74 5 5 5 5 1 Stanford University 64 707 706 60 53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 5 5 5 1 University of California Berkeley 82 421 461 68 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 5 5 5 6 University of Washington 73 345 295 58 47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 7 University of California San Diego 63 341 330 49 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 8 Georgia Institute of Technology 105 203 198 75 53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 8 Princeton University 48 360 345 36 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 8 University of Illinois Urbana Champaign 76 265 248 56 43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 11 University of California Los Angeles 44 317 316 35 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 11 University of Pennsylvania 67 256 238 49 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 13 Columbia University 52 249 288 42 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 13 New York University 44 288 254 38 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 4 13 University of Michigan Ann Arbor 67 251 254 47 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 16 Harvard University 34 277 286 28 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 17 Duke University 56 216 195 37 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 17 Johns Hopkins University 32 258 309 23 17 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4 17 University of Maryland College Park 56 219 186 37 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4 4 17 University of Wisconsin Madison 45 285 225 28 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 21 Northeastern University 77 165 180 45 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='6 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='2 21 University of Southern California 40 272 238 26 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='9 21 University of Texas Austin 53 182 184 38 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE1T4oBgHgl3EQfYgTD/content/2301.03140v1.pdf'} +page_content='7 24 Brown University 31 224 233 22 17 3.' 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Raizen and Logan E. Hillberry +Department of Physics, The University of Texas at Austin, Austin, Texas, 78712, USA +Dmitry Budker +Johannes Gutenberg-Universit¨at Mainz, Helmholtz-Institut Mainz, +GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany and +Department of Physics, University of California, Berkeley, California 94720, USA +Simon M. Rochester +Rochester Scientific, LLC, El Cerrito, California 94530, USA +(Dated: January 31, 2023) +We propose a highly efficient and fast method of translational cooling for high-angular-momentum +atoms or molecules. Internal-state optical pumping and stimulated optical transitions, combined +with magnetic forces, can be used to compress phase-space density, and the efficiency of each com- +pression step increases with the angular momentum. Entropy is removed by spontaneously emitted +photons, and particle number is conserved. This method may be an attractive alternative to evap- +orative cooling of atoms and molecules in order to produce quantum degenerate gases. +INTRODUCTION +Laser cooling, first proposed almost half a century +ago, remains the standard approach for producing ultra- +cold atoms. This method relies on momentum transfer +from light to atoms as photons are repeatedly scattered, +enabling the production and study of ultracold atomic +gases. Many improvements on basic laser cooling have +advanced the state of the art, including Sisyphus cooling +[1–4], and subrecoil cooling [5, 6]. +While laser cooling works extremely well, the require- +ment of a closed, two-level transition has limited the ap- +plicability of the method to a subset of elements in the +periodic table. For those atoms, after many years of re- +finement, laser cooling has reached saturation in its per- +formance due to multiple scattering of resonant photons +which create an effective repulsive interaction between +the atoms, pushing them apart. +An important figure- +of-merit is the phase-space density, a dimensionless pa- +rameter which is the product of number density and the +third power of the average de Broglie wavelength. Laser +cooling typically produces a phase-space density of 10−6. +This is also the starting point for the formation of Bose- +Einstein condensates through evaporative cooling in a +trap, and the creation of the so-called atom laser [7–9]. +Evaporative cooling is even more restrictive than laser +cooling, as it relies on elastic collisions between atoms to +maintain thermal equilibrium as the hottest atoms are +ejected. +Inelastic channels create unwanted losses and +often make evaporative cooling impossible. Even when +working optimally, evaporative cooling is a slow process +and results in a significant loss of atom number. Cooling +of molecules has become the focus of much effort in re- +cent years, but is severely hampered by the difficulty of +implementing the above cooling methods. +In recent years, alternative approaches to producing +Optical +pumping +Stimulated +transitions +Magnetic +forces +(a) +(b) +FIG. 1. A schematic depiction of the MOP-cooling sequence +for ensembles with (a) angular momentum J = 1 and (b) +J = 2. The boxes represent atoms, initially trapped in a flat, +hard-wall potential. +The colors represent the atom’s mag- +netic state (orange: mJ = −J to purple: mJ = 0 to teal: +mJ = +J). A cycle begins with optically pumping all atoms +to the same state. Then, stimulated transitions correlate the +magnetic states with position along the direction of compres- +sion. A sequence of one-dimensional magnetic kicks pushes +atoms of oppositely-signed magnetic states together. The cy- +cle is closed by optically pumping the compressed atoms back +into the same magnetic state. A cycle’s compression factor is +limited by the number of available magnetic states. +cold atoms and molecules were developed (see [10] and +references therein). The starting point for much of this +work is a supersonic molecular beam where desired atoms +or molecules can be entrained in the flow and stopped in +a series of pulsed magnetic or electric fields. +Alterna- +tively, cold atoms and molecules are produced by buffer- +arXiv:2301.13121v1 [physics.atom-ph] 30 Jan 2023 + +2 +gas cooling. After stopping, these atoms and molecules +can be trapped, typically in a magnetic field configura- +tion that confines the low-field seekers to the center of +the trap. In parallel, cooling of atoms with a one-way +wall was proposed and demonstrated [11, 12], and re- +lies on photon entropy, not momentum as in laser cool- +ing. The one-way wall is the first practical realization +of Maxwell’s demon for an ensemble of atoms. +While +one-way-wall cooling demonstrated a large increase in +phase-space density, it did not conserve atom number. +To address this limitation, we proposed [13] a variation +which we called magneto-optical (MOP) cooling, relying +on cycles of optical pumping and magnetic kicks. +In this paper, we present a new and highly efficient +version of MOP cooling that can work for high-angular- +momentum systems of atoms and molecules and offers +an attractive alternative to laser cooling and evaporative +cooling. Our method is depicted schematically in Fig. 1 +and is described in detail in the following section. We +conclude the paper with a discussion of possible limita- +tions to our new method, how those limitations may be +overcome, and the significance of MOP cooling in the +atomic physics toolbox. +MOP COOLING +MOP cooling is a conceptually new method that does +not rely on the momentum of the photon, making it +completely different from laser cooling. +The key ben- +efit of this approach is its universality and simplicity, +since it relies only on optical pumping of an atomic or +molecular internal magnetic state, combined with mag- +netic forces from pulsed coils. We evaluate the efficacy of +MOP cooling through numeric simulations. In this sec- +tion, our simulation methodology is described, followed +by a brief review of MOP cooling for a spin-1/2 system, +as was originally proposed [13]. Finally, a new and much +more efficient version of MOP cooling for high-angular- +momentum systems is considered. +Our +MOP-cooling +simulations +track +the +three- +dimensional positions x, velocities v, and magnetic states +mJ ∈ {−2J, −2J + 1, . . . , 2J} of a sample of N = 105 +atoms. Positions are initialized from a flat distribution +of a 0.5 cm width. +Velocities are initialized from the +Maxwell-Boltzmann distribution corresponding to a tem- +perature of 25×Trec where Trec = h2/2mkBλ2 is the recoil +temperature imposed by, e.g., a magneto-optical trap op- +erating at wavelength λ to trap a species of mass m. Here +h is Planck’s constant and kB is Boltzmann’s constant. +A fourth-order Runge-Kutta algorithm updates the po- +sition and velocity of each atom subject to the force +F(t) = −mJgJµB∇ |B(x, t)| where gJ is the Land´e g- +factor of the atom, µB is the Bohr magneton, and B is +the pulsed magnetic field arranged to provide the one- +dimensional kick. The internal state of each atom is set +in accordance with the MOP-cooling cycle. The magnetic +field is produced with two sets of coaxial coil pairs, one +in the Maxwell configuration to provide a strong gradient +[14] and the other in the Helmholtz configuration to shift +the zero-crossing of the field away from the trap center, +thereby providing a nearly-one-dimensional kick. We use +superposition of the exact solution for a current-carrying +loop to model the full coil geometry. B is evaluated on +a dense grid for unit current. Vector interpolation allows +us to evaluate ∇ |B(x)| at arbitrary positions. +Time- +dependent current pulses are modeled by scaling the field +gradient interpolation result by I(t) = I0 sin[(t−t0)/2πτ] +for t ∈ [t0, t0 + τ] and I(t) = 0 otherwise, where I0 is the +peak current, t0 is the pulse delay, and τ is the pulse +width. +Table I reports the atomic properties used in this +study. Additionally, the simulated coil parameters are +as follows. +There are 7 turns × 2 layers, or 14 loops +per Helmholtz coil, each with a nominal radius of +RHH = 3.5 cm, axially-separated by RHH, and carry- +ing identically-oriented currents . There are 5 turns × +2 layers, or 10 loops per Maxwell coil, each with a nom- +inal radius of RHH/ +√ +3, separated by RHH, and carry- +ing oppositely-oriented currents. The peak currents are +I0,HH = 1000 A for the Helmholtz coils and I0,M = 500 A +for the Maxwell coils. +The shared midpoint between +the coil pairs is displaced by 0.2 cm in the positive z- +direction from the initial center of mass of the atomic +sample (taken as the origin of the coordinate system). +We find that this region provides a more uniform and +one-dimensional kick. +TABLE I. Atomic properties used for simulation purposes. +The Ref. column provides an example of magneto-optical trap +operation for each species. Values for gJ are provided in [15], +except for Li for which we adopt the electron’s value. +Atom +m +J +gJ +λ +Trec Ref. +(10−26 kg) +(nm) +(µK) +Li +1.15 +1/2 +2.002 32 +671 +76.2 +[16] +Cr +8.63 +3 +2.001 83 +425 +25.3 +[17] +Er +27.8 +6 +1.163 81 +583 +4.2 +[18] +Dy +27.0 +8 +1.241 59 +626 +3.7 +[19] +For a specific example, consider atomic lithium (Li) +trapped using standard techniques [16]. +At sufficient +magnetic fields, the electronic spin (J = 1/2) decou- +ples from the nuclear spin; the two electronic mJ states +are denoted |1/2⟩ and |−1/2⟩ and it is in this high-field +regime that we propose MOP cooling. The cooling se- +quence starts with suddenly turning off the trap so that +the atoms are free. In step (1) of MOP cooling, all of the +atoms are optically pumped to the |1/2⟩ state. +Then, +half of the cloud is transferred to the |−1/2⟩ state by +stimulated transitions , i.e., stimulated Raman adiabatic +passage (STIRAP) sequences [20, 21], thereby creating + +3 +−2 +0 +2 +vz (cm/s) +(a) +−50 +0 +50 +(b) +−50 +0 +50 +(c) +−2 +0 +2 +(d) +−0.25 +0.00 +0.25 +z (cm) +−0.1 +0.0 +0.1 +0.2 +vz (cm/s) +(e) +−0.25 +0.00 +0.25 +z (cm) +−20 +0 +20 +(f) +−0.25 +0.00 +0.25 +z (cm) +−20 +0 +20 +(g) +−0.25 +0.00 +0.25 +z (cm) +−0.1 +0.0 +0.1 +0.2 +(h) +−J +0 +J +mJ +FIG. 2. +MOP cooling simulations for Li (a-d) and Dy (e-h) visualized in phase space. Each column of plots represents a +snapshot in the cycle. First, the magnetic states are correlated with position along the z-axis through optical pumping followed +by spatially-resolved coherent population transfer via stimulated transitions. The vertical dashed black lines in panels (a) and +(e) mark the ideal boundary between different magnetic states. Second, a one-dimensional magnetic kick accelerates atoms to a +velocity that is proportional to their magnetic state. Third, after waiting an optimized delay time the phase space distribution +has been compressed in real space but remains extended in velocity space. Finally, a reverse kick returns the atom’s velocity +distribution to near its original extent. More precisely, we find the standard deviation of the ensemble’s velocity is, at most, +about 11% of its initial value for all four species simulated. +two spatially-distinct populations [see Fig. 2 (a)]. In step +(2), a magnetic-field-gradient kick is applied to the cloud, +thereby causing the two halves to merge [Fig. 2 (b-c)], +and then a reverse kick returns the atoms to their origi- +nal velocity distribution [Fig. 2 (d)]. As proposed in [13] +and demonstrated experimentally in [22], such magnetic +kicks can be applied along a single axis while minimally +affecting the other two dimensions. +Two-dimensional +slices (Fig. 3) of the magnetic field gradient used in our +simulations clearly show the one-dimensional-nature of +the magnetic kick. In step (3), all atoms are optically- +pumped back to the |1/2⟩ state, thereby completing the +cooling cycle. In principle, a factor of 2× in phase-space +compression is possible. +We now generalize the method to a system with an +arbitrary total angular momentum J. There are 2J + 1 +states in this case, and we assume that the atoms are +trapped in a hard-walled flat potential. Just as in the +case of Li above, we turn off the trap to start the cooling +sequence. Using optical pumping and stimulated transi- +tions in step (1), the cloud is divided into 2J + 1 com- +ponents, where the leftmost section is prepared in state +|J, −J⟩, then |J, –J + 1⟩, and so on, to the rightmost sec- +tion in state |J, J⟩. In step (2), a magnetic field gradi- +ent kick is applied to the cloud, causing each sub-section +to move at a velocity that is proportional to the mag- +netic quantum number. +Thus, the leftmost section in +state |J, J⟩ will move the fastest to the right, and lower +magnetic-quantum-numbered sections will move slower. +The cloud will collapse to a single section after an opti- +mal delay time, and a reverse magnetic kick will restore +the original velocity distributions. Finally, in step (3), +all atoms would be optically pumped to the |J, J⟩ state, +and the cloud can be re-trapped. +Figures 2 (e - h) show snapshots of the simulated phase +space for MOP cooling of Dy atoms (J = 8). Compar- +ing the final spatial distribution of Li [Fig. 2 (d)] to that +of Dy [Fig. 2 (h)] clearly demonstrates how MOP cool- +ing may leverage high-angular momentum systems for +efficient phase-space compression. The delay time is op- +timized in our numeric simulations and the results are +shown for a variety of atoms in Fig. 4. +As a figure of +merit, we compute the compression factor as the ratio +of standard deviations between the initial and final z- +coordinate distributions in the cloud. The inset of Fig. 4 +shows the peak compression factor for each species is +bound by the geometric limit 2J + 1. In the following +section we consider limitations of MOP cooling and how +they may be overcome in an experiment. +DISCUSSION +In MOP cooling, a maximum compression factor of +2J + 1 per cycle may be approached. However, in prac- +tice, the efficiency will be lower due to deviations from +a flat density distribution, imperfect kicking fields, and +photon-recoil heating during the optical pumping (OP) +stage. +The flat-density initial condition is quite different from + +4 +−0.5 +0.0 +0.5 +y (cm) +−0.5 +0.0 +0.5 +z (cm) +(a) x = −0.25 cm +944 +952 +952 +952 +960 +960 +−0.5 +0.0 +0.5 +y (cm) +(b) +x = 0.00 cm +936 +944 +952 +952 +952 +952 +960 +960 +−0.5 +0.0 +0.5 +y (cm) +(c) +x = 0.25 cm +944 +952 +952 +952 +960 +960 +−0.5 +0.0 +0.5 +x (cm) +−0.5 +0.0 +0.5 +y (cm) +(d) z = −0.25 cm +15 +30 +45 +45 +45 +45 +−0.5 +0.0 +0.5 +x (cm) +(e) +z = 0.00 cm +10 +20 +30 +40 +40 +40 +40 +−0.5 +0.0 +0.5 +x (cm) +(f) +z = 0.25 cm +8 +16 +24 +32 +40 +40 +40 +40 +900 +920 +940 +960 +980 +∇|B| (G/cm) +0 +20 +40 +60 +80 +∇|B| (G/cm) +FIG. 3. Two-dimensional slices of the magnetic field gradient used for MOP cooling simulations, evaluated at peak current. +The red dashed line marks the initial extent of the atomic cloud. The quadrupole field provided by Maxwell coils is symmetric +under a sign change of any coordinate axis. However, the bias field provided by the Helmholtz coils breaks the symmetry along +the z-axis by shifting the center of the quadrupole off of the coordinate origin. The magnitude of the total field |B| varies +primarily along z in a nearly-linear fashion. +0 +10 +20 +30 +40 +Unkick delay (ms) +0 +5 +10 +15 +Compression factor +0 +4 +8 +J +0 +5 +10 +15 +Li +Cr +Er +Dy +FIG. 4. Optimizing the wait time between the MOP cool- +ing kick and unkick according to the compression factor. The +open circle marks the optimal delay time for each species sub- +ject to the initial conditions and magnetic forces described in +the text. The inset shows the peak compression factor vs the +species’ angular momentum J, compared to the geometric +limit 2J + 1 (black line). +that usually encountered with laser cooled atoms in a +magneto-optical trap, but turns out to be important for +the gains in efficiency that are predicted. For example, +our simulations predict a compression factor of 1.9 for +Li initialized with with a flat density or 1.66 for a Gaus- +sian distribution. For Dy initialized with a flat density, +a peak compression factor of 13.8 is observed, while for +an initially-Gaussian distribution the factor reduces to +only 4.3. To obtain a flat “boxlike” distribution in an ex- +periment, the atoms can first be confined in a magnetic +quadrupole trap. An optical box can be created around +the atoms using a time-averaged optical dipole potential +from beams that are moving rapidly in two dimensions. +Such potentials were created in the past to study opti- +cal billiards [23, 24] and BECs in painted potentials [25]. +After trapping, the box can be adiabatically expanded in +three dimensions to a desired size, which will result in a +nearly flat density profile of atoms. The optical setup for +state preparation of each segment would enable multiple +cycles of cooling in each dimension. Adiabatic expansion +would lower the kinetic energy and MOP cooling would +compress the cloud spatially. The process can be dynam- +ically controlled with motorized zoom lenses [26]. +Ideally, a complete cycle of MOP cooling leaves the +velocity distribution of the atomic sample unchanged. +This means that the momentum imparted on the atoms +in the kick phase must be nulled in the reverse-kick + +5 +phase. +Inhomogeneities in the kicking field will result +in a nonzero mean velocity for the cloud. +Such inho- +mogeneities are noticeable in Fig. 3 (b) that shows the +peak magnetic field gradient in the x = 0 plane. +For +instance, atoms near z = 0.25 will be kicked downward +with less force than their upward motion-arresting kick +that is applied once they are near z = 0. This is why +Fig. 2 (h) shows a net positive velocity, which is partic- +ularly evident for mJ = J (teal). In our simulation, the +final mean velocity of the Dy atoms is ∼ 0.03 cm/s. In +practice, this is not a significant limitation because the +resulting net velocity is still well within the capture range +of any reasonable trap as it is smaller than the thermal +velocity. Over the same time scale t, the relative ther- +mal expansion of the cloud σ(t)/σ0 ≈ +� +1 + t2kBT/mσ2 +0 +is insignificant compared to the MOP cooling compres- +sion factor σ0/σ(t) ≈ 2J + 1. Moreover, there exist ad- +vanced wiring-design-optimization techniques to gener- +ate uniform bias or gradient fields with minimal induc- +tance [14, 27]. Though originally developed for magnetic +resonance imaging, MOP cooling could benefit from such +analyses to improve switching times of the pulsed mag- +netic fields and increase the uniformity of the required +biased gradients. +A more significant mean velocity is incurred due to +free fall in the Earth gravitational field. For example, +in the ∼ 19 ms of delay time required for MOP cooling +of Cr, the cloud accelerates to nearly 19 cm/s and dis- +places 0.18 cm. +Depending on the details of the trap, +such velocity or displacement may result in significant +atom loss. To mitigate the free fall effects, one could use +the MOP cooling setup to apply an additional uniform +kicks against gravity. +In general, optical pumping the cloud to the same mag- +netic state is a lossy step due to, for instance, atoms de- +caying to unobserved trap states. Fortunately, there ex- +ist efficient techniques for optical pumping as discussed +in [28], where it is shown that it is possible to perform +optical pumping with only one spontaneous photon emit- +ted per atom. +The same physics lets us understand +MOP cooling’s high phase-space compression efficiency +in terms of the photon entropy carried away from the +ensemble by spontaneous emission [29]. +The entropy associated with the motional degrees of +freedom is given by Smo. = kB ln V , where V is the phase- +space volume. +This volume is reduced by a factor of +(at most) 2J + 1 at each cooling step, so the maximum +entropy reduction per step is ∆Smo. = kB ln (2J + 1). +The magnetic kicks are reversible evolution, so they do +not produce a net change in entropy. Therefore the en- +tropy change must occur during the optical pumping +step. +Optical pumping of an unpolarized ensemble to +produce a pure state reduces the polarization entropy +by ∆Spol. = kB ln (2J + 1). Thus for each step, (1) OP +takes unpolarized state to pure state, reducing polariza- +tion entropy by kB ln (2J + 1); (2) kicks take pure state +to unpolarized state by overlapping ensembles, increas- +ing polarization entropy by kB ln (2J + 1) and reducing +motional entropy by the same amount. The cooling will +be limited by recoil heating. +From Ref. [28], we know +that OP can theoretically be done with only one sponta- +neously emitted photon per atom. If this is achieved, the +temperature limit will correspond to the recoil energy, +i.e., the standard recoil limit Trec. For less efficient OP +the temperature limit will be higher. +CONCLUSION +In this paper, we proposed a highly efficient method +for phase-space compression of high-angular momentum +atomic and molecular samples. +This work extends an +earlier MOP-cooling proposal from Li to general high- +angular-momentum systems. We numerically tested our +new MOP-cooling protocol on four atomic species of in- +creasing angular momenta that have each already been +cooled using traditional techniques. We find, for exam- +ple, an impressive compression factor of 13.5 is conceiv- +ably attainable in less than 13 ms for the case of Dy. +ACKNOWLEDGEMENTS +The +work +of +MGR +was +supported +by +the +Sid +W. Richardson Foundation. The work of DB was sup- +ported by the Deutsche Forschungsgemeinschaft (DFG) +- Project ID 423116110 and by the Cluster of Excellence +Precision Physics, Fundamental Interactions, and Struc- +ture of Matter (PRISMA+ EXC 2118/1) funded by the +DFG within the German Excellence Strategy (Project ID +39083149). + +6 +[1] P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, +P. L. Gould, and H. J. Metcalf, Physical review letters +61, 169 (1988). +[2] J. Dalibard and C. Cohen-Tannoudji, JOSA B 6, 2023 +(1989). +[3] P. J. Ungar, D. S. Weiss, E. Riis, and S. 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Res. 4, 013218 (2022). + diff --git a/M9FPT4oBgHgl3EQflTVS/content/tmp_files/load_file.txt b/M9FPT4oBgHgl3EQflTVS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3432244263339525fece1dfda5392b8c2486870 --- /dev/null +++ b/M9FPT4oBgHgl3EQflTVS/content/tmp_files/load_file.txt @@ -0,0 +1,535 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf,len=534 +page_content='Efficient cooling of high-angular-momentum systems Mark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Raizen and Logan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Hillberry Department of Physics, The University of Texas at Austin, Austin, Texas, 78712, USA Dmitry Budker Johannes Gutenberg-Universit¨at Mainz, Helmholtz-Institut Mainz, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany and Department of Physics, University of California, Berkeley, California 94720, USA Simon M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Rochester Rochester Scientific, LLC, El Cerrito, California 94530, USA (Dated: January 31, 2023) We propose a highly efficient and fast method of translational cooling for high-angular-momentum atoms or molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Internal-state optical pumping and stimulated optical transitions, combined with magnetic forces, can be used to compress phase-space density, and the efficiency of each com- pression step increases with the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Entropy is removed by spontaneously emitted photons, and particle number is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This method may be an attractive alternative to evap- orative cooling of atoms and molecules in order to produce quantum degenerate gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' INTRODUCTION Laser cooling, first proposed almost half a century ago, remains the standard approach for producing ultra- cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This method relies on momentum transfer from light to atoms as photons are repeatedly scattered, enabling the production and study of ultracold atomic gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Many improvements on basic laser cooling have advanced the state of the art, including Sisyphus cooling [1–4], and subrecoil cooling [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' While laser cooling works extremely well, the require- ment of a closed, two-level transition has limited the ap- plicability of the method to a subset of elements in the periodic table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For those atoms, after many years of re- finement, laser cooling has reached saturation in its per- formance due to multiple scattering of resonant photons which create an effective repulsive interaction between the atoms, pushing them apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' An important figure- of-merit is the phase-space density, a dimensionless pa- rameter which is the product of number density and the third power of the average de Broglie wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Laser cooling typically produces a phase-space density of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This is also the starting point for the formation of Bose- Einstein condensates through evaporative cooling in a trap, and the creation of the so-called atom laser [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Evaporative cooling is even more restrictive than laser cooling, as it relies on elastic collisions between atoms to maintain thermal equilibrium as the hottest atoms are ejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Inelastic channels create unwanted losses and often make evaporative cooling impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Even when working optimally, evaporative cooling is a slow process and results in a significant loss of atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Cooling of molecules has become the focus of much effort in re- cent years, but is severely hampered by the difficulty of implementing the above cooling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In recent years, alternative approaches to producing Optical pumping Stimulated transitions Magnetic forces (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A schematic depiction of the MOP-cooling sequence for ensembles with (a) angular momentum J = 1 and (b) J = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The boxes represent atoms, initially trapped in a flat, hard-wall potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The colors represent the atom’s mag- netic state (orange: mJ = −J to purple: mJ = 0 to teal: mJ = +J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A cycle begins with optically pumping all atoms to the same state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Then, stimulated transitions correlate the magnetic states with position along the direction of compres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A sequence of one-dimensional magnetic kicks pushes atoms of oppositely-signed magnetic states together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The cy- cle is closed by optically pumping the compressed atoms back into the same magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A cycle’s compression factor is limited by the number of available magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' cold atoms and molecules were developed (see [10] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The starting point for much of this work is a supersonic molecular beam where desired atoms or molecules can be entrained in the flow and stopped in a series of pulsed magnetic or electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Alterna- tively, cold atoms and molecules are produced by buffer- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='13121v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='atom-ph] 30 Jan 2023 2 gas cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' After stopping, these atoms and molecules can be trapped, typically in a magnetic field configura- tion that confines the low-field seekers to the center of the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In parallel, cooling of atoms with a one-way wall was proposed and demonstrated [11, 12], and re- lies on photon entropy, not momentum as in laser cool- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The one-way wall is the first practical realization of Maxwell’s demon for an ensemble of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' While one-way-wall cooling demonstrated a large increase in phase-space density, it did not conserve atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' To address this limitation, we proposed [13] a variation which we called magneto-optical (MOP) cooling, relying on cycles of optical pumping and magnetic kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In this paper, we present a new and highly efficient version of MOP cooling that can work for high-angular- momentum systems of atoms and molecules and offers an attractive alternative to laser cooling and evaporative cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Our method is depicted schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 1 and is described in detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We conclude the paper with a discussion of possible limita- tions to our new method, how those limitations may be overcome, and the significance of MOP cooling in the atomic physics toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' MOP COOLING MOP cooling is a conceptually new method that does not rely on the momentum of the photon, making it completely different from laser cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The key ben- efit of this approach is its universality and simplicity, since it relies only on optical pumping of an atomic or molecular internal magnetic state, combined with mag- netic forces from pulsed coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We evaluate the efficacy of MOP cooling through numeric simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In this sec- tion, our simulation methodology is described, followed by a brief review of MOP cooling for a spin-1/2 system, as was originally proposed [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Finally, a new and much more efficient version of MOP cooling for high-angular- momentum systems is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Our MOP-cooling simulations track the three- dimensional positions x, velocities v, and magnetic states mJ ∈ {−2J, −2J + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' , 2J} of a sample of N = 105 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Positions are initialized from a flat distribution of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 cm width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Velocities are initialized from the Maxwell-Boltzmann distribution corresponding to a tem- perature of 25×Trec where Trec = h2/2mkBλ2 is the recoil temperature imposed by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=', a magneto-optical trap op- erating at wavelength λ to trap a species of mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Here h is Planck’s constant and kB is Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A fourth-order Runge-Kutta algorithm updates the po- sition and velocity of each atom subject to the force F(t) = −mJgJµB∇ |B(x, t)| where gJ is the Land´e g- factor of the atom, µB is the Bohr magneton, and B is the pulsed magnetic field arranged to provide the one- dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The internal state of each atom is set in accordance with the MOP-cooling cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The magnetic field is produced with two sets of coaxial coil pairs, one in the Maxwell configuration to provide a strong gradient [14] and the other in the Helmholtz configuration to shift the zero-crossing of the field away from the trap center, thereby providing a nearly-one-dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We use superposition of the exact solution for a current-carrying loop to model the full coil geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' B is evaluated on a dense grid for unit current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Vector interpolation allows us to evaluate ∇ |B(x)| at arbitrary positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Time- dependent current pulses are modeled by scaling the field gradient interpolation result by I(t) = I0 sin[(t−t0)/2πτ] for t ∈ [t0, t0 + τ] and I(t) = 0 otherwise, where I0 is the peak current, t0 is the pulse delay, and τ is the pulse width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Table I reports the atomic properties used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Additionally, the simulated coil parameters are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' There are 7 turns × 2 layers, or 14 loops per Helmholtz coil, each with a nominal radius of RHH = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 cm, axially-separated by RHH, and carry- ing identically-oriented currents .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' There are 5 turns × 2 layers, or 10 loops per Maxwell coil, each with a nom- inal radius of RHH/ √ 3, separated by RHH, and carry- ing oppositely-oriented currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The peak currents are I0,HH = 1000 A for the Helmholtz coils and I0,M = 500 A for the Maxwell coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The shared midpoint between the coil pairs is displaced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='2 cm in the positive z- direction from the initial center of mass of the atomic sample (taken as the origin of the coordinate system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We find that this region provides a more uniform and one-dimensional kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Atomic properties used for simulation purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' column provides an example of magneto-optical trap operation for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Values for gJ are provided in [15], except for Li for which we adopt the electron’s value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Atom m J gJ λ Trec Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' (10−26 kg) (nm) (µK) Li 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='15 1/2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='002 32 671 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='2 [16] Cr 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='63 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='001 83 425 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='3 [17] Er 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='8 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='163 81 583 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='2 [18] Dy 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='241 59 626 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='7 [19] For a specific example, consider atomic lithium (Li) trapped using standard techniques [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' At sufficient magnetic fields, the electronic spin (J = 1/2) decou- ples from the nuclear spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' the two electronic mJ states are denoted |1/2⟩ and |−1/2⟩ and it is in this high-field regime that we propose MOP cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The cooling se- quence starts with suddenly turning off the trap so that the atoms are free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In step (1) of MOP cooling, all of the atoms are optically pumped to the |1/2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Then, half of the cloud is transferred to the |−1/2⟩ state by stimulated transitions , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=', stimulated Raman adiabatic passage (STIRAP) sequences [20, 21], thereby creating 3 −2 0 2 vz (cm/s) (a) −50 0 50 (b) −50 0 50 (c) −2 0 2 (d) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 z (cm) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='2 vz (cm/s) (e) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 z (cm) −20 0 20 (f) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 z (cm) −20 0 20 (g) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 z (cm) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='2 (h) −J 0 J mJ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' MOP cooling simulations for Li (a-d) and Dy (e-h) visualized in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Each column of plots represents a snapshot in the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' First, the magnetic states are correlated with position along the z-axis through optical pumping followed by spatially-resolved coherent population transfer via stimulated transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The vertical dashed black lines in panels (a) and (e) mark the ideal boundary between different magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Second, a one-dimensional magnetic kick accelerates atoms to a velocity that is proportional to their magnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Third, after waiting an optimized delay time the phase space distribution has been compressed in real space but remains extended in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Finally, a reverse kick returns the atom’s velocity distribution to near its original extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' More precisely, we find the standard deviation of the ensemble’s velocity is, at most, about 11% of its initial value for all four species simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' two spatially-distinct populations [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In step (2), a magnetic-field-gradient kick is applied to the cloud, thereby causing the two halves to merge [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (b-c)], and then a reverse kick returns the atoms to their origi- nal velocity distribution [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' As proposed in [13] and demonstrated experimentally in [22], such magnetic kicks can be applied along a single axis while minimally affecting the other two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Two-dimensional slices (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 3) of the magnetic field gradient used in our simulations clearly show the one-dimensional-nature of the magnetic kick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In step (3), all atoms are optically- pumped back to the |1/2⟩ state, thereby completing the cooling cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In principle, a factor of 2× in phase-space compression is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We now generalize the method to a system with an arbitrary total angular momentum J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' There are 2J + 1 states in this case, and we assume that the atoms are trapped in a hard-walled flat potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Just as in the case of Li above, we turn off the trap to start the cooling sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Using optical pumping and stimulated transi- tions in step (1), the cloud is divided into 2J + 1 com- ponents, where the leftmost section is prepared in state |J, −J⟩, then |J, –J + 1⟩, and so on, to the rightmost sec- tion in state |J, J⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In step (2), a magnetic field gradi- ent kick is applied to the cloud, causing each sub-section to move at a velocity that is proportional to the mag- netic quantum number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Thus, the leftmost section in state |J, J⟩ will move the fastest to the right, and lower magnetic-quantum-numbered sections will move slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The cloud will collapse to a single section after an opti- mal delay time, and a reverse magnetic kick will restore the original velocity distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Finally, in step (3), all atoms would be optically pumped to the |J, J⟩ state, and the cloud can be re-trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Figures 2 (e - h) show snapshots of the simulated phase space for MOP cooling of Dy atoms (J = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Compar- ing the final spatial distribution of Li [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (d)] to that of Dy [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (h)] clearly demonstrates how MOP cool- ing may leverage high-angular momentum systems for efficient phase-space compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The delay time is op- timized in our numeric simulations and the results are shown for a variety of atoms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' As a figure of merit, we compute the compression factor as the ratio of standard deviations between the initial and final z- coordinate distributions in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 4 shows the peak compression factor for each species is bound by the geometric limit 2J + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In the following section we consider limitations of MOP cooling and how they may be overcome in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' DISCUSSION In MOP cooling, a maximum compression factor of 2J + 1 per cycle may be approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' However, in prac- tice, the efficiency will be lower due to deviations from a flat density distribution, imperfect kicking fields, and photon-recoil heating during the optical pumping (OP) stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The flat-density initial condition is quite different from 4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 y (cm) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 z (cm) (a) x = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 cm 944 952 952 952 960 960 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 y (cm) (b) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 cm 936 944 952 952 952 952 960 960 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 y (cm) (c) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 cm 944 952 952 952 960 960 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 x (cm) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 y (cm) (d) z = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 cm 15 30 45 45 45 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 x (cm) (e) z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='00 cm 10 20 30 40 40 40 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 x (cm) (f) z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 cm 8 16 24 32 40 40 40 40 900 920 940 960 980 ∇|B| (G/cm) 0 20 40 60 80 ∇|B| (G/cm) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Two-dimensional slices of the magnetic field gradient used for MOP cooling simulations, evaluated at peak current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The red dashed line marks the initial extent of the atomic cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The quadrupole field provided by Maxwell coils is symmetric under a sign change of any coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' However, the bias field provided by the Helmholtz coils breaks the symmetry along the z-axis by shifting the center of the quadrupole off of the coordinate origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The magnitude of the total field |B| varies primarily along z in a nearly-linear fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 0 10 20 30 40 Unkick delay (ms) 0 5 10 15 Compression factor 0 4 8 J 0 5 10 15 Li Cr Er Dy FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Optimizing the wait time between the MOP cool- ing kick and unkick according to the compression factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The open circle marks the optimal delay time for each species sub- ject to the initial conditions and magnetic forces described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The inset shows the peak compression factor vs the species’ angular momentum J, compared to the geometric limit 2J + 1 (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' that usually encountered with laser cooled atoms in a magneto-optical trap, but turns out to be important for the gains in efficiency that are predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For example, our simulations predict a compression factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='9 for Li initialized with with a flat density or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='66 for a Gaus- sian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For Dy initialized with a flat density, a peak compression factor of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='8 is observed, while for an initially-Gaussian distribution the factor reduces to only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' To obtain a flat “boxlike” distribution in an ex- periment, the atoms can first be confined in a magnetic quadrupole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' An optical box can be created around the atoms using a time-averaged optical dipole potential from beams that are moving rapidly in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Such potentials were created in the past to study opti- cal billiards [23, 24] and BECs in painted potentials [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' After trapping, the box can be adiabatically expanded in three dimensions to a desired size, which will result in a nearly flat density profile of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The optical setup for state preparation of each segment would enable multiple cycles of cooling in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Adiabatic expansion would lower the kinetic energy and MOP cooling would compress the cloud spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The process can be dynam- ically controlled with motorized zoom lenses [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Ideally, a complete cycle of MOP cooling leaves the velocity distribution of the atomic sample unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This means that the momentum imparted on the atoms in the kick phase must be nulled in the reverse-kick 5 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Inhomogeneities in the kicking field will result in a nonzero mean velocity for the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Such inho- mogeneities are noticeable in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 3 (b) that shows the peak magnetic field gradient in the x = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For instance, atoms near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='25 will be kicked downward with less force than their upward motion-arresting kick that is applied once they are near z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This is why Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 2 (h) shows a net positive velocity, which is partic- ularly evident for mJ = J (teal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In our simulation, the final mean velocity of the Dy atoms is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='03 cm/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In practice, this is not a significant limitation because the resulting net velocity is still well within the capture range of any reasonable trap as it is smaller than the thermal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Over the same time scale t, the relative ther- mal expansion of the cloud σ(t)/σ0 ≈ � 1 + t2kBT/mσ2 0 is insignificant compared to the MOP cooling compres- sion factor σ0/σ(t) ≈ 2J + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Moreover, there exist ad- vanced wiring-design-optimization techniques to gener- ate uniform bias or gradient fields with minimal induc- tance [14, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Though originally developed for magnetic resonance imaging, MOP cooling could benefit from such analyses to improve switching times of the pulsed mag- netic fields and increase the uniformity of the required biased gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' A more significant mean velocity is incurred due to free fall in the Earth gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For example, in the ∼ 19 ms of delay time required for MOP cooling of Cr, the cloud accelerates to nearly 19 cm/s and dis- places 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='18 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Depending on the details of the trap, such velocity or displacement may result in significant atom loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' To mitigate the free fall effects, one could use the MOP cooling setup to apply an additional uniform kicks against gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' In general, optical pumping the cloud to the same mag- netic state is a lossy step due to, for instance, atoms de- caying to unobserved trap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Fortunately, there ex- ist efficient techniques for optical pumping as discussed in [28], where it is shown that it is possible to perform optical pumping with only one spontaneous photon emit- ted per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The same physics lets us understand MOP cooling’s high phase-space compression efficiency in terms of the photon entropy carried away from the ensemble by spontaneous emission [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The entropy associated with the motional degrees of freedom is given by Smo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' = kB ln V , where V is the phase- space volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This volume is reduced by a factor of (at most) 2J + 1 at each cooling step, so the maximum entropy reduction per step is ∆Smo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' = kB ln (2J + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The magnetic kicks are reversible evolution, so they do not produce a net change in entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Therefore the en- tropy change must occur during the optical pumping step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Optical pumping of an unpolarized ensemble to produce a pure state reduces the polarization entropy by ∆Spol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' = kB ln (2J + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Thus for each step, (1) OP takes unpolarized state to pure state, reducing polariza- tion entropy by kB ln (2J + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' (2) kicks take pure state to unpolarized state by overlapping ensembles, increas- ing polarization entropy by kB ln (2J + 1) and reducing motional entropy by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The cooling will be limited by recoil heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' From Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' [28], we know that OP can theoretically be done with only one sponta- neously emitted photon per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' If this is achieved, the temperature limit will correspond to the recoil energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=', the standard recoil limit Trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' For less efficient OP the temperature limit will be higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a highly efficient method for phase-space compression of high-angular momentum atomic and molecular samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' This work extends an earlier MOP-cooling proposal from Li to general high- angular-momentum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We numerically tested our new MOP-cooling protocol on four atomic species of in- creasing angular momenta that have each already been cooled using traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' We find, for exam- ple, an impressive compression factor of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content='5 is conceiv- ably attainable in less than 13 ms for the case of Dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The work of MGR was supported by the Sid W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Richardson Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' The work of DB was sup- ported by the Deutsche Forschungsgemeinschaft (DFG) Project ID 423116110 and by the Cluster of Excellence Precision Physics, Fundamental Interactions, and Struc- ture of Matter (PRISMA+ EXC 2118/1) funded by the DFG within the German Excellence Strategy (Project ID 39083149).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' 6 [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FPT4oBgHgl3EQflTVS/content/2301.13121v1.pdf'} +page_content=' Lett, R.' 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file mode 100644 index 0000000000000000000000000000000000000000..34ce95e0e1afdb480be4f673166bab872587d347 --- /dev/null +++ b/OdAzT4oBgHgl3EQfWfyE/content/tmp_files/2301.01302v1.pdf.txt @@ -0,0 +1,2013 @@ +MNRAS 000, 1–13 (2021) +Preprint 5 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A Comparative Analysis of the Chemical Compositions of +Gaia-Enceladus/Sausage and Milky Way Satellites using APOGEE +Laura Fernandes1,★ Andrew C. Mason1, Danny Horta1, Ricardo P. Schiavon1, Christian Hayes2, +Sten Hasselquist3, Diane Feuillet4, Rachael L. Beaton5,6, Henrik Jönsson7, Shobhit Kisku1, +Ivan Lacerna8,9, Jianhui Lian10, Dante Minniti11,12, Sandro Villanova13 +1 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK +2 NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, B.C., Canada, V9E 2E7 +3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +4 Lund Observatory, Department of Astronomy and Theoretical Physics, Box 43, SE-221 00 Lund, Sweden +5 Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544 +6 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101 +7 Materials Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden +8 Instituto de Astronomía y Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiapó, Chile +9 Millennium Institute of Astrophysics, Nuncio Monsenor Sotero Sanz 100, Of. 104, Providencia, Santiago, Chile +10 Department of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA +11 Departamento de Ciencias Físicas, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernández Concha 700, Las Condes, Santiago, Chile +12 Vatican Observatory, Vatican City State, V-00120, Italy +13 Departamento de Astronomía, Universidad de Concepción, Casilla 160-C, Concepción, Chile +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We use data from the 17th data release of the Apache Point Observatory Galactic Evolution Experiment (APOGEE 2) to contrast +the chemical composition of the recently discovered Gaia Enceladus/Sausage system (GE/S) to those of ten Milky Way (MW) +dwarf satellite galaxies: LMC, SMC, Boötes I, Carina, Draco, Fornax, Sagittarius, Sculptor, Sextans and Ursa Minor. Our main +focus is on the distributions of the stellar populations of those systems in the [Mg/Fe]-[Fe/H] and [Mg/Mn]-[Al/Fe] planes, +which are commonly employed in the literature for chemical diagnosis and where dwarf galaxies can be distinguished from +in situ populations. We show that, unlike MW satellites, a GE/S sample defined purely on the basis of orbital parameters falls +almost entirely within the locus of “accreted” stellar populations in chemical space, which is likely caused by an early quenching +of star formation in GE/S. Due to a more protracted history of star formation, stars in the metal-rich end of the MW satellite +populations are characterized by lower [Mg/Mn] than those of their GE/S counterparts. The chemical compositions of GE/S stars +are consistent with a higher early star formation rate than MW satellites of comparable and even higher mass, suggesting that +star formation in the early universe was strongly influenced by other parameters in addition to mass. We find that the direction +of the metallicity gradient in the [Mg/Mn]–[Al/Fe] plane of dwarf galaxies is an indicator of the early star formation rate of the +system. +Key words: Galaxies: dwarf spheroidal – stars: abundances +1 INTRODUCTION +Our current understanding of galaxy formation is immersed in the +framework of the Lambda Cold Dark Matter (ΛCDM) model. In this +model, galaxies are an ensemble of stars, gas, and dust bound by +massive dark matter haloes, built largely by the merging of smaller +systems (e.g., White & Rees 1978; Blumenthal et al. 1984; Frenk +& White 2012). In this context, the Milky Way (MW) is of great +importance, as the galaxy for which we can obtain the most detailed +information. Naturally, one should expect that our knowledge of the +MW should be used to constrain galaxy formation theory. While +★ E-mail: Lfernandes2@msn.com (LF); R.P.Schiavon@ljmu.ac.uk (RPS) +most of the stellar mass in the MW is contained in its bar and both +thin and thick disks (Morris & Serabyn 1996; Barbuy et al. 2018; +Nataf 2017; Rich 2013; Beraldo e Silva et al. 2021), these structures +are enveloped by a large stellar halo which contains about 1.3 × 109 +𝑀⊙ in the form of stars (Mackereth & Bovy 2020a). +Galaxy mergers were very common in the early universe. Local +evidence of the merging activity of the MW has been accumulating +over the years, with the identification of the remnants of multiple +accretion events, starting with the discovery of the Sagittarius dwarf +spheroidal (Sgr dSph) by Ibata et al. (1994). In addition, various +stellar streams have been identified as remnants of past accretion +events (e.g., Belokurov et al. 2006). Due to the stellar halo’s long +dynamical timescale, such relatively recent accretion events can be +© 2021 The Authors +arXiv:2301.01302v1 [astro-ph.GA] 3 Jan 2023 + +2 +L. Fernandes et al. +identified in the form of spatial substructure. However, discerning +the remnants of earlier accretion requires additional information, in +the form of, e.g., kinematics, chemical compositions, and ages (e.g., +Nissen & Schuster 2010, 2011; Schuster et al. 2012; Hayes et al. +2018) +With the advent of large spectroscopic surveys, we entered the +golden age of Galactic archaeology. The information acquired by +past, ongoing and future surveys such as RAVE, (Steinmetz et al. +2006), SEGUE (Yanny et al. 2009), LAMOST (Cui et al. 2012), +GALAH (De Silva et al. 2015), Gaia mission (Gaia Collaboration +et al. 2016), WEAVE (Dalton 2016), APOGEE (Majewski et al. +2017), H3 (Conroy et al. 2019), MOONS (Cirasuolo et al. 2011; +Gonzalez et al. 2020), 4MOST (de Jong et al. 2019) is very quickly +advancing our understanding of the history of the MW. Chemical +compositions and orbital information is becoming available for +millions of stars in the MW and its Local Group companions. In +addition, with the launching of the Gaia satellite (Gaia Collaboration +et al. 2016), detailed phase space information has become available, +enabling the identification of further substructures in the stellar halo +of the Galaxy, associated with various accretion events, including +Gaia-Enceladus/Sausage, a massive dwarf galaxy accreted to the +MW about ∼10Gyr ago (Belokurov et al. 2018; Haywood et al. 2018; +Helmi et al. 2018; Mackereth et al. 2019), as well as Heracles (Horta +et al. 2021a), and various other substructures (e.g., Koppelman et al. +2019a; Belokurov et al. 2018; Kruijssen et al. 2020; Naidu et al. +2020; Horta et al. 2022a). +Hawkins et al. (2015) and Das et al. (2020) have recently proposed +that a combination of the abundances of Mn, Mg, Al, and Fe can +be used to chemically discriminate accreted populations from their +counterparts formed in situ. However, Horta et al. (2021a) used +chemical evolution models to show that the region occupied by +accreted populations in the [Mg/Mn] vs [Al/Fe] plane has a non- +negligible presence of stars formed in situ. +It is difficult to disentangle metal-poor in situ populations +from their accreted counterparts based on purely kinematic or +orbital criteria. To minimise inter-sample contamination, one has +to often resort to additional criteria based on chemical compositions, +introducing sample biases that prevent a clean analysis of the +chemical compositions of accreted and in situ populations. That +difficulty can be overcome by resorting to data from external systems, +on the assumption that the chemistry of their stellar populations is +dominated by intrinsic evolutionary effects. +In this paper we contrast the distribution of stars from GE/S system +in key chemical composition planes with those of dwarf satellites of +the MW. The latter include the Sagittarius dSph, which is currently in +the process of being engulfed by the Milky Way, as well as gas rich, +irregular dwarf galaxies, such as the Large and Small Magellanic +clouds (LMC & SMC); the gas-deficient, classical dwarf spheroidal +galaxies (dSph), Carina, Draco, Fornax, Sculptor, Sextans, and Ursa- +Minor; and the ultra-faint dwarf spheroidal galaxy, Boötes I. Our +goal is to make inferences regarding the history of star formation and +chemical enrichment of GE/S by contrasting it with MW satellites. +We note that Hasselquist et al. (2021) have performed a similar +comparative analysis of GE/S and massive satellites of the MW. +In this paper we extend the analysis to lower mass satellites, with +a focus on the distribution of dwarf galaxy stellar populations on +the [Mg/Mn]-[Al/Fe] plane and how it can be used to constrain the +history of star formation of those systems. In addition, we examine +the behaviour of chemical evolution models calculated for a range of +relevant input parameters on that same chemical plane. +This paper is organised as follows. Section 2 presents the data +and sample adopted. In Section 3 we present an examination of the +Figure 1. Selecting GE/S candidates in IoM in the plane of APOGEE DR17 +stars, the number density is represented by the grey-scale colour bar and does +not include the following stellar populations; thin-thick disk selection (𝐿Z > +0, eccentricity < 0.3), globular clusters [VAC by Schiavon et al. (2021, in +prep)], the large and small Magellanic clouds, Sagittarius dSph and low-mass +dSph galaxies. The GE/S stars in this selection (pink thistle circles) are within +-1.6 < E/105 < -1.3 km2 s-2 and -0.4 < 𝐿z/103 < 0.3 kpc km s-1. The selection +criteria of disk stars (light steel blue circles) in this analysis have been placed +on top of the plane for reference. +distribution of the stellar populations of the various systems in key +chemical planes. In Section 4 we contrast the chemical composition +characteristics of GE/S with those of MW satellites of various +masses, using chemical evolution models to guide the discussion. +Our conclusions are presented in Section 5. +2 DATA AND SAMPLE +2.1 APOGEE DR17 +The second generation Apache Point Observatory Galactic Evolution +Experiment (APOGEE-2, Majewski et al. 2017) is part of the Sloan +Digital Sky Survey IV (Blanton et al. 2017). APOGEE-2 surveys the +stellar populations of the MW with high resolution (R ∼ 22,500), high +S/N, spectroscopy in the H–band spectral region (𝜆 ∼ 1.51–1.70 𝜇m), +using two twin multi–fibre spectrographs (Wilson et al. 2019). Less +hampered by interstellar dust than optical surveys its observations +cover both Northern and Southern hemispheres, based primarily on +the 2.5 m Sloan Foundation telescope (Gunn et al. 2006) at Apache +Point Observatory and the 2.5 m Iréné du Pont telescope (Bowen & +Vaughan 1973) at Las Campanas Observatory. +The data employed in this paper consist of chemical compositions, +stellar parameters and integrals of motion obtained from the 17th +data release (DR17, Abdurro’uf et al. 2021, Holtzman et al. in +prep.) of SDSS-IV/APOGEE-2. Chemical compositions and stellar +parameters were generated by the APOGEE Stellar Parameter and +Chemical Abundance Pipeline (ASPCAP, García Pérez et al. 2016), +whereas the integrals of motion result from application of the +galpy package (Bovy 2015; Mackereth & Bovy 2018) to 6D phase +space information resulting from combination of Gaia eDR3 proper +motions (Gaia Collaboration et al. 2021), APOGEE-2 DR17 radial +velocities (Nidever et al. 2015, Holtzman et al. in prep.), and +astroNN machine learning-based distances (Leung & Bovy 2019). +Calculations were performed adopting a McMillan (2017) potential +for the Milky Way. +MNRAS 000, 1–13 (2021) + +103 +-1.0 +S +102 +-1.5 +K +(kpc +-2.0 +105 +101 +E +-2.5 +Stars in GE/S System +Disk Selection +APOGEE DR17 +Reduced Sample Selection +-3.0 +100 +-2 +-1 +-3 +0 +2 +Lz/103 +(kpc km Chemistry of GE/S vs Milky Way Satellites +3 +Table 1. Total sample selection. +Stellar +No. of Stars +Structures +in Sample +Disks +193,220 +LMC +4,610 +SMC +1,660 +Sgr +291 +Boötes I +3 +Carina +35 +Draco +31 +Fornax +140 +Sculptor +85 +Sextans +18 +Ursa Minor +25 +GE/S +1,952 +2.2 SELECTION CRITERIA +Our goal is to examine the chemical compositions of stars belonging +to 10 dwarf satellites of the Milky Way, namely LMC, SMC, +Boötes I, Carina, Draco, Fornax, Sculptor, Sagittarius, Sextans and +Ursa Minor, contrasting their distributions in chemical diagnostic +planes with those of the accreted system GE/S and the MW high- +and low-𝛼 disks. The APOGEE-2 DR17 catalogue contains data for +733,901 stars, selected according to criteria extensively discussed +by Zasowski et al. (2013), Zasowski et al. (2017), and Santana +et al. (2021). Before selecting stars belonging to the above systems, +we must apply a number of criteria to certify the quality of the +data for our analysis. We first cleaned the sample from stars with +unreliable parameters by removing all stars with STARFLAG or +ASPCAPFLAG=BAD (see definitions in Holtzman et al. 2015). We +then limit the data to red giant stars with S/N>50, stellar effective +temperatures (𝑇eff) between 3750 – 5500 K and surface gravity +(log(g)) < 3.0. +Next, we removed from the sample a total of 7,562 stars that are +deemed to be members of globular clusters, as listed in the Value +Added Catalogue by Schiavon et al. (2022, in prep.). Application of +the above filters left us with a sample of 300,389 stars. The data for +the objects of interest were extracted from this surviving catalogue +according to the criteria described in the following sub-sections. The +final sample sizes for each system are given in Table 1. +2.2.1 Magellanic Clouds +Our sample selection for the LMC and SMC members mimics that +of Nidever et al. (2020) and is summarised in Table 2. We focus on +the bright and faint red giant branch (RGB) stellar populations in the +MCs (see their figures 3 and 5). In this way we expect to restrict our +sample to stars in approximately the same evolutionary stage as those +in the MW and other satellites. +2.2.2 Sgr dSph +Our sample for the Sgr dSph stars comes from the study by +Hasselquist et al. (2017) and was selected by the methods described +in Majewski et al. (2013). Further sampling of Sgr core and stream +Table 2. MCs Sample Selection. Table summaries the sky positions (𝛼,𝛿), +projected distance on the sky (𝑑𝑝𝑟𝑜 𝑗), Gaia proper motions (PM), radial +velocities (RV), and magnitudes (H) for the LMC and SMC. +LMC +SMC +𝛼 𝛿 : (80.893860, -69.756126) +𝛼 𝛿 : (13.18667, -72.8286) +𝑑𝑝𝑟𝑜 𝑗 <= 12 +𝑑𝑝𝑟𝑜 𝑗 <= 8 +RV >= 125 +RV >= 100 +2.7 <= 𝛼PM >= 1 +2.0 <= 𝛼PM >= 0 +2 <= 𝛿PM >= -1.2 +-0.5 <= 𝛿PM >= -2.0 +J-K >= 0.5 +J-K >= 0.5 +12.35 < H < 14.6 +12.9 < H < 14.7 +members are covered extensively in Hasselquist et al. (2019a) and +Hayes et al. (2020)1. Table 3 summaries the selection criteria. +2.2.3 Dwarf Spheroidal Galaxies +APOGEE-2 has targeted a number of dwarf spheroidal galaxies. The +field placement and target selection criteria adopted are described by +Zasowski et al. (2017) and Santana et al. (2021). To identify dwarf +spheroidal members we first selected all stars observed within the +fields of each dwarf galaxy and filtered out foreground contaminants +on the basis of radial velocity and surface gravity. Stars considered +members are giants (log 𝑔 < 3.0) whose heliocentric radial velocities +differ from the central values for each galaxy by less than twice +its velocity dispersion. By proceeding in this way we prioritise +sample purity over completeness. Radial velocities and velocity +dispersions of the sample dSph galaxies are taken from Table 1 +from McConnachie & Venn (2020) and Table 4 from McConnachie +(2012)2 with the exception of Boötes I, for which values were taken +from Martin et al. (2007). +2.2.4 Gaia-Enceladus/Sausage +Stars from the accreted system GE/S are distributed throughout +the MW and can be discriminated through a range of chemical, +kinematical and/or orbital selection criteria. In order to obtain a +GE/S sample devoid of chemical composition biases, we base our +selection purely on integral of motion (IoM) measurements. +After removal from the sample of all stars associated to the MW +satellites, the stellar populations of the accreted system GE/S are +selected on the basis of their position in the energy vs. angular +momentum (E-𝐿z) plane. +A star is considered a member of the Gaia-Enceladus system if its +energy and angular momentum fall within the following intervals: +• –1.6 < E/105 < –1.3 km2 s−2 +• –0.4 < 𝐿z/103 < 0.3 kpc km s−1 +This region of E-𝐿z is shown in Figure 1. The above selection +criteria mimic those adopted in previous work (e.g., Koppelman +et al. 2019b; Massari et al. 2019; Feuillet et al. 2021; Horta et al. +2021a). They are designed to take advantage of the overdensity +1 http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/ +ApJ/889/63 +2 http://www.astro.uvic.ca/~alan/Nearby_Dwarf_Database. +html +MNRAS 000, 1–13 (2021) + +4 +L. Fernandes et al. +Table 3. Sgr Sample Selection. Table summaries the sky positions (𝛼,𝛿), +radial velocities (RV), surface gravity (log g), effective temperature (𝑇eff), +and signal-to-noise (S/N) selection criteria for the Sagittarius dSph galaxy +core and tail. +Sagittarius dSph Galaxy +𝛼 𝛿 : (284, -30) +90 km s−1 < RV < 220 km s−1 +d > 5 kpc +J - K0 > 0.8 +S/N > 70 +3550 K < 𝑇eff < 4200 K +log g < 4 +in the E-𝐿Z plane around 𝐿Z ∼ 0 and at relatively high energy, +that is easily identifiable in Figure 1. We deliberately adopted a +relatively high lower energy limit for our GE/S selection with an eye +towards minimising disk contamination. Yet because we imposed +no chemical composition cuts, we expect a small contamination of +our GE/S sample by disk stars. See discussion in Section 3.1. This +contamination by disk stars is further enhanced by the existence of +“Splash” stars, which are early disk stars whose orbits were perturbed +by the collision with GE/S (Belokurov et al. 2020). For a discussion +of the impact of selection criteria on the chemical properties of GE/S, +see Horta et al. (2022b). +2.2.5 Thin & Thick Disk +The stellar populations of the the MW disk are selected using orbital +parameters in IoM, focusing on stars with circular, prograde orbits. +We thus retain disk stars with the following criteria: +• 𝐿z > 0 +• eccentricity < 0.3 +• S/N > 70 +The eccentricity cut employed is used to select stellar populations +with disc orbits. The adopted eccentricity threshold is arbitrary. +Placing the cut at, e.g., eccentricity < 0.2 or < 0.4 would cause +no impact on our analysis. The detailed choice is not critical because +the disc population is used solely as a reference for comparison with +the dwarf satellite data. +3 CHEMICAL PROPERTIES +In the next two sections we analyse the chemical properties of the +dwarf galaxies and GE/S members alongside the MW disks in the +chemical planes of Mg, Mn, Al and [Fe/H]. +3.1 Magnesium +Magnesium is an 𝛼-element synthesised during carbon burning +in massive stars, and injected into the interstellar medium during +supernovae type II (SNe II) explosions (Kobayashi et al. 2006; +Woosley & Weaver 1995). The distribution of the stellar populations +on the 𝛼-Fe plane provides important clues on the star formation +history and IMF of the system (e.g., Matteucci & Greggio 1986; +Wheeler et al. 1989; McWilliam 1997). +In Figure 2, the distributions of the stellar populations in the dwarf +galaxies and GE/S stars are shown in the chemical plane of [Fe/H] +Table 4. Properties of dSph Galaxies in the sample, including ID, stellar +mass, velocity dispersion, radial velocity, and original references. All data +from compilation by McConnachie (2012). +Galaxy +M★ +R.V. +𝜎 +References +(106𝑀⊙) +(km s−1) +(km s−1) +Boötes I +0.029 +99.9 ± 2.4 +6.5+2.1 +−1.3 +(1) +Carina +0.38 +222.9 ± 0.1 +6.6 ± 1.2 +(2)(6)(7) +Draco +0.29 +-291 ± 0.1 +9.1 ± 1.2 +(2)(3)(4) +Fornax +20 +55.3 ± 0.1 +11.7 ± 0.9 +(2)(6)(7)(9) +Sculptor +2.3 +111.4 ± 0.12 +9.2 ± 1.4 +(2)(6)(7)(8) +Sextans +0.44 +224.2 ± 0.1 +7.9 ± 1.3 +(2)(6)(7) +Ursa Minor +0.29 +-246.9 ± 0.1 +9.5 ± 1.2 +(2)(4)(5) +References: (1) Martin et al. (2007), (2) Grcevich & Putman (2009), +(3) Walker et al. (2007), (4) Wilkinson et al. (2004), (5) Walker +et +al. +(2009b), +(6) +Walker +et +al. +(2009a), +(7) +Walker +et +al. +(2008), +(8) +Carignan +et +al. +(1998), +(9) +Bouchard +et +al. +(2006) +- [Mg/Fe] across eleven panels, each displayed alongside the stellar +populations of the low- and high-𝛼 disks of the MW, whose chemical +compositions range roughly between ∼ −1.2 < [Fe/H] < +0.65 +and −0.2 +< +[Mg/Fe] +< ++0.4. These numbers are in good +agreement with independent studies based on abundance analysis +of high resolution optical spectra, such as those by Aguado et al. +(2021), Matsuno et al. (2021), and Carrillo et al. (2022). +We start by pointing out that, for the reasons discussed in +Section 2.2, our GE/S sample contains a small, yet non-negligible, +contamination by in situ stars, which can be easily spotted as they fall +on the loci defined by the low- and high-𝛼 disks. It is fair to assume +that the metal-rich stars in our GE/S sample that overlap with the high- +and low-𝛼 disc sequences are contaminants because, on one hand, +their chemical properties associate them strongly with the disc, and +moreover it has been argued by Mackereth et al. (2018) and Mason +et al. (2022, in prep.) that low mass galaxies do not host a bimodal +𝛼 distribution. It is also likely that our sample may be contaminated +on the metal-poor end, as in situ and accreted populations overlap +on the [Mg/Fe]-[Fe/H] plane at low metallicity, (e.g., Horta et al. +2020). However, the disc sequence becomes very thinly populated +at [Fe/H]<∼–1.2, and moreover our selection criteria prioritizing +high energy stars also helps minimising disk contaminations. Thus, +unless otherwise stated, our discussion henceforth ignores metal-rich +contaminants overlapping with the high- and loq-𝛼 disk sequences +on both chemical planes under study. +The latter are characterised by predominantly low [Fe/H] and lower +[Mg/Fe] than high-𝛼 disk stars at [Fe/H]>∼–1, and overall decreasing +with increasing metallicity. On the metal-poor end ([Fe/H]<∼–1.8), +GE/S stars reach [Mg/Fe] values comparable or even slightly higher +than those of the high-𝛼 disk, whereas on the metal-rich end +([Fe/H]∼–0.6) GE/S stars have slightly lower [Mg/Fe] than that of +low-𝛼 disk stars of same metallicity. The slope of the [Mg/Fe]-[Fe/H] +relation undergoes a slight change, forming the so-called “𝛼-knee” +at [Fe/H]∼–1.2 (see also Mackereth et al. 2019; Horta et al. 2021a), +which indicates the increased contribution of SN Ia to the chemical +enrichment of the interstellar medium. +By and large, the locus occupied by GE/S stars in the Mg-Fe +plane is somewhat similar to that where stars from the massive MW +satellites LMC, SMC, and Sgr dSph are located (see also Hasselquist +et al. 2021). The distributions however differ in important details. +All three of the massive satellites show, at a given [Fe/H], a positive +MNRAS 000, 1–13 (2021) + +Chemistry of GE/S vs Milky Way Satellites +5 +Figure 2. Stellar populations of the dwarf galaxies and GE/S in the plane of MW disk (marginal density 2D hexagonal binning - the grey-scale of each hexbin +denotes the number of points) in the chemical plane of [Mg/Fe] versus [Fe/H]. From top to bottom; the selected GE/S stellar population (thistle) as shown +in figure 1, LMC (light blue), SMC (coral), Sgr dSph (blue), Sculptor (green), Fornax (teal), Carina (crimson), Draco (cyan), Sextans (fuchsia), Ursa Minor +(brown) and Boötes I (purple). +change in the slope of the [Mg/Fe]-[Fe/H] relation, whereby [Mg/Fe] +starts increasing with increasing metallicity. This is likely associated +with the occurrence of a burst of star formation in those systems, +which causes an increase in the contribution of SNII to the chemical +enrichment of the interstellar medium, resulting in a jump in [Mg/Fe] +for increasing [Fe/H] (for a discussion, see Mason et al. 2022, in +prep.). Because star formation in GE/S was quenched at the time +of accretion, no similar change in slope can be seen in its stellar +populations. +In contrast, for most low-mass dwarf spheroidal galaxies (except +for Fornax), [Mg/Fe] decreases monotonically with increasing +[Fe/H]. The slope of the relation is steeper than that of GE/S and +the more massive satellites, and the mean [Mg/Fe] is substantially +lower in low-mass satellites. At [Fe/H]∼–1.2 the stellar populations +of low mass satellites are lower in [Mg/Fe] by ∼0.2–0.4 dex than +those from their massive counterparts. +The absence of a clear “knee” in the 𝛼-Fe plane of low mass +satellites is likely due to it being located in those systems at +metallicities that are lower than the values spanned by our sample. +Indeed, previous studies of stars in Draco, Sextans and Ursa Minor +(Shetrone et al. 2003), Sculptor (Hill et al. 2019), and Sextans (Theler +et al. 2020) suggest the presence of a “knee” at [Fe/H]∼–2. Based on a +compilation of chemical compositions from various works (Cohen & +Huang 2009, 2010; Starkenburg et al. 2013), Hendricks et al. (2014) +quote [Fe/H]𝑘𝑛𝑒𝑒 ∼ −1.9 for Sculptor, Ursa Minor, and Fornax +and [Fe/H]𝑘𝑛𝑒𝑒 <∼ −2.5 for Draco and Carina (see also de Boer et al. +2014). The low metallicity limit of our sample prohibits any statement +on the existence of a change of slope for the latter two galaxies. As +for the others, while our data do not rule out the presence of a +change of slope in the Mg-Fe relation for Sculptor, Ursa Minor, and +Fornax at [Fe/H]∼ −2, they cannot confirm it either, due to increased +uncertainties and relatively small sample sizes in the low metallicity +end. +The above result, taken together with the overall lower [Mg/Fe] of +lower mass satellites, suggests that the contribution by SN Ia to the +chemical enrichment of the interstellar medium of those satellites was +more dominant than in their massive counterparts, possibly indicating +a lower star formation rate (SFR) throughout their histories (e.g., +Mason et al. 2022, in prep.). The one exception is Fornax, for which +a sharp positive change of slope can be seen at [Fe/H]∼–1.2, similar +to the case of more massive satellites, suggesting also in the case of +Fornax the occurrence of a burst of star formation in the recent past. +MNRAS 000, 1–13 (2021) + +0.75F +GE/S +0.50 +0.25 +0.00 +103 +-0.25 +-0.50 +-2.4 +-1.8 +-1.2 +-0.6 +0.0 +0.6 +0.75 +0.75 +0.75 +LMC +SMC +Sgr dSph +0.50 +0.50 +0.50 +0.25 +0.25 +0.25 +0.00 +0.00 +0.00 +-0.25 +-0.25 +-0.25 +102 +-0.50 +-0.50 +-0.50 +-0.6 +0.6 +-2.4 +-1.8 +-0.6 +0.6 +-0.6 +-2.4 +-1.8 +1.2 +0.0 +-1.2 +0.0 +-2.4 +-1.8 +-1.2 +0.0 +0.6 +0.75 +0.75 +Sculptor +●Fornax +0.50 +0.50 +F +Mg/Fel +0.25 +0.25 +0.00 +0.00 +-0.25 +-0.25 +-0.50 +-0.50 +-2.4 +-1.8 +-1.2 +-0.6 +0.0 +0.6 +-2.4 +-1.8 +-1.2 +-0.6 +0.0 +0.6 +101 +0.75 +0.75 +0.75 +●Draco +Sextans +●Carina +0.50 +0.50 +0.50 +0.25 +0.25 +0.25 +0.00 +0.00 +0.00 +-0.25 +-0.25 +-0.25 +. +:. +-0.50 +-0.50 +-0.50 +-0.6 +0.6 +-0.6 +0.6 +-0.6 +-2.4 +-1.8 +1.2 +0.0 +-2.4 +-1.8 +-1.2 +0.0 +-2.4 +-1.8 +-1.2 +0.0 +0.6 +0.75F +0.75F +●UrsaMinoｒ +Bootes +100 +0.50 +0.50 +F +0.25 +0.25 +0.00 +0.00 +-0.25 +-0.25 +-0.50 +-0.50 +-2.4 +-1.8 +-1.2 +-0.6 +0.0 +0.6 +-2.4 +-1.8 +-1.2 +-0.6 +0.0 +0.6 +[Fe/H]6 +L. Fernandes et al. +Most importantly for the goals of this study, when we contrast the +position of GE/S in the Mg-Fe plane with those of dwarf satellites of +the MW, we conclude that it has undergone an intense early history of +star formation leading up to the build up of a relatively metal-rich and +𝛼-enhanced stellar population, akin to those of the massive satellites +of the MW. This is not surprising, considering current estimates for +the original mass of the GE/S progenitor (∼ 108−109M⊙, e.g., Helmi +et al. 2018; Belokurov et al. 2018; Mackereth et al. 2019; Deason +et al. 2019; Mackereth & Bovy 2020b; Feuillet et al. 2020). +3.2 Aluminium & Manganese [Al, Mn] +It is well known that stellar population diagnosis is substantially +improved by the consideration of the abundances of elements +associated with distinct nucleosynthetic pathways. In that vein, it +has been proposed by Hawkins et al. (2015) and Das et al. (2020) +that the combination of the abundances of Fe, Mg, Mn, and Al +can aid in the discrimination of accreted stellar populations in the +Galactic halo. Manganese is an Fe-peak element generated in Type +Ia supernovae (SNIa) (Iwamoto et al. 1999; Hillebrandt & Niemeyer +2000; Weinberg et al. 2019). According to Hawkins et al. (2015), +manganese is a more pure indicator of enrichment by SNe Ia than +iron, which makes that element particularly useful for chemical +diagnosis. Unlike the case of [Mg/Fe], [Mn/Fe] correlates positively +with metallicity (Kobayashi et al. 2020). This sharp distinction +between the dependence of Mg and Mn on metallicity makes the +ratio between these two elements a powerful discriminator between +stellar populations with different chemical evolution histories. +Aluminium, in turn, is referred to as an odd-Z element. Although +similarly to magnesium, Al is produced predominantly by SNe II +(Buchmann et al. 1984; Prantzos & Diehl 1996), it can be contributed +relevantly by a number of other nucleosynthetic sources. Aluminium +primarily forms during H burning phases in the CNO, NeNa and +MgAl cycles (Samland 1998; Guelin et al. 1995). A small amount +of Al is also created in white dwarf binary collisions (Nofar et al. +1991; Weiss & Truran 1990). Other sources of Al are observed from +the winds of Wolf-Rayet (Limongi & Chieffi 2006) and AGB stars +(Nomoto et al. 1984). Furthermore, viable sources of 26Al in the ISM +are thought to originate from the accretion of hydrogen-rich gas in +white dwarf binaries following novae explosions. (Gehrz et al. 1998; +Kamiński et al. 2018; Clayton 1984). +Das et al. (2020) used the SDSS-III/APOGEE DR14 sample +(Abolfathi et al. 2018; Holtzman et al. 2018) to show that stars +belonging to the GE/S system occupy a distinct locus in the [Al/Fe] +vs. [Mg/Mn] plane, characterised by low [Al/Fe] and high [Mg/Mn]. +More recently, Horta et al. (2021a) used chemical evolution models +to show that this particular locus of chemical space is actually the +home of chemically unevolved stellar populations. In other words, +any early stellar populations inhabit that region of chemical space, +regardless of where they are formed. As chemical evolution proceeds, +the elemental abundances of in situ populations move away from that +locus of chemical space, whereas the star formation of early accreted +systems is quenched, so that chemical compositions are frozen in +their early, pre-accretion state. +4 STELLAR POPULATIONS: CHEMICALLY EVOLVED +OR UN-EVOLVED +In this section we examine the distribution of GE/S stars and MW +dwarf satellites in the [Al/Fe] vs. [Mg/Mn] plane, in order to check +whether the above scenario is supported by an entirely empirical +examination of the data. For that purpose we compare the distribution +of GE/S stars in that plane with those of the low- and high-mass +satellites of the MW. +4.1 The detailed chemistry of a kinematically selected GE/S +Our first goal is to check whether the locus of a GE/S sample selected +purely on the basis of kinematics would be concentrated in the +“chemically unevolved” region of the [Mg/Mn]-[Al/Fe] space. We +recall that, for this analysis to be meaningful, it is critical that the +selection of the stars from all systems is entirely free of any chemical +composition criterion (for details, see Section 2.2.4). +Inspection of Figure 3 shows that the majority of GE/S stars +selected purely on the basis of orbital parameters fall within the +“chemically unevolved” locus of the [Mg/Mn]-[Al/Fe] space where +82% of GE/S stars are located. It is important, however, to keep in +mind that this line is arbitrary so that it is possible that some stars +in the “evolved” region actually belong to GE/S. There is a small +amount of contamination by high-𝛼 disk stars, spreading towards the +high [Al/Fe] and high [Mg/Mn] (upper right) locus of the chemical +space. In contrast to the case of [Mg/Fe] abundance ratios, all the +dwarf galaxies and GE/S are characterised by similarly low [Al/Fe]. +In fact, at [Fe/H]<∼–1.0, the bulk [Al/Fe] abundances in the dwarf +galaxies are lower than those in the low- and high-𝛼 disks. +Our GE/S sample stars fall squarely within the “chemically +unevolved” region of the plot, which we interpret as indicating an +early quenching of star formation taking place during the merger of +that galaxy with the MW (see discussion in Feuillet et al. 2021). This +result confirms our interpretation of the distribution of the stellar +populations in the [Mg/Mn]-[Al/Fe] plane, as well as the chemical +evolution calculations presented in Horta et al. (2021b). +4.2 GE/S vs Dwarf Satellites +We next compare the distribution of GE/S stars in the [Al/Fe] vs. +[Mg/Mn] plane with those of the massive MW satellites (MCs and +Sgr dSph). In contrast to the case of GE/S, a substantial fraction of the +stars in the MCs and the Sgr dSph lie outside that locus. This is not +surprising, as these massive satellites continued forming stars long +after star formation in GE/S had ceased. The chemically evolved, +more metal-rich, stars in those massive satellites spread towards the +low [Mg/Mn] region of the plane, at approximately constant [Al/Fe]. +It is also worth noticing that those among the stars belonging to the +massive satellite that do inhabit the “chemically unevolved” locus of +the plot are located towards a substantially lower mean [Mg/Mn] than +the stars from GE/S. While the GE/S stars within the “chemically +unevolved” region have mean [Mg/Mn] ∼ 0.50, those in the MCs and +Sgr dSph have mean values between ∼ 0.35 and 0.25 dex lower. We +speculate whether this difference is due to a selection effect caused +by the fact that the massive satellites samples may be biased towards +the high metallicity end of the MDF. +Differences between the loci occupied by GE/S and lower mass +satellites are even more pronounced, particularly for Draco, Carina, +Sextans and Ursa Minor, whose sample stars are predominantly +located outside the “chemically unevolved” region. While this +difference may partly reflect selection biases, it is qualitatively +consistent with these lower mass satellites having undergone an +evolutionary history characterised by a low, yet more prolonged, +SFR than those of GE/S and the massive MW satellites, which would +naturally lead to a stronger contribution to enrichment by SN Ia and +MNRAS 000, 1–13 (2021) + +Chemistry of GE/S vs Milky Way Satellites +7 +Figure 3. Diagnostic plot [Al/Fe] - [Mg/Mn]. Stellar populations of the dwarf galaxies and GE/S in the plane of MW disk (marginal density 2D hexagonal +binning - the grey-scale of each hexbin denotes the number of points). From top to bottom; the selected GE/S stellar population (thistle) as shown in figure 1, +LMC (light blue), SMC (coral), Sgr dSph (blue), Sculptor (green), Fornax (teal), Carina (crimson), Draco (cyan), Sextans (fuchsia), Ursa Minor (brown) and +the disk selection (steel blue) as shown in figure 1. Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars and the unevolved region. +a consequently lower mean [Mg/Mn]. This is further discussed in +Section 4.5. +4.3 Comparison with Chemical Evolution Models +In Figure 4 we build on the diagnostic [Al/Fe]-[Mg/Mn] plane with +the addition of two chemical evolution models calculated using +the flexCE code from Andrews et al. (2017). The orange line +shows the evolution of a model made to match the properties of +the stellar populations of the solar neighbourhood, representing an +in situ population, and the yellow model line shows a chemical +evolution model built to match the chemical properties of Gaia- +Enceladus/Sausage, characterising the chemical evolution of a +relatively massive satellite galaxy. The parameters adopted for the +chemical evolution models are shown in Table 5. The in situ MW +model is outlined in Horta et al. (2020). The model for GE/S was +built to match the distribution of the data on the Si-Fe plane. For +details on the model parameters, see Tables 3 and 4 of Hasselquist +et al. (2021). The models evolve for approximately 13 Gyr: the +filled circles on the models mark the evolutionary times at 0.3, 1 +and 5 Gyr. The black cross marks the position at which the iron +abundance reaches [Fe/H]=–1. The star formation efficiencies in +the two models differ by an order of magnitude, at 1.5 Gyr−1 in +the in situ case (Horta et al. 2021b) and 0.14 Gyr−1 for the best +fitting GE/S model (Hasselquist et al. 2021). As a result, the solar +neighbourhood model reaches [Fe/H]=–1 a mere 0.12 Gyr after the +beginning of the evolution, whereas the GE/S model takes ∼ 1.18 +Gyr to reach the same metallicity. The models provide a qualitatively +good description of the data for the in situ, accreted, and satellite +stellar populations on the [Mg/Mn] vs. [Al/Fe] plane. These model +calculations are an important tool for the interpretation of the data. +In both cases, the early chemical enrichment drives the evolution +towards the right due to the contribution by Type II/core collapse +supernovae, because [Mg/Mn] remains approximately constant while +[Al/Fe] grows due to the metallicity dependence of Al yields. By +the same token, downward/left evolution on this plane reflects the +increased contribution by SN Ia. +The data for the various accreted/satellite stellar populations in +Figure 4 are colour-coded by metallicity. One can immediately +notice a difference between GE/S and all the MW satellites, which +MNRAS 000, 1–13 (2021) + +1.2 +Unevolved +GE/S +0.9 +Region +situ +0.6 +α +103 +0.3 +0.0 +low α +-Evolved +In - situ +-0.3 +Region +-0.6 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +Unevolved +Unevolved +Unevolved +OLMC +● SMC +dSph +0.9 +Region +0.9 +Region +0.9 +Region +0.6 +situ +0.6 + situ +situ +0.6 +α +h +α +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +low α +Low α +low α +102 +In - situ +In - situ +In - situ +-0.3 +-Evolved +-0.3 +Evolved +-0.3 +Evolved +Region +Region +Region +-0.6. +-0.6,L +-0.6. +1.2 +-0.8 +-0.4 +0.0 +0.4 +.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +Unevolved +Unevolved + Sculptor +CFornax +0.9TRegion +0.9tRegion + situ +0.6 + situ +0.6 +gh α +α +0.3 +0.3 +0.0 +0.0 +low α +low α +-0.3 Evolved +In - situ +-0.3Evolved +In - situ +Region +Region +101 +-0.6 +-0.6 +-1.2 +-0.8 +-0.4 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +1.2 +2 +1.2 +Unevolved +Unevolved +/ODraco +Unevolved +Carina +OSextans +0.9 +0.9 +Region +0.9 +Region +Region +situ +situ +situ +0.6 +0.6 +0.6 +sh α +0.3 +0.3 +0.3 +. +C +0.0 +0.0 +0.0 +low α +Low α +low α +In - situ +In - situ +In - situ +-0.3 +LEvolved +-0.3 +-Evolved +-0.3 +-Evolved +Region +Region +Region +-0.6. +-0.6L +-0.6. +-0.8 +-0.4 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +100 +Unevolved +OUrsaMinor +Unevolved +Disk +0.9Region +0.9tRegion +0.6 +situ +0.6 +gh α +highα +0.3 +0.3 +0.0 +0.0 +low α +10w.α +· +-0.3 +Evolved· +In - situ +-0.3 +-Evoived +In - situ +Region +Region +-0.6 +-0.6 +-0.4 +-1.2 +-0.8 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +[Al/Fe]8 +L. Fernandes et al. +Figure 4. Diagnostic plot [Al/Fe] - [Mg/Mn], colour-coded by metallicity. The MW Chemical Evolution Model (orange) and Dwarf Chemical Evolution Model +(yellow) in the [Mg/Mn] vs [Al/Fe] abundance plane. Coloured circles mark the evolutionary times at 0.3, 1.0 and 5.0 Gyrs. Black cross is the position at which +the models reach [Fe/H] = ∼-1: t= 0.1 Gyrs for the MW chemical evolution model and t = 1.2 Gyrs for the dwarf chemical evolution model. From top to bottom; +GE/S, LMC, SMC, Sgr dSph, Sculptor, Fornax, Carina, Draco, Sextans, Ursa Minor and the disk. Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars and +the unevolved region. +Table 5. Summary of parameters used in the chemical evolution models. +Model +MW +GE/S +LMC-like +Sgr dSph-like +Sculptor-like +Parameters +Initial Gas Mass +2 × 1010M⊙ +3 × 109M⊙ +3 × 109M⊙ +3 × 109M⊙ +3 × 109M⊙ +Inflow Mass Scale +3.5 × 1011M⊙ +6 × 1010M⊙ +6 × 1010M⊙ +6 × 1010M⊙ +6 × 1010M⊙ +Outflow Mass Loading Factor +2.5 +6 +5.4 +20 +40 +Star Formation Efficiency +1 × 10−9yr−1 +1.5 × 10−10yr−1 +1.25 × 10−11yr−1 +2.5 × 10−11yr−1 +1 × 10−11yr−1 +Exponential Inflow Timescale +6 Gyr +2.5 Gyr +2.5 Gyr +2.5 Gyr +2.5 Gyr +have lower [Al/Fe] on average than GE/S. Indeed, the model is a +good match to GE/S, by construction, while failing to reproduce +the main locus of satellites, which is particularly noteworthy in +[Al/Fe]. Given the clear dependence of the value of [Al/Fe] on star +formation efficiency indicated by the models, this difference suggests +that GE/S formed stars more vigorously early in its history than the +MW satellites. In Section 4.5 we discuss the behaviour of chemical +evolution models on this plane in more detail. +MNRAS 000, 1–13 (2021) + +[Fe/H] +-2.5 +-2.0 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.2 +Unevolved +.GE/S +0.9TRegion +0.6 +situ +0.3 +103 +0.0 +low α +-0.3 LEvolved +In - situ +Region +-0.6, L +J +1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +1.2 +Unevolved +Unevolved +Unevolved +LMC +SMC +dSph +0.9TRegion +0.9 +Region +0.9 +Region +situ +0.6 +situ + situ +0.6 +0.6 +α +α +α +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +102 +low α +low α +low α +-Evolved +-0.3 +-0.3 FEvolved +-0.3 +-Evolved +In - situ +In - situ +In - situ +Region: +Region +Region +-0.6. +-0.6.1 +-0.6 +-0.8 +-0.4 +0.0 +0.4 +1.2 +-0.8 +-0.4 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +Unevolved +Sculptor +Unevolved +Fornax +0.9TRegion +0.9ERegion +· +Mn] +situ +situ +0.6 +0.6 +2 +α +gh ++ +0.3 +0.3 +/ +8 +90 +0.0 +0.0 +M. +low α +·low α +-0.3 +-Evolved +-0.3 FEvolved +In - situ +In - situ +Region +Region +101 +-0.6 +-0.6.l +1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +1.2 +Unevolved +Unevolved +Unevolved +Carina +Draco +Sextans +0.9TRegion +0.9 +Region +0.6 +situ + situ + situ +0.6 +0.6 +α +gh +gh α +. +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +low α +low α +low α +-0.3 FEvolved' +-0.3 LEvolved +-0.3 +FEvolved +In - situ +In - situ +In - situ +Region +Region +Region +. +-0.6 +-0.6. +-0.6 +.2 +-0.8 +-0.4 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +100 + Unevolved +Unevolved +UrsaMinｏr +Disk +0.9 +ERegion +0.9 +FRegion +-situ +0.6 +In - situ +0.6 +α +high:α +gh +0.3 +0.3 +0.0 +0.0 +low α +"low α +0.3 +Evolved +0.3 +Evoived +In - situ +In - situ +Region +Region +0.6. +a +0.6, +-1.2 +-0.8 +-0.4 +0.0 +0.4 +-1.2 +-0.8 +-0.4 +0.0 +0.4 +[Al/Fe]Chemistry of GE/S vs Milky Way Satellites +9 +Figure 5. Diagnostic plot [Al/Fe] - [Mg/Mn], colour-coded by metallicity. From top to bottom; GE/S stellar population (excluding the contaminating thick- +disk/high-𝛼 stars), LMC, SMC, Sgr dSph, Sculptor, Fornax and the disk selection as shown in figure 1. Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars +and the unevolved region. In each panel, the direction of the arrow represents the metallicity vector of the stellar population. +4.4 Metallicity gradients on the [Mg/Mn]-[Al/Fe] plane +We call attention to an important feature in the distribution of the +data for different stellar populations in Figure 4. As the points are +colour-coded by metallicity, one can see that the colour gradient on +the [Mg/Mn] vs [Al/Fe] plane, varies widely from galaxy to galaxy, +with more metal-rich stars being distributed towards the lower right +in more massive systems and towards the lower left in low mass +satellites. This suggests that GE/S and the MCs seem to have evolved +more quickly in [Al/Fe] than all the other surviving satellites of the +MW (this is true even after correction for contamination by in situ +stars, see Section 4.4). +We quantified this effect by measuring the metallicity gradients +of our sample galaxies in the [Mg/Mn]-[Al/Fe] plane as follows. +We first reduced the sample in two ways. For GE/S, we minimised +contamination by in situ stars by selecting them in the Mg-Fe plane +in the same way as (Horta et al. 2021a, see their appendix). In +addition, we reduced the overall sample to stars with the most +reliable elemental abundances. The APOGEE/ASPCAP abundance +measurements for Fe, Mg, and Al are quite reliable in the range of +metallicities spanned by our data. However, the abundance of Mn +relies on a few relatively weak lines, becoming more uncertain in +the low metallicity end. We examined the spectra of sample stars +in a range of metallicities and S/N ratios and decided to restrict the +sample to stars with [Fe/H]>–1.6 and, for those with [Fe/H]<–1.5 we +only kept spectra with S/N>70. As a result the samples for Carina, +Draco, Sextans, and Ursa Minor become too small and they are not +included in this analysis. On the basis of this reduced sample we fit +linear relations to the data for each galaxy in the [Mg/Mn]-[Fe/H] +and [Al/Fe]-[Fe/H] planes and derive the coefficients of the relations +in the [Mg/Mn]-[Al/Fe] plane. +The result is displayed in Figure 5, where arrows indicating the +direction (but not the modulus) of the metallicity gradients are +overlaid on the reduced samples described above. The arrows confirm +the visual impression about the direction of chemical evolution on +the diagnostic plane of Figures 3-5. In massive Milky Way satellites +the arrow points towards the lower right, whereas in lower mass +satellites and the Milky Way disk it points towards the lower left. +Along the same lines, the direction of the metallicity gradient in the +disk population points strongly towards the lower left of the chemical +plane. That is also the case of the disk of the Milky Way. +As indicated by the models in Figure 4, the overall direction of the +metallicity gradient vector in the [Mg/Mn]–[Al/Fe] plane seems to +be dependent on the star formation history of the system, which in +turn is associated with its mass. It is therefore instructive to examine +the behaviour of stellar populations on this chemical plane on the +basis of chemical evolution models sampling a wider range of input +parameters. That is the topic of the next Sub-section. +4.5 Chemical Evolution on the [Mg/Mn]-[Al/Fe] plane +In this Section we further examine the hypothesis that the metallicity +gradient in the [Mg/Mn]-[Al/Fe] plane is sensitive to the SFR. +For that purpose we calculated further chemical evolution models +building on those presented in Figure 4, based on flexCE (Andrews +et al. 2017). The same yields and SNe Ia delay time distribution +are adopted in all calculations. In order to produce a spectrum of +models that replicate the behaviour of dwarfs of different masses in +this chemical plane, we adopt a range of values for the star formation +MNRAS 000, 1–13 (2021) + +1.2 +fUnevolved +GE/S +0.9tRegion +:h- situ +0.6 +0.3 +[Fe/H] +0.0 +low α +0.5 +-0.3LEvolved +In - situ +Region +-0.6,L +T +-1.2 +-0.8 +-0.4 +0.0 +0.4 +1.2 +1.2 +0.0 +Unevolved +tUnevolved +Unevolved +LMC +SMC +Sgr +dSph +0.9ERegion +0.9FRegion +0.9 +Region +0.6 +Insitu +In - situ +In - situ +0.6 +0.6 +high α +high α +high α +-0.5 +0.3 +0.3 +0.3 +0.0 +0.0 +0.0 +low α +low α +low α +-0.3 FEvolved +-0.3 LEvolved +-0.3 +Evolved +In - situ +In - situ +In - situ +Region +Region +Region +-1.0 +-0.6. +-0.6 +-0.6 +-0.8 +-0.4 +0.0 +0.4 +-0.8 +-0.4 +0.0 +0.4 +1.2 +-0.8 +-0.4 +0.0 +0.4 +.2 +1.2 +Mn] +1.2 +1.2 +Unevolved +Unevolved +Sculptor +Fornax +0.9TRegion +0.9tRegion +-1.5 +/ +In - situ +In - situ +0.6 +0.6 +M +high α +high α +0.3 +0.3E +-2.0 +0.0 +0.0F +low α +·low α +-0.3LEvolved +-0.3LEvolved +In - situ +In - situ +Region +Region +-0.61 +0.6. +-0.8 +-0.4 +0.0 +0.4 +.2 +-0.8 +-0.4 +0.0 +0.4 +-2.5 +1.2 +Unevolved +Disk +0.9tRegion + situ +0.6F +high:α +0.3= +0.0 +low α +-0.3 LEvoived +In - situ +Region +-0.6,L +-0.8 +-0.4 +0.0 +0.4 +[Al/Fe]10 +L. Fernandes et al. +efficiency (SFE) and the wind mass-loading factor (𝜂), which are +respectively positively and negatively correlated with galaxy mass. +For GE/S and MW we adopt the same models as discussed in +Section 4.3. Table 5 summarises how these parameters differ between +the models. They are chosen to cover, in a qualitative fashion, the +range of properties characteristic of the dSph galaxies included in +our sample. Specific parameters adopted are tuned to approximately +match the properties of the MCs (MC-like), Sgr dSph (Sgr-like) and +the lower mass galaxies (Sculptor-like). +The results are displayed in Figure 6. The top panel shows +evolutionary tracks on the [Mg/Mn]-[Al/Fe] plane, whereas the star +formation histories are plotted in the bottom panel. As in Figure 4, +evolutionary times of 0.3, 1.0, and 5.0 Gyr are indicated for each +model. +An examination of the behaviour of these various models is +quite informative. We start by considering the fiducial MW model +(Table 5), represented by the solid blue line. It is the one for which +the SFR is by far the largest, exceeding those of the other models +by at least an order of magnitude. On account of such a very +high SFR, enrichment is initially entirely dominated by massive +stars. Thus the [Mg/Mn] ratio remains very high in the first few +100 Myr of evolution, whereas [Al/Fe] builds up very quickly. With +the increasing contribution by SN Ia at 𝑡 >∼300 Myr, [Mg/Mn] starts +declining steadily. As the star formation rate declines further, so does +the [Mg/Mn] ratio, and eventually the contribution by SN Ia becomes +important enough that the [Al/Fe] ratio starts declining, after about +1 Gyr of evolution. +The remainder of this discussion contrasts the behaviour of various +models within the first Gyr of cosmic evolution, where enrichment of +the elements involved is dominated by CCSNe and SN Ia. In all of the +dwarf-like models, the initial SFR is substantially lower than for the +MW model. As a result, within the first few 100 Myr the evolution in +[Al/Fe] is slower while [Mg/Mn] is more strongly influenced by the +SN Ia enrichment. Hence, with decreasing SFR the early evolutionary +tracks switch from pointing straight to the right to instead pointing +at increasing degrees towards the lower right, and finally towards +the lower left. Indeed, as suggested by the discussion in the previous +section, within the first Gyr of evolution, there is a clear correlation +between the orientation of the track on the [Mg/Mn]-[Al/Fe] plane +and the SFR. +This qualitative analysis can inform an interpretation of broad +trends that one can promptly grasp from even a perfunctory +evaluation of the data. For instance, it is easy to see from Figures 3 +and 4 that the data for GE/S are shifted towards higher [Al/Fe] and +[Mg/Mn] than those of the MCs, which in turn have higher values +than the lower mass MW satellites. The numbers are summarised in +Table 6. The mean values for [Mg/Mn] and [Al/Fe] are substantially +larger in GE/S than in all the MW satellites which, according to +our interpretation of the models, suggests a stronger SFR in the early +stages of evolution, in agreement with the results by Hasselquist et al. +(2019b). The same conclusion can be drawn when comparing the +massive MW satellites with their less massive counterparts, whose +data suggest a slower rate of chemical evolution, associated with a +weaker star formation rate. +Subsequent evolution, beyond ∼ 1 Gyr seems to be dictated by the +slope of the SFR. Models with strongly decreasing SFR (e.g., MW, +Sausage-, and Sgr-like) tend to evolve more strongly towards lower +[Mg/Mn] with more or less constant [Al/Fe], whereas those with +more approximately constant SFR (e.g., Sculptor- and LMC-like) +display a slight turn over of [Al/Fe] and slower decline in [Mg/Mn]. +These trends ultimately reflect the balance between the contribution +-1.0 +-0.5 +0.0 +0.5 +[Al/Fe] +-0.5 +0.0 +0.5 +1.0 +[Mg/Mn] +Un-Evolved Region +In-situ +high +In-situ +low +Evolved Region +0 +2 +4 +6 +t [Gyr] +10-3 +10-2 +10-1 +100 +101 +SFR [M yr−1] +Andrews+17 +fiducial model (MW-like) +LMC-like +Sausage-like +Sgr dSph-like +Sculptor-like +Figure 6. Chemical evolution and star formation histories for a series of +one-zone open box models ran using the flexCE code. Top panel: tracks of +[Mg/Mn](t) plotted as a function of [Al/Fe](t). Bottom panel: star formation +rate as a function of time, SFR(t). The solid blue tracks illustrate the fiducial +model of Andrews et al. (2017) as shown in Fig. 4. For the other models, +for a fixed history of gas inflow we vary the star formation efficiency and +the wind mass loading factor (𝜂) so as to qualitatively capture the behaviour +of a selection of the Local Group dwarfs in our sample with varying stellar +masses. Parameters adopted are listed in Table 5. +Table 6. Mean abundances and their dispersions of MW satelites in our +sample. +System +<[Al/Fe]> +𝜎[Al/Fe] +<[Mg/Mn]> +𝜎[Mg/Mn] +GE/S +-0.21 +0.17 +0.50 +0.20 +GE/S (clean) +-0.23 +0.14 +0.51 +0.20 +SMC +-0.42 +0.13 +0.27 +0.14 +LMC +-0.31 +0.12 +0.23 +0.13 +Sgr +-0.46 +0.08 +0.14 +0.10 +Sculptor +-0.64 +0.20 +0.28 +0.31 +Fornax +-0.59 +0.17 +0.17 +0.18 +MNRAS 000, 1–13 (2021) + +Chemistry of GE/S vs Milky Way Satellites +11 +by CCSNe and SN Ia (and AGB stars in the case of Al) to the +chemical enrichment of the interstellar medium (see Mason et al. +2022, in prep., for a detailed discussion). +The contrast between the histories of star formation of GE/S +and massive MW satellites, and in particular the LMC and SMC +is interesting in light of the fact that the mass of GE/S, according to +various studies, is of the order of a few to several times 108 𝑀⊙ (e.g., +?Deason et al. 2019; Mackereth et al. 2019), which is comparable +to that of the SMC (∼ 5 ×108 𝑀⊙) and smaller than that of the +LMC (∼ 1.5×109 𝑀⊙, McConnachie 2012). That galaxies with +similar masses have undergone such vastly different histories of star +formation indicates a physical variable other than mass is at play at +regulating the star formation histories of dwarf galaxies. Hasselquist +et al. (2021) suggest it is the environment in which the dwarf galaxies +formed and evolved. We tackle this problem from the point of view of +numerical cosmological simulations in a forthcoming paper (Mason +et al. 2022, in prep.). +5 CONCLUSIONS +We present a comparative study of the distribution in chemical +diagnostic +planes +of +the +stellar +populations +of +the +Gaia- +Enceladus/Sausage (GE/S) system and those of satellites of the Milky +Way. Our main conclusions are the following: +• We investigate the location on the [Mg/Mn] vs. [Al/Fe] plane +of a GE/S sample defined purely on the basis of orbital properties. +When selected in this way, GE/S stars lie almost entirely within +the locus of that chemical plane deemed to contain “accreted” +populations by (Das et al. 2020; Hawkins et al. 2015). While this +result validates previous use of that method for the identification of +stellar populations formed ex situ, caution is recommended, since +old populations formed in situ share the same locus, as shown by +Horta et al. (2021a). We therefore propose adopting a “chemically +unevolved” nomenclature when referring to that particular locus of +chemical space. +• The stellar populations of the satellites of the Milky Way are +mostly divided between the chemically evolved and unevolved loci on +this plane. The chemically evolved, more metal-rich stars are located +towards the region of lower [Mg/Mn] and approximately constant, or +slightly different, [Al/Fe]. +• The distribution of GE/S stars on the [Mg/Mn]–[Al/Fe] plane +differs from those of MW satellites in an important respect. The +chemical evolution of its stellar populations in this plane suggest a +higher early star formation rate than MW satellites with comparable +or even higher masses, as suggested by Hasselquist et al. (2019b). +• The direction of the metallicity vector on the [Mg/Mn]–[Al/Fe] +plane is an indicator of the early star formation rate of a system. +Higher mass galaxies and/or those undergoing high star formation +rates evolve more quickly in [Al/Fe] than in [Mg/Mn]. The existence +of this trend is suggested by the APOGEE data on the stellar +populations of the systems under study, and is boldly confirmed by +the predictions of analytical chemical evolution models. The ensuing +interpretation of our data on MW satellites in the light of such models +leads to the conclusion that the early star formation rates of these +systems was strongly affected by parameters other than galaxy mass. +ACKNOWLEDGEMENTS +Funding for the Sloan Digital Sky Survey IV has been provided by the +Alfred P. Sloan Foundation, the U.S. Department of Energy Office +of Science, and the Participating Institutions. +SDSS-IV acknowledges support and resources from the Center for +High Performance Computing at the University of Utah. The SDSS +website is www.sdss.org. +SDSS-IV is managed by the Astrophysical Research Consortium +for the Participating Institutions of the SDSS Collaboration including +the Brazilian Participation Group, the Carnegie Institution for +Science, Carnegie Mellon University, Center for Astrophysics | +Harvard & Smithsonian, the Chilean Participation Group, the +French Participation Group, Instituto de Astrofísica de Canarias, +The Johns Hopkins University, Kavli Institute for the Physics +and Mathematics of the Universe (IPMU) / University of Tokyo, +the Korean Participation Group, Lawrence Berkeley National +Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), +Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max- +Planck-Institut für Astrophysik (MPA Garching), Max-Planck- +Institut für Extraterrestrische Physik (MPE), National Astronomical +Observatories of China, New Mexico State University, New York +University, University of Notre Dame, Observatário Nacional / +MCTI, The Ohio State University, Pennsylvania State University, +Shanghai Astronomical Observatory, United Kingdom Participation +Group, Universidad Nacional Autónoma de México, University of +Arizona, University of Colorado Boulder, University of Oxford, +University of Portsmouth, University of Utah, University of Virginia, +University of Washington, University of Wisconsin, Vanderbilt +University, and Yale University. +This work has made use of data from the European Space +Agency (ESA) mission Gaia (https://www.cosmos.esa.int/ +gaia), processed by the Gaia Data Processing and Analysis +Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/ +dpac/consortium). Funding for the DPAC has been provided by +national institutions, in particular the institutions participating in the +Gaia Multilateral Agreement. +Software used in this research: Astropy (Astropy Collaboration +et al. 2018; Astropy Collaboration et al. 2013), SciPy (Virtanen +et al. 2020), NumPy (Oliphant 2006; Harris et al. 2020), Matplotlib +(Hunter 2007), Galpy (Bovy 2015), TOPCAT (Taylor 2005), flexCE +(Andrews et al. 2017). +DATA AVAILABILITY +This research was made possible with data from the SDSS- +IV/APOGEE-2, 17th data release and Gaia eDR3, publicly available +at https://www.sdss.org/dr17/irspec/spectro_data/ and +https://gea.esac.esa.int/archive/, respectively. +MNRAS 000, 1–13 (2021) + +12 +L. Fernandes et al. +REFERENCES +Abdurro’uf et al., 2021, arXiv e-prints, p. arXiv:2112.02026 +Abolfathi B., et al., 2018, ApJS, 235, 42 +Aguado D. S., et al., 2021, ApJ, 908, L8 +Andrews B. H., Weinberg D. H., Schönrich R., Johnson J. 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S., et al., 2019, The Messenger, 175, 3 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2021) + diff --git a/OdAzT4oBgHgl3EQfWfyE/content/tmp_files/load_file.txt b/OdAzT4oBgHgl3EQfWfyE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b57370227cb7fc9330eb2e49a72c1a03bd4fe795 --- /dev/null +++ b/OdAzT4oBgHgl3EQfWfyE/content/tmp_files/load_file.txt @@ -0,0 +1,1815 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf,len=1814 +page_content='MNRAS 000, 1–13 (2021) Preprint 5 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 A Comparative Analysis of the Chemical Compositions of Gaia-Enceladus/Sausage and Milky Way Satellites using APOGEE Laura Fernandes1,★ Andrew C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mason1, Danny Horta1, Ricardo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Schiavon1, Christian Hayes2, Sten Hasselquist3, Diane Feuillet4, Rachael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Beaton5,6, Henrik Jönsson7, Shobhit Kisku1, Ivan Lacerna8,9, Jianhui Lian10, Dante Minniti11,12, Sandro Villanova13 1 Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK 2 NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Canada, V9E 2E7 3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 4 Lund Observatory, Department of Astronomy and Theoretical Physics, Box 43, SE-221 00 Lund, Sweden 5 Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University, Princeton, NJ 08544 6 The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Pasadena, CA 91101 7 Materials Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden 8 Instituto de Astronomía y Ciencias Planetarias, Universidad de Atacama, Copayapu 485, Copiapó, Chile 9 Millennium Institute of Astrophysics, Nuncio Monsenor Sotero Sanz 100, Of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 104, Providencia, Santiago, Chile 10 Department of Physics and Astronomy, University of Utah, 115 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1400 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Salt Lake City, UT 84112, USA 11 Departamento de Ciencias Físicas, Facultad de Ciencias Exactas, Universidad Andres Bello, Fernández Concha 700, Las Condes, Santiago, Chile 12 Vatican Observatory, Vatican City State, V-00120, Italy 13 Departamento de Astronomía, Universidad de Concepción, Casilla 160-C, Concepción, Chile Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We use data from the 17th data release of the Apache Point Observatory Galactic Evolution Experiment (APOGEE 2) to contrast the chemical composition of the recently discovered Gaia Enceladus/Sausage system (GE/S) to those of ten Milky Way (MW) dwarf satellite galaxies: LMC, SMC, Boötes I, Carina, Draco, Fornax, Sagittarius, Sculptor, Sextans and Ursa Minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Our main focus is on the distributions of the stellar populations of those systems in the [Mg/Fe]-[Fe/H] and [Mg/Mn]-[Al/Fe] planes, which are commonly employed in the literature for chemical diagnosis and where dwarf galaxies can be distinguished from in situ populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We show that, unlike MW satellites, a GE/S sample defined purely on the basis of orbital parameters falls almost entirely within the locus of “accreted” stellar populations in chemical space, which is likely caused by an early quenching of star formation in GE/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Due to a more protracted history of star formation, stars in the metal-rich end of the MW satellite populations are characterized by lower [Mg/Mn] than those of their GE/S counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The chemical compositions of GE/S stars are consistent with a higher early star formation rate than MW satellites of comparable and even higher mass, suggesting that star formation in the early universe was strongly influenced by other parameters in addition to mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We find that the direction of the metallicity gradient in the [Mg/Mn]–[Al/Fe] plane of dwarf galaxies is an indicator of the early star formation rate of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Key words: Galaxies: dwarf spheroidal – stars: abundances 1 INTRODUCTION Our current understanding of galaxy formation is immersed in the framework of the Lambda Cold Dark Matter (ΛCDM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In this model, galaxies are an ensemble of stars, gas, and dust bound by massive dark matter haloes, built largely by the merging of smaller systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', White & Rees 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Blumenthal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Frenk & White 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In this context, the Milky Way (MW) is of great importance, as the galaxy for which we can obtain the most detailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Naturally, one should expect that our knowledge of the MW should be used to constrain galaxy formation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' While ★ E-mail: Lfernandes2@msn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='com (LF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='Schiavon@ljmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='uk (RPS) most of the stellar mass in the MW is contained in its bar and both thin and thick disks (Morris & Serabyn 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Nataf 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Rich 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Beraldo e Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021), these structures are enveloped by a large stellar halo which contains about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 × 109 𝑀⊙ in the form of stars (Mackereth & Bovy 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Galaxy mergers were very common in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Local evidence of the merging activity of the MW has been accumulating over the years, with the identification of the remnants of multiple accretion events, starting with the discovery of the Sagittarius dwarf spheroidal (Sgr dSph) by Ibata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In addition, various stellar streams have been identified as remnants of past accretion events (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Due to the stellar halo’s long dynamical timescale, such relatively recent accretion events can be © 2021 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='01302v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='GA] 3 Jan 2023 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' identified in the form of spatial substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' However, discerning the remnants of earlier accretion requires additional information, in the form of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', kinematics, chemical compositions, and ages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Nissen & Schuster 2010, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Schuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018) With the advent of large spectroscopic surveys, we entered the golden age of Galactic archaeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The information acquired by past, ongoing and future surveys such as RAVE, (Steinmetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2006), SEGUE (Yanny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2009), LAMOST (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2012), GALAH (De Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2015), Gaia mission (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2016), WEAVE (Dalton 2016), APOGEE (Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2017), H3 (Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019), MOONS (Cirasuolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020), 4MOST (de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019) is very quickly advancing our understanding of the history of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Chemical compositions and orbital information is becoming available for millions of stars in the MW and its Local Group companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In addition, with the launching of the Gaia satellite (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2016), detailed phase space information has become available, enabling the identification of further substructures in the stellar halo of the Galaxy, associated with various accretion events, including Gaia-Enceladus/Sausage, a massive dwarf galaxy accreted to the MW about ∼10Gyr ago (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Haywood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mackereth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019), as well as Heracles (Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021a), and various other substructures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Koppelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2015) and Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020) have recently proposed that a combination of the abundances of Mn, Mg, Al, and Fe can be used to chemically discriminate accreted populations from their counterparts formed in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' However, Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021a) used chemical evolution models to show that the region occupied by accreted populations in the [Mg/Mn] vs [Al/Fe] plane has a non- negligible presence of stars formed in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is difficult to disentangle metal-poor in situ populations from their accreted counterparts based on purely kinematic or orbital criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' To minimise inter-sample contamination, one has to often resort to additional criteria based on chemical compositions, introducing sample biases that prevent a clean analysis of the chemical compositions of accreted and in situ populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' That difficulty can be overcome by resorting to data from external systems, on the assumption that the chemistry of their stellar populations is dominated by intrinsic evolutionary effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In this paper we contrast the distribution of stars from GE/S system in key chemical composition planes with those of dwarf satellites of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The latter include the Sagittarius dSph, which is currently in the process of being engulfed by the Milky Way, as well as gas rich, irregular dwarf galaxies, such as the Large and Small Magellanic clouds (LMC & SMC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the gas-deficient, classical dwarf spheroidal galaxies (dSph), Carina, Draco, Fornax, Sculptor, Sextans, and Ursa- Minor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' and the ultra-faint dwarf spheroidal galaxy, Boötes I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Our goal is to make inferences regarding the history of star formation and chemical enrichment of GE/S by contrasting it with MW satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We note that Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021) have performed a similar comparative analysis of GE/S and massive satellites of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In this paper we extend the analysis to lower mass satellites, with a focus on the distribution of dwarf galaxy stellar populations on the [Mg/Mn]-[Al/Fe] plane and how it can be used to constrain the history of star formation of those systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In addition, we examine the behaviour of chemical evolution models calculated for a range of relevant input parameters on that same chemical plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Section 2 presents the data and sample adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In Section 3 we present an examination of the Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Selecting GE/S candidates in IoM in the plane of APOGEE DR17 stars, the number density is represented by the grey-scale colour bar and does not include the following stellar populations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' thin-thick disk selection (𝐿Z > 0, eccentricity < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3), globular clusters [VAC by Schiavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021, in prep)], the large and small Magellanic clouds, Sagittarius dSph and low-mass dSph galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The GE/S stars in this selection (pink thistle circles) are within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 < E/105 < -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 km2 s-2 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 < 𝐿z/103 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 kpc km s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The selection criteria of disk stars (light steel blue circles) in this analysis have been placed on top of the plane for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' distribution of the stellar populations of the various systems in key chemical planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In Section 4 we contrast the chemical composition characteristics of GE/S with those of MW satellites of various masses, using chemical evolution models to guide the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Our conclusions are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2 DATA AND SAMPLE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 APOGEE DR17 The second generation Apache Point Observatory Galactic Evolution Experiment (APOGEE-2, Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2017) is part of the Sloan Digital Sky Survey IV (Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' APOGEE-2 surveys the stellar populations of the MW with high resolution (R ∼ 22,500), high S/N, spectroscopy in the H–band spectral region (𝜆 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='51–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='70 𝜇m), using two twin multi–fibre spectrographs (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Less hampered by interstellar dust than optical surveys its observations cover both Northern and Southern hemispheres, based primarily on the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 m Sloan Foundation telescope (Gunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2006) at Apache Point Observatory and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 m Iréné du Pont telescope (Bowen & Vaughan 1973) at Las Campanas Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The data employed in this paper consist of chemical compositions, stellar parameters and integrals of motion obtained from the 17th data release (DR17, Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021, Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=') of SDSS-IV/APOGEE-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Chemical compositions and stellar parameters were generated by the APOGEE Stellar Parameter and Chemical Abundance Pipeline (ASPCAP, García Pérez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2016), whereas the integrals of motion result from application of the galpy package (Bovy 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mackereth & Bovy 2018) to 6D phase space information resulting from combination of Gaia eDR3 proper motions (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021), APOGEE-2 DR17 radial velocities (Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2015, Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' ), and astroNN machine learning-based distances (Leung & Bovy 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Calculations were performed adopting a McMillan (2017) potential for the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' MNRAS 000, 1–13 (2021) 103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 S 102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 K (kpc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 105 101 E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Stars in GE/S System Disk Selection APOGEE DR17 Reduced Sample Selection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 100 2 1 3 0 2 Lz/103 (kpc km Chemistry of GE/S vs Milky Way Satellites 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Total sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Stellar No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' of Stars Structures in Sample Disks 193,220 LMC 4,610 SMC 1,660 Sgr 291 Boötes I 3 Carina 35 Draco 31 Fornax 140 Sculptor 85 Sextans 18 Ursa Minor 25 GE/S 1,952 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 SELECTION CRITERIA Our goal is to examine the chemical compositions of stars belonging to 10 dwarf satellites of the Milky Way, namely LMC, SMC, Boötes I, Carina, Draco, Fornax, Sculptor, Sagittarius, Sextans and Ursa Minor, contrasting their distributions in chemical diagnostic planes with those of the accreted system GE/S and the MW high- and low-𝛼 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The APOGEE-2 DR17 catalogue contains data for 733,901 stars, selected according to criteria extensively discussed by Zasowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2013), Zasowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2017), and Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Before selecting stars belonging to the above systems, we must apply a number of criteria to certify the quality of the data for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We first cleaned the sample from stars with unreliable parameters by removing all stars with STARFLAG or ASPCAPFLAG=BAD (see definitions in Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We then limit the data to red giant stars with S/N>50, stellar effective temperatures (𝑇eff) between 3750 – 5500 K and surface gravity (log(g)) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Next, we removed from the sample a total of 7,562 stars that are deemed to be members of globular clusters, as listed in the Value Added Catalogue by Schiavon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Application of the above filters left us with a sample of 300,389 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The data for the objects of interest were extracted from this surviving catalogue according to the criteria described in the following sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The final sample sizes for each system are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 Magellanic Clouds Our sample selection for the LMC and SMC members mimics that of Nidever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020) and is summarised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We focus on the bright and faint red giant branch (RGB) stellar populations in the MCs (see their figures 3 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In this way we expect to restrict our sample to stars in approximately the same evolutionary stage as those in the MW and other satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Sgr dSph Our sample for the Sgr dSph stars comes from the study by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2017) and was selected by the methods described in Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Further sampling of Sgr core and stream Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' MCs Sample Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table summaries the sky positions (𝛼,𝛿), projected distance on the sky (𝑑𝑝𝑟𝑜 𝑗), Gaia proper motions (PM), radial velocities (RV), and magnitudes (H) for the LMC and SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' LMC SMC 𝛼 𝛿 : (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='893860, -69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='756126) 𝛼 𝛿 : (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='18667, -72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8286) 𝑑𝑝𝑟𝑜 𝑗 <= 12 𝑑𝑝𝑟𝑜 𝑗 <= 8 RV >= 125 RV >= 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='7 <= 𝛼PM >= 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 <= 𝛼PM >= 0 2 <= 𝛿PM >= -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 <= 𝛿PM >= -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 J-K >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 J-K >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='35 < H < 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 < H < 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='7 members are covered extensively in Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2019a) and Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table 3 summaries the selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Dwarf Spheroidal Galaxies APOGEE-2 has targeted a number of dwarf spheroidal galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The field placement and target selection criteria adopted are described by Zasowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2017) and Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' To identify dwarf spheroidal members we first selected all stars observed within the fields of each dwarf galaxy and filtered out foreground contaminants on the basis of radial velocity and surface gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Stars considered members are giants (log 𝑔 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0) whose heliocentric radial velocities differ from the central values for each galaxy by less than twice its velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' By proceeding in this way we prioritise sample purity over completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Radial velocities and velocity dispersions of the sample dSph galaxies are taken from Table 1 from McConnachie & Venn (2020) and Table 4 from McConnachie (2012)2 with the exception of Boötes I, for which values were taken from Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 Gaia-Enceladus/Sausage Stars from the accreted system GE/S are distributed throughout the MW and can be discriminated through a range of chemical, kinematical and/or orbital selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In order to obtain a GE/S sample devoid of chemical composition biases, we base our selection purely on integral of motion (IoM) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' After removal from the sample of all stars associated to the MW satellites, the stellar populations of the accreted system GE/S are selected on the basis of their position in the energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' angular momentum (E-𝐿z) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' A star is considered a member of the Gaia-Enceladus system if its energy and angular momentum fall within the following intervals: –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 < E/105 < –1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 km2 s−2 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 < 𝐿z/103 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 kpc km s−1 This region of E-𝐿z is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The above selection criteria mimic those adopted in previous work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Koppelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' They are designed to take advantage of the overdensity 1 http://vizier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='fr/viz-bin/VizieR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='-source=J/ ApJ/889/63 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='uvic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='ca/~alan/Nearby_Dwarf_Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' html MNRAS 000, 1–13 (2021) 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Sgr Sample Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table summaries the sky positions (𝛼,𝛿), radial velocities (RV), surface gravity (log g), effective temperature (𝑇eff), and signal-to-noise (S/N) selection criteria for the Sagittarius dSph galaxy core and tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Sagittarius dSph Galaxy 𝛼 𝛿 : (284, -30) 90 km s−1 < RV < 220 km s−1 d > 5 kpc J - K0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 S/N > 70 3550 K < 𝑇eff < 4200 K log g < 4 in the E-𝐿Z plane around 𝐿Z ∼ 0 and at relatively high energy, that is easily identifiable in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We deliberately adopted a relatively high lower energy limit for our GE/S selection with an eye towards minimising disk contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Yet because we imposed no chemical composition cuts, we expect a small contamination of our GE/S sample by disk stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' See discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This contamination by disk stars is further enhanced by the existence of “Splash” stars, which are early disk stars whose orbits were perturbed by the collision with GE/S (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For a discussion of the impact of selection criteria on the chemical properties of GE/S, see Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Thin & Thick Disk The stellar populations of the the MW disk are selected using orbital parameters in IoM, focusing on stars with circular, prograde orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We thus retain disk stars with the following criteria: 𝐿z > 0 eccentricity < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 S/N > 70 The eccentricity cut employed is used to select stellar populations with disc orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The adopted eccentricity threshold is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Placing the cut at, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', eccentricity < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 or < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 would cause no impact on our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The detailed choice is not critical because the disc population is used solely as a reference for comparison with the dwarf satellite data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 3 CHEMICAL PROPERTIES In the next two sections we analyse the chemical properties of the dwarf galaxies and GE/S members alongside the MW disks in the chemical planes of Mg, Mn, Al and [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 Magnesium Magnesium is an 𝛼-element synthesised during carbon burning in massive stars, and injected into the interstellar medium during supernovae type II (SNe II) explosions (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Woosley & Weaver 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The distribution of the stellar populations on the 𝛼-Fe plane provides important clues on the star formation history and IMF of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Matteucci & Greggio 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' McWilliam 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In Figure 2, the distributions of the stellar populations in the dwarf galaxies and GE/S stars are shown in the chemical plane of [Fe/H] Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Properties of dSph Galaxies in the sample, including ID, stellar mass, velocity dispersion, radial velocity, and original references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' All data from compilation by McConnachie (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Galaxy M★ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 𝜎 References (106𝑀⊙) (km s−1) (km s−1) Boötes I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='029 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 (1) Carina 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='38 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 (2)(6)(7) Draco 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='29 291 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 (2)(3)(4) Fornax 20 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 (2)(6)(7)(9) Sculptor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 (2)(6)(7)(8) Sextans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='44 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 (2)(6)(7) Ursa Minor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='29 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 (2)(4)(5) References: (1) Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2007), (2) Grcevich & Putman (2009), (3) Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2007), (4) Wilkinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2004), (5) Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2009b), (6) Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2009a), (7) Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2008), (8) Carignan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (1998), (9) Bouchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2006) [Mg/Fe] across eleven panels, each displayed alongside the stellar populations of the low- and high-𝛼 disks of the MW, whose chemical compositions range roughly between ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 < [Fe/H] < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='65 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 < [Mg/Fe] < +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' These numbers are in good agreement with independent studies based on abundance analysis of high resolution optical spectra, such as those by Aguado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021), Matsuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021), and Carrillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We start by pointing out that, for the reasons discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2, our GE/S sample contains a small, yet non-negligible, contamination by in situ stars, which can be easily spotted as they fall on the loci defined by the low- and high-𝛼 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is fair to assume that the metal-rich stars in our GE/S sample that overlap with the high- and low-𝛼 disc sequences are contaminants because, on one hand, their chemical properties associate them strongly with the disc, and moreover it has been argued by Mackereth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2018) and Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=') that low mass galaxies do not host a bimodal 𝛼 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is also likely that our sample may be contaminated on the metal-poor end, as in situ and accreted populations overlap on the [Mg/Fe]-[Fe/H] plane at low metallicity, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' However, the disc sequence becomes very thinly populated at [Fe/H]<∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2, and moreover our selection criteria prioritizing high energy stars also helps minimising disk contaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Thus, unless otherwise stated, our discussion henceforth ignores metal-rich contaminants overlapping with the high- and loq-𝛼 disk sequences on both chemical planes under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The latter are characterised by predominantly low [Fe/H] and lower [Mg/Fe] than high-𝛼 disk stars at [Fe/H]>∼–1, and overall decreasing with increasing metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' On the metal-poor end ([Fe/H]<∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8), GE/S stars reach [Mg/Fe] values comparable or even slightly higher than those of the high-𝛼 disk, whereas on the metal-rich end ([Fe/H]∼–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6) GE/S stars have slightly lower [Mg/Fe] than that of low-𝛼 disk stars of same metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The slope of the [Mg/Fe]-[Fe/H] relation undergoes a slight change, forming the so-called “𝛼-knee” at [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 (see also Mackereth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021a), which indicates the increased contribution of SN Ia to the chemical enrichment of the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' By and large, the locus occupied by GE/S stars in the Mg-Fe plane is somewhat similar to that where stars from the massive MW satellites LMC, SMC, and Sgr dSph are located (see also Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The distributions however differ in important details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' All three of the massive satellites show, at a given [Fe/H], a positive MNRAS 000, 1–13 (2021) Chemistry of GE/S vs Milky Way Satellites 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Stellar populations of the dwarf galaxies and GE/S in the plane of MW disk (marginal density 2D hexagonal binning - the grey-scale of each hexbin denotes the number of points) in the chemical plane of [Mg/Fe] versus [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' From top to bottom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the selected GE/S stellar population (thistle) as shown in figure 1, LMC (light blue), SMC (coral), Sgr dSph (blue), Sculptor (green), Fornax (teal), Carina (crimson), Draco (cyan), Sextans (fuchsia), Ursa Minor (brown) and Boötes I (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' change in the slope of the [Mg/Fe]-[Fe/H] relation, whereby [Mg/Fe] starts increasing with increasing metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This is likely associated with the occurrence of a burst of star formation in those systems, which causes an increase in the contribution of SNII to the chemical enrichment of the interstellar medium, resulting in a jump in [Mg/Fe] for increasing [Fe/H] (for a discussion, see Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Because star formation in GE/S was quenched at the time of accretion, no similar change in slope can be seen in its stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In contrast, for most low-mass dwarf spheroidal galaxies (except for Fornax), [Mg/Fe] decreases monotonically with increasing [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The slope of the relation is steeper than that of GE/S and the more massive satellites, and the mean [Mg/Fe] is substantially lower in low-mass satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' At [Fe/H]∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 the stellar populations of low mass satellites are lower in [Mg/Fe] by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 dex than those from their massive counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The absence of a clear “knee” in the 𝛼-Fe plane of low mass satellites is likely due to it being located in those systems at metallicities that are lower than the values spanned by our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Indeed, previous studies of stars in Draco, Sextans and Ursa Minor (Shetrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2003), Sculptor (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019), and Sextans (Theler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020) suggest the presence of a “knee” at [Fe/H]∼–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Based on a compilation of chemical compositions from various works (Cohen & Huang 2009, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2013), Hendricks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2014) quote [Fe/H]𝑘𝑛𝑒𝑒 ∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 for Sculptor, Ursa Minor, and Fornax and [Fe/H]𝑘𝑛𝑒𝑒 <∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 for Draco and Carina (see also de Boer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The low metallicity limit of our sample prohibits any statement on the existence of a change of slope for the latter two galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As for the others, while our data do not rule out the presence of a change of slope in the Mg-Fe relation for Sculptor, Ursa Minor, and Fornax at [Fe/H]∼ −2, they cannot confirm it either, due to increased uncertainties and relatively small sample sizes in the low metallicity end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The above result, taken together with the overall lower [Mg/Fe] of lower mass satellites, suggests that the contribution by SN Ia to the chemical enrichment of the interstellar medium of those satellites was more dominant than in their massive counterparts, possibly indicating a lower star formation rate (SFR) throughout their histories (e.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 [Fe/H]6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Most importantly for the goals of this study, when we contrast the position of GE/S in the Mg-Fe plane with those of dwarf satellites of the MW, we conclude that it has undergone an intense early history of star formation leading up to the build up of a relatively metal-rich and 𝛼-enhanced stellar population, akin to those of the massive satellites of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This is not surprising, considering current estimates for the original mass of the GE/S progenitor (∼ 108−109M⊙, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mackereth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mackereth & Bovy 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Aluminium & Manganese [Al, Mn] It is well known that stellar population diagnosis is substantially improved by the consideration of the abundances of elements associated with distinct nucleosynthetic pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In that vein, it has been proposed by Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2015) and Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020) that the combination of the abundances of Fe, Mg, Mn, and Al can aid in the discrimination of accreted stellar populations in the Galactic halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Manganese is an Fe-peak element generated in Type Ia supernovae (SNIa) (Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hillebrandt & Niemeyer 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Weinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' According to Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2015), manganese is a more pure indicator of enrichment by SNe Ia than iron, which makes that element particularly useful for chemical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Unlike the case of [Mg/Fe], [Mn/Fe] correlates positively with metallicity (Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This sharp distinction between the dependence of Mg and Mn on metallicity makes the ratio between these two elements a powerful discriminator between stellar populations with different chemical evolution histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Aluminium, in turn, is referred to as an odd-Z element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Although similarly to magnesium, Al is produced predominantly by SNe II (Buchmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Prantzos & Diehl 1996), it can be contributed relevantly by a number of other nucleosynthetic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Aluminium primarily forms during H burning phases in the CNO, NeNa and MgAl cycles (Samland 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Guelin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' A small amount of Al is also created in white dwarf binary collisions (Nofar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Weiss & Truran 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Other sources of Al are observed from the winds of Wolf-Rayet (Limongi & Chieffi 2006) and AGB stars (Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Furthermore, viable sources of 26Al in the ISM are thought to originate from the accretion of hydrogen-rich gas in white dwarf binaries following novae explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (Gehrz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Kamiński et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Clayton 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020) used the SDSS-III/APOGEE DR14 sample (Abolfathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018) to show that stars belonging to the GE/S system occupy a distinct locus in the [Al/Fe] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' [Mg/Mn] plane, characterised by low [Al/Fe] and high [Mg/Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' More recently, Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021a) used chemical evolution models to show that this particular locus of chemical space is actually the home of chemically unevolved stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In other words, any early stellar populations inhabit that region of chemical space, regardless of where they are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As chemical evolution proceeds, the elemental abundances of in situ populations move away from that locus of chemical space, whereas the star formation of early accreted systems is quenched, so that chemical compositions are frozen in their early, pre-accretion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4 STELLAR POPULATIONS: CHEMICALLY EVOLVED OR UN-EVOLVED In this section we examine the distribution of GE/S stars and MW dwarf satellites in the [Al/Fe] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' [Mg/Mn] plane, in order to check whether the above scenario is supported by an entirely empirical examination of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For that purpose we compare the distribution of GE/S stars in that plane with those of the low- and high-mass satellites of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 The detailed chemistry of a kinematically selected GE/S Our first goal is to check whether the locus of a GE/S sample selected purely on the basis of kinematics would be concentrated in the “chemically unevolved” region of the [Mg/Mn]-[Al/Fe] space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We recall that, for this analysis to be meaningful, it is critical that the selection of the stars from all systems is entirely free of any chemical composition criterion (for details, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Inspection of Figure 3 shows that the majority of GE/S stars selected purely on the basis of orbital parameters fall within the “chemically unevolved” locus of the [Mg/Mn]-[Al/Fe] space where 82% of GE/S stars are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is important, however, to keep in mind that this line is arbitrary so that it is possible that some stars in the “evolved” region actually belong to GE/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' There is a small amount of contamination by high-𝛼 disk stars, spreading towards the high [Al/Fe] and high [Mg/Mn] (upper right) locus of the chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In contrast to the case of [Mg/Fe] abundance ratios, all the dwarf galaxies and GE/S are characterised by similarly low [Al/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In fact, at [Fe/H]<∼–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0, the bulk [Al/Fe] abundances in the dwarf galaxies are lower than those in the low- and high-𝛼 disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Our GE/S sample stars fall squarely within the “chemically unevolved” region of the plot, which we interpret as indicating an early quenching of star formation taking place during the merger of that galaxy with the MW (see discussion in Feuillet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This result confirms our interpretation of the distribution of the stellar populations in the [Mg/Mn]-[Al/Fe] plane, as well as the chemical evolution calculations presented in Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 GE/S vs Dwarf Satellites We next compare the distribution of GE/S stars in the [Al/Fe] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' [Mg/Mn] plane with those of the massive MW satellites (MCs and Sgr dSph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In contrast to the case of GE/S, a substantial fraction of the stars in the MCs and the Sgr dSph lie outside that locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This is not surprising, as these massive satellites continued forming stars long after star formation in GE/S had ceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The chemically evolved, more metal-rich, stars in those massive satellites spread towards the low [Mg/Mn] region of the plane, at approximately constant [Al/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is also worth noticing that those among the stars belonging to the massive satellite that do inhabit the “chemically unevolved” locus of the plot are located towards a substantially lower mean [Mg/Mn] than the stars from GE/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' While the GE/S stars within the “chemically unevolved” region have mean [Mg/Mn] ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='50, those in the MCs and Sgr dSph have mean values between ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='25 dex lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We speculate whether this difference is due to a selection effect caused by the fact that the massive satellites samples may be biased towards the high metallicity end of the MDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Differences between the loci occupied by GE/S and lower mass satellites are even more pronounced, particularly for Draco, Carina, Sextans and Ursa Minor, whose sample stars are predominantly located outside the “chemically unevolved” region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' While this difference may partly reflect selection biases, it is qualitatively consistent with these lower mass satellites having undergone an evolutionary history characterised by a low, yet more prolonged, SFR than those of GE/S and the massive MW satellites, which would naturally lead to a stronger contribution to enrichment by SN Ia and MNRAS 000, 1–13 (2021) Chemistry of GE/S vs Milky Way Satellites 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Diagnostic plot [Al/Fe] - [Mg/Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Stellar populations of the dwarf galaxies and GE/S in the plane of MW disk (marginal density 2D hexagonal binning - the grey-scale of each hexbin denotes the number of points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' From top to bottom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the selected GE/S stellar population (thistle) as shown in figure 1, LMC (light blue), SMC (coral), Sgr dSph (blue), Sculptor (green), Fornax (teal), Carina (crimson), Draco (cyan), Sextans (fuchsia), Ursa Minor (brown) and the disk selection (steel blue) as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars and the unevolved region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' a consequently lower mean [Mg/Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This is further discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Comparison with Chemical Evolution Models In Figure 4 we build on the diagnostic [Al/Fe]-[Mg/Mn] plane with the addition of two chemical evolution models calculated using the flexCE code from Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The orange line shows the evolution of a model made to match the properties of the stellar populations of the solar neighbourhood, representing an in situ population, and the yellow model line shows a chemical evolution model built to match the chemical properties of Gaia- Enceladus/Sausage, characterising the chemical evolution of a relatively massive satellite galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The parameters adopted for the chemical evolution models are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The in situ MW model is outlined in Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The model for GE/S was built to match the distribution of the data on the Si-Fe plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For details on the model parameters, see Tables 3 and 4 of Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The models evolve for approximately 13 Gyr: the filled circles on the models mark the evolutionary times at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3, 1 and 5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The black cross marks the position at which the iron abundance reaches [Fe/H]=–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The star formation efficiencies in the two models differ by an order of magnitude, at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Gyr−1 in the in situ case (Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021b) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='14 Gyr−1 for the best fitting GE/S model (Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As a result, the solar neighbourhood model reaches [Fe/H]=–1 a mere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='12 Gyr after the beginning of the evolution, whereas the GE/S model takes ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='18 Gyr to reach the same metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The models provide a qualitatively good description of the data for the in situ, accreted, and satellite stellar populations on the [Mg/Mn] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' [Al/Fe] plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' These model calculations are an important tool for the interpretation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In both cases, the early chemical enrichment drives the evolution towards the right due to the contribution by Type II/core collapse supernovae, because [Mg/Mn] remains approximately constant while [Al/Fe] grows due to the metallicity dependence of Al yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' By the same token, downward/left evolution on this plane reflects the increased contribution by SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The data for the various accreted/satellite stellar populations in Figure 4 are colour-coded by metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' One can immediately notice a difference between GE/S and all the MW satellites, which MNRAS 000, 1–13 (2021) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved GE/S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 α 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α Evolved In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved Unevolved Unevolved OLMC SMC dSph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 situ situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 α h α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α Low α low α 102 In - situ In - situ In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 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+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' 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+page_content='6 gh α α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evolved In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3Evolved In - situ Region Region 101 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region Region situ situ situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 sh α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α Low α low α In - situ In - situ In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 LEvolved 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 100 Unevolved OUrsaMinor Unevolved Disk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9tRegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 gh α highα 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α 10w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evolved· In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evoived In - situ Region Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 [Al/Fe]8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Diagnostic plot [Al/Fe] - [Mg/Mn], colour-coded by metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The MW Chemical Evolution Model (orange) and Dwarf Chemical Evolution Model (yellow) in the [Mg/Mn] vs [Al/Fe] abundance plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Coloured circles mark the evolutionary times at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 Gyrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Black cross is the position at which the models reach [Fe/H] = ∼-1: t= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='1 Gyrs for the MW chemical evolution model and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Gyrs for the dwarf chemical evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' From top to bottom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' GE/S, LMC, SMC, Sgr dSph, Sculptor, Fornax, Carina, Draco, Sextans, Ursa Minor and the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars and the unevolved region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Summary of parameters used in the chemical evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Model MW GE/S LMC-like Sgr dSph-like Sculptor-like Parameters Initial Gas Mass 2 × 1010M⊙ 3 × 109M⊙ 3 × 109M⊙ 3 × 109M⊙ 3 × 109M⊙ Inflow Mass Scale 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 × 1011M⊙ 6 × 1010M⊙ 6 × 1010M⊙ 6 × 1010M⊙ 6 × 1010M⊙ Outflow Mass Loading Factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 20 40 Star Formation Efficiency 1 × 10−9yr−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 × 10−10yr−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='25 × 10−11yr−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 × 10−11yr−1 1 × 10−11yr−1 Exponential Inflow Timescale 6 Gyr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Gyr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Gyr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Gyr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Gyr have lower [Al/Fe] on average than GE/S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Indeed, the model is a good match to GE/S, by construction, while failing to reproduce the main locus of satellites, which is particularly noteworthy in [Al/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Given the clear dependence of the value of [Al/Fe] on star formation efficiency indicated by the models, this difference suggests that GE/S formed stars more vigorously early in its history than the MW satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 we discuss the behaviour of chemical evolution models on this plane in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' MNRAS 000, 1–13 (2021) [Fe/H] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved Unevolved Unevolved LMC SMC dSph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9TRegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 situ situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 α α α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved Sculptor Unevolved Fornax 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+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' low α low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 FEvolved In - situ In - situ Region Region 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6.' metadata={'source': 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LEvolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 FEvolved In - situ In - situ In - situ Region Region Region .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 100 Unevolved Unevolved UrsaMinｏr Disk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 ERegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 FRegion situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 α high:α gh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α "low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evoived In - situ In - situ Region Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 [Al/Fe]Chemistry of GE/S vs Milky Way Satellites 9 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Diagnostic plot [Al/Fe] - [Mg/Mn], colour-coded by metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' From top to bottom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' GE/S stellar population (excluding the contaminating thick- disk/high-𝛼 stars), LMC, SMC, Sgr dSph, Sculptor, Fornax and the disk selection as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Black lines separate in-situ high-𝛼, in-situ low-𝛼 stars and the unevolved region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In each panel, the direction of the arrow represents the metallicity vector of the stellar population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 Metallicity gradients on the [Mg/Mn]-[Al/Fe] plane We call attention to an important feature in the distribution of the data for different stellar populations in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As the points are colour-coded by metallicity, one can see that the colour gradient on the [Mg/Mn] vs [Al/Fe] plane, varies widely from galaxy to galaxy, with more metal-rich stars being distributed towards the lower right in more massive systems and towards the lower left in low mass satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This suggests that GE/S and the MCs seem to have evolved more quickly in [Al/Fe] than all the other surviving satellites of the MW (this is true even after correction for contamination by in situ stars, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We quantified this effect by measuring the metallicity gradients of our sample galaxies in the [Mg/Mn]-[Al/Fe] plane as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We first reduced the sample in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For GE/S, we minimised contamination by in situ stars by selecting them in the Mg-Fe plane in the same way as (Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2021a, see their appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In addition, we reduced the overall sample to stars with the most reliable elemental abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The APOGEE/ASPCAP abundance measurements for Fe, Mg, and Al are quite reliable in the range of metallicities spanned by our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' However, the abundance of Mn relies on a few relatively weak lines, becoming more uncertain in the low metallicity end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We examined the spectra of sample stars in a range of metallicities and S/N ratios and decided to restrict the sample to stars with [Fe/H]>–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 and, for those with [Fe/H]<–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 we only kept spectra with S/N>70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As a result the samples for Carina, Draco, Sextans, and Ursa Minor become too small and they are not included in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' On the basis of this reduced sample we fit linear relations to the data for each galaxy in the [Mg/Mn]-[Fe/H] and [Al/Fe]-[Fe/H] planes and derive the coefficients of the relations in the [Mg/Mn]-[Al/Fe] plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The result is displayed in Figure 5, where arrows indicating the direction (but not the modulus) of the metallicity gradients are overlaid on the reduced samples described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The arrows confirm the visual impression about the direction of chemical evolution on the diagnostic plane of Figures 3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In massive Milky Way satellites the arrow points towards the lower right, whereas in lower mass satellites and the Milky Way disk it points towards the lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Along the same lines, the direction of the metallicity gradient in the disk population points strongly towards the lower left of the chemical plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' That is also the case of the disk of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As indicated by the models in Figure 4, the overall direction of the metallicity gradient vector in the [Mg/Mn]–[Al/Fe] plane seems to be dependent on the star formation history of the system, which in turn is associated with its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is therefore instructive to examine the behaviour of stellar populations on this chemical plane on the basis of chemical evolution models sampling a wider range of input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' That is the topic of the next Sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 Chemical Evolution on the [Mg/Mn]-[Al/Fe] plane In this Section we further examine the hypothesis that the metallicity gradient in the [Mg/Mn]-[Al/Fe] plane is sensitive to the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For that purpose we calculated further chemical evolution models building on those presented in Figure 4, based on flexCE (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The same yields and SNe Ia delay time distribution are adopted in all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In order to produce a spectrum of models that replicate the behaviour of dwarfs of different masses in this chemical plane, we adopt a range of values for the star formation MNRAS 000, 1–13 (2021) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 fUnevolved GE/S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9tRegion :h- situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3LEvolved In - situ Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6,L T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 Unevolved tUnevolved Unevolved LMC SMC Sgr dSph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9ERegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9FRegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9 Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 Insitu In - situ In - situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 high α high α high α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α low α low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 FEvolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 LEvolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 Evolved In - situ In - situ In - situ Region Region Region 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Mn] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved Unevolved Sculptor Fornax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9TRegion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9tRegion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 / In 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low α low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3LEvolved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3LEvolved In - situ In - situ Region Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='2 Unevolved Disk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='9tRegion situ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6F high:α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 low α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3 LEvoived In - situ Region 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='6,L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='4 [Al/Fe]10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' efficiency (SFE) and the wind mass-loading factor (𝜂), which are respectively positively and negatively correlated with galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For GE/S and MW we adopt the same models as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table 5 summarises how these parameters differ between the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' They are chosen to cover, in a qualitative fashion, the range of properties characteristic of the dSph galaxies included in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Specific parameters adopted are tuned to approximately match the properties of the MCs (MC-like), Sgr dSph (Sgr-like) and the lower mass galaxies (Sculptor-like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The results are displayed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The top panel shows evolutionary tracks on the [Mg/Mn]-[Al/Fe] plane, whereas the star formation histories are plotted in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As in Figure 4, evolutionary times of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 Gyr are indicated for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' An examination of the behaviour of these various models is quite informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We start by considering the fiducial MW model (Table 5), represented by the solid blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' It is the one for which the SFR is by far the largest, exceeding those of the other models by at least an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' On account of such a very high SFR, enrichment is initially entirely dominated by massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Thus the [Mg/Mn] ratio remains very high in the first few 100 Myr of evolution, whereas [Al/Fe] builds up very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' With the increasing contribution by SN Ia at 𝑡 >∼300 Myr, [Mg/Mn] starts declining steadily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As the star formation rate declines further, so does the [Mg/Mn] ratio, and eventually the contribution by SN Ia becomes important enough that the [Al/Fe] ratio starts declining, after about 1 Gyr of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The remainder of this discussion contrasts the behaviour of various models within the first Gyr of cosmic evolution, where enrichment of the elements involved is dominated by CCSNe and SN Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' In all of the dwarf-like models, the initial SFR is substantially lower than for the MW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' As a result, within the first few 100 Myr the evolution in [Al/Fe] is slower while [Mg/Mn] is more strongly influenced by the SN Ia enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hence, with decreasing SFR the early evolutionary tracks switch from pointing straight to the right to instead pointing at increasing degrees towards the lower right, and finally towards the lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Indeed, as suggested by the discussion in the previous section, within the first Gyr of evolution, there is a clear correlation between the orientation of the track on the [Mg/Mn]-[Al/Fe] plane and the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This qualitative analysis can inform an interpretation of broad trends that one can promptly grasp from even a perfunctory evaluation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For instance, it is easy to see from Figures 3 and 4 that the data for GE/S are shifted towards higher [Al/Fe] and [Mg/Mn] than those of the MCs, which in turn have higher values than the lower mass MW satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The numbers are summarised in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The mean values for [Mg/Mn] and [Al/Fe] are substantially larger in GE/S than in all the MW satellites which, according to our interpretation of the models, suggests a stronger SFR in the early stages of evolution, in agreement with the results by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The same conclusion can be drawn when comparing the massive MW satellites with their less massive counterparts, whose data suggest a slower rate of chemical evolution, associated with a weaker star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Subsequent evolution, beyond ∼ 1 Gyr seems to be dictated by the slope of the SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Models with strongly decreasing SFR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', MW, Sausage-, and Sgr-like) tend to evolve more strongly towards lower [Mg/Mn] with more or less constant [Al/Fe], whereas those with more approximately constant SFR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', Sculptor- and LMC-like) display a slight turn over of [Al/Fe] and slower decline in [Mg/Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' These trends ultimately reflect the balance between the contribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 [Al/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='0 [Mg/Mn] Un-Evolved Region In-situ high In-situ low Evolved Region 0 2 4 6 t [Gyr] 10-3 10-2 10-1 100 101 SFR [M yr−1] Andrews+17 fiducial model (MW-like) LMC-like Sausage-like Sgr dSph-like Sculptor-like Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Chemical evolution and star formation histories for a series of one-zone open box models ran using the flexCE code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Top panel: tracks of [Mg/Mn](t) plotted as a function of [Al/Fe](t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Bottom panel: star formation rate as a function of time, SFR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The solid blue tracks illustrate the fiducial model of Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2017) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' For the other models, for a fixed history of gas inflow we vary the star formation efficiency and the wind mass loading factor (𝜂) so as to qualitatively capture the behaviour of a selection of the Local Group dwarfs in our sample with varying stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Parameters adopted are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mean abundances and their dispersions of MW satelites in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' System <[Al/Fe]> 𝜎[Al/Fe] <[Mg/Mn]> 𝜎[Mg/Mn] GE/S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='20 GE/S (clean) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='20 SMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='14 LMC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='13 Sgr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='10 Sculptor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='31 Fornax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='18 MNRAS 000, 1–13 (2021) Chemistry of GE/S vs Milky Way Satellites 11 by CCSNe and SN Ia (and AGB stars in the case of Al) to the chemical enrichment of the interstellar medium (see Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', for a detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The contrast between the histories of star formation of GE/S and massive MW satellites, and in particular the LMC and SMC is interesting in light of the fact that the mass of GE/S, according to various studies, is of the order of a few to several times 108 𝑀⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Mackereth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2019), which is comparable to that of the SMC (∼ 5 ×108 𝑀⊙) and smaller than that of the LMC (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='5×109 𝑀⊙, McConnachie 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' That galaxies with similar masses have undergone such vastly different histories of star formation indicates a physical variable other than mass is at play at regulating the star formation histories of dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021) suggest it is the environment in which the dwarf galaxies formed and evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We tackle this problem from the point of view of numerical cosmological simulations in a forthcoming paper (Mason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2022, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 5 CONCLUSIONS We present a comparative study of the distribution in chemical diagnostic planes of the stellar populations of the Gaia- Enceladus/Sausage (GE/S) system and those of satellites of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Our main conclusions are the following: We investigate the location on the [Mg/Mn] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' [Al/Fe] plane of a GE/S sample defined purely on the basis of orbital properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' When selected in this way, GE/S stars lie almost entirely within the locus of that chemical plane deemed to contain “accreted” populations by (Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' While this result validates previous use of that method for the identification of stellar populations formed ex situ, caution is recommended, since old populations formed in situ share the same locus, as shown by Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' We therefore propose adopting a “chemically unevolved” nomenclature when referring to that particular locus of chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The stellar populations of the satellites of the Milky Way are mostly divided between the chemically evolved and unevolved loci on this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The chemically evolved, more metal-rich stars are located towards the region of lower [Mg/Mn] and approximately constant, or slightly different, [Al/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The distribution of GE/S stars on the [Mg/Mn]–[Al/Fe] plane differs from those of MW satellites in an important respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The chemical evolution of its stellar populations in this plane suggest a higher early star formation rate than MW satellites with comparable or even higher masses, as suggested by Hasselquist et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The direction of the metallicity vector on the [Mg/Mn]–[Al/Fe] plane is an indicator of the early star formation rate of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Higher mass galaxies and/or those undergoing high star formation rates evolve more quickly in [Al/Fe] than in [Mg/Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The existence of this trend is suggested by the APOGEE data on the stellar populations of the systems under study, and is boldly confirmed by the predictions of analytical chemical evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The ensuing interpretation of our data on MW satellites in the light of such models leads to the conclusion that the early star formation rates of these systems was strongly affected by parameters other than galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Department of Energy Office of Science, and the Participating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High Performance Computing at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The SDSS website is www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the Carnegie Institution for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Center for Astrophysics | Harvard & Smithsonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the Chilean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the French Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' the Korean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Leibniz Institut für Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Max-Planck-Institut für Astronomie (MPIA Heidelberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Max- Planck-Institut für Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Max-Planck- Institut für Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Observatário Nacional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Universidad Nacional Autónoma de México,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='int/ gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='int/web/gaia/ dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Software used in this research: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2013), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020), NumPy (Oliphant 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2020), Matplotlib (Hunter 2007), Galpy (Bovy 2015), TOPCAT (Taylor 2005), flexCE (Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content=' DATA AVAILABILITY This research was made possible with data from the SDSS- IV/APOGEE-2, 17th data release and Gaia eDR3, publicly available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} +page_content='org/dr17/irspec/spectro_data/ and https://gea.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAzT4oBgHgl3EQfWfyE/content/2301.01302v1.pdf'} diff --git a/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/2301.03156v1.pdf.txt b/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/2301.03156v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd1d6830b3bb20bedd85ce242ff803971c77e98c --- /dev/null +++ b/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/2301.03156v1.pdf.txt @@ -0,0 +1,2688 @@ +FINITE TOPOLOGIES FOR FINITE GEOMETRIES +OLIVER KNILL +Abstract. Without leaving finite mathematics and using finite topological spaces only, we +give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. +Besides exploring the definition in various contexts, we add some remarks like that the +general Lefschetz formula works for any continuous map on any finite topological space. +We also noted that any higher order Wu characteristic as well as their cohomology are +topological invariants which are not homotopy invariants. Energy theorems allow to express +these topological invariants in terms of interaction energies of local open sets. +1. About +1.1. +When exploring finite geometries using finite topological spaces, one is challenged with +the fact that homeomorphic finite topological spaces have the same cardinality, which is too +rigid. A finite metric space produces the discrete topology which is too fine. For the geodesic +metric on a graph for example, the topology is totally disconnected and so does not reflect +at all the connectivity of the graph. As probably first realized in 1937 by Alexandroff [1], +non-Hausdorff finite topologies still can capture essential parts of a topology that is usually +only explored using geometric realizations. How can one avoid geometric realizations and +still have a workable definition of homeomorphism? We have explored a related notion in [23] +using covers but use now classical finite topologies, accepting the fact that all reasonable finite +topological spaces are Alexandroff and naturally non-Hausdorff if they capture connectivity +properties of the space under consideration. +1.2. +This working document has grown a bit longer than anticipated but has been a seed +for further results like Green function formulas. As a remedy, we added summaries at the +end of each section. The write-up is a contribution to a program of replacing continuum +geometries in a finite set-up but with as little changes in notation as possible. We want to +avoid geometric realizations because using the continuum is a rather serious step. It is not +just a philosophical obsession: the mathematics of topological manifolds has told lessons like +that there are finite geometries G - and example is the simplicial complex obtained by taking +the join of a discrete circle with a homology 3-sphere - which produces a geometric realization +which is classically homeomorphic to the standard 5-sphere H even so from any finite point of +view they are not homeomorphic. They are geometries which have homeomorphic geometric +realizations but should not be considered homeomorphics. In some sense the mathematics +of the Hauptvermutung [52] has indicated that the continuum can lead to identifications +which are not expeected. A finite topology more honestly preserves details which geometric +realizations do not see any more. +One could of course use piecewise linear geometry to +capture what finite topology does, but also from a computer science point of view, it is +desirable to have finite objects and finite data to deal with only. All geometric objects are +Date: January 8, 2023. +Key words and phrases. Topology, Simplicial complexes, Graphs. +1 +arXiv:2301.03156v1 [math.CO] 9 Jan 2023 + +FINITE TOPOLOGY +faithfully implemented using a finite amount of information. The axiom of infinity is never +used. +1.3. +Finite topologies by definition are always Alexandroff spaces [1], meaning that every +point x has a smallest neighborhood U(x). When working with a sheaf over such a topology, +we do not need to conceptualize direct limit constructions like “stalks” or “germs”. In the +case of simplicial complexes, the smallest atomic “Planck units of space” containing x is +known as the “star” U(x) of the simplex x. +As connection calculus [36] illustrates, the +topology of intersections of stars U(x) ∩ U(y) can be complicated, even so each U(x), U(y) +is contractible. Indeed, the Euler characteristic of U(x) ∩ U(y) agrees up to a sign with +the matrix entries of the inverse g(x, y) of the connection matrix L(x, y), which is 1 if the +simplices intersect and 0 else. In analogy with the Green functions in classical frame works, +these numbers g(x, y) must be thought of the potential energies between x and y. They can +be rather arbitrary for large dimensional spaces. One of the reasons why small dimensional +topology is so much different from larger dimensional ones is that there can be surprises +in the topology of local “atomic parts of space”. While simple in small dimensions, the +intersection U(x)∩U(y) can be entangled in a rather complicated way in higher dimensions. +Figure 1. +The figure shows a finite simple graph G and its second Barycen- +tric refinement G2. The Whitney simplicial complex G of G has 17 sets x, +leading to 17 basis elements U(x) = U(x). These open sets in G are minimal +and called the stars of x. The closure U(x) is the unit ball and its boundary +S(x) = B(x)\U(x) is the unit sphere. The basis B generates a finite topology +O with 3032 open sets and 3032 closed sets. Most of the 217 = 131072 possible +subsets of G are neither open nor closed. The topology is not Hausdorff: one +can not separate points which intersect. As every finite topological space, it +is Alexandroff: every point x has a smallest neighborhood U(x). +2 + +OLIVER KNILL +1.4. +Graphs and simplicial complexes and finite topological spaces all provide model frame +works in finite geometry. The categories are closely related: if one of them is given, one can +construct relatives in the other classes. One can get from a graph to a simplicial complex +with the Whitney functor by assigning to the graph the vertex sets of complete subgraphs. +The ˇCech nerve construction produces from a topological space a simplicial complex. 1 From +a simplicial complex, one can then construct a graph in which vertices are the sets of the +complex and where two sets are connected if one is contained in the other. +Switching +forth and back between complexes and graphs produces a Barycentric refinement of the +topology. After identifying Barycentric refined complexes, simplicial complexes or graphs +can serve the same purpose. We always construct the topology O on the simplicial complex +G and denote individual points in G with x. The sub-simplicial complexes of G are then the +closed sets. In algebraic geometry, a similar constructions has led to the Zariski topology. +1.5. +While we deal here only with finite sets, most could be generalized to locally finite +complexes, meaning that there is an upper bound on the number of elements in an atom +U(x) and again define the basis by the smallest neighborhoods U(x). This local finite as- +sumption corresponds in the continuum to the step to restrict to paracompact topological +spaces, spaces where every open cover has a locally finite refinement. As for references in +topology, see [5, 14, 46, 58] for topology, and especially [2], a text already using abstract sim- +plicial complexes (introduced 1907 by Dehn and Heegaard [6]) and not the more commonly +used definition using geometric realization. Veblen [61] in 1922 defined a neighborhood of +a k-simplex but still used geometric realizations and cites Poincar´e for introducing the notion +of homeomorphism. Veblen also used already the terminology of “stars”. Every topology +on a finite set is always an Alexandroff topology. This is a topology, where points have +smallest neighborhoods or alternatively where arbitrary intersections of sets are open too. +The notion of “Stars” was established in combinatorial topology like [1] but also entered +some calculus textbooks like Whitney [63]. Still, as most texts, even Alexandroff look at +it primarily at geometric realizations of cell complexes. Alexandroff topology generalizes +the co-finite topology for 0-dimensional complexes or the order topology of a general +partially ordered set with a basis U(x) = {y, y ≥ x} [62]. +1.6. +As for simplicial complexes, see [8] or [47]. Simplicial complexes appearing in graph +theory are covered in [17]. +The most important link between graphs and complexes is +definitely the Whitney functor which assigns to a graph a complex which exactly has the +topological properties which the graph suggests without leading to ambiguities like what we +consider to be a face. In algebraic topology, the actual topology generated by the star basis +has not obtained the attention it deserves but it appears, for example in [58] on page 311. +The topology defined in [57] is defined on the vertex set and of different nature, as even +connectivity properties are different. See [43] for a review on finite topological spaces. Also +[59] is completely unrelated because the star of a vertex is defined by Stallings as the set of +edges attached to it and graphs considered one-dimensional objects, a common perspective +in the 20th century. There are quite a few other discrete frame works in finite geometry. +We should mention Ivashchenko [15, 10] who translated Whitehead’s homotopy notion into +concrete procedures in graph theory. It has been simplified in [7] and crucial for defining +what a “sphere” is combinatorially. The Morse approach is used in Forman’s discrete Morse +theory [11, 12]. In discrete combinatorics, one sometimes also looks a abstract simplicial +1The Whitney complex is also known as the face complex, clique complex, flag complex, or face poset. +3 + +FINITE TOPOLOGY +complexes, in which the ∅ is included. We use the frame work, where ∅ is not considered +to be a simplex but where it is considered to be a (−1)-dimensional sphere. The empty +set itself is a simplicial complex, as it fulfills the axiom, but it also does not contain the +empty set. All these definitions are compatible with the continuum, where d-spheres also +have Euler characteristic 1 + (−1)d even for d = −1 and where simplices all are contractible +and have Euler characteristic 1. +1.7. +We have explored the problem of defining homeomorphism within finite mathematics +for a few years already, the first time more seriously in 2014 [23]. Stars came up for us +especially in the context of the Green star identity, which explicitly gives the matrix +entries g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) of the inverse of the connection matrix L(x, y) = +χ({x ∩ y}) attached to a simplicial complex G with simplices x with ω(x) = (−1)dim(x) and +Euler characteristic χ(A) = � +x∈A ω(x) of a subset A of G. The connection matrix L +involves the Euler characteristic of closed sets like {x}. Its inverse matrix g involves the +Euler characteristic of open sets U(x). See [29, 37, 36]. The sets U(x) = W +(x) are open +sets and {x} = W −(x) are closed sets. An important role plays the closure B(x) = U(x) +called the unit ball. Its boundary S(x) = δU(x) = U(x)\U(x) called unit sphere. They +all are closed sets and so simplicial complexes. +Figure 2. +A visualization of the star U(x) in two examples. Unlike what +the picture suggests, we do not look at geometric realizations however. The +star of a point x ∈ G consists of all the simplices which contain x: that is +U(x) = {y, x ⊂ y}. +1.8. +In the case of a finite simple graph G, we could also look at the topology O on +its Whitney complex G, where the closed sets are the simplicial complexes coming +from subgraphs of G. This is a slightly rougher finite topology, because not all simplicial +complexes are Whitney complexes of graphs. We are currently under the impression that +the notion of homeomorphism discussed here is new. It has shifted quite a bit while we +were writing this text. +A useful definition needs to be simple and lead to all expected +results. The goal had been to get a definition of homeomorphism which generalizes the +definition used for one-dimensional complexes in topological graph theory [13], and which +has the ability to identify different triangulations of obvious manifolds like the icosahedron +4 + +OLIVER KNILL +and octahedron. In one dimensions, the notion of homeomorphism is old and enters for +example the Kuratowski theorem: two graphs G, H are called graph homeomorphic if +there exists a graph isomorphism between some edge refined versions of G and H. The +story relating the finite and infinite is a bit tricky: more than a hundred years of work in +the context of the Hauptvermutung has lead to surprises in the relation between finite +and infinite models. Finite topology can contain more information than the topology to +geometric realizations. Using the topology from realizations in Euclidean spaces allows for +some surprising homeomorphisms in higher dimensions. +1.9. +Having a topology on simplicial complexes is useful as it allows to reformulate classical +results in a more familiar language but within a finite frame work. For example: given +any simplicial complex G and any continuous map f : G → G, then the Lefschetz fixed +point formula [22] � +x∈F if(x) = χf(G) holds, where F is the set of fixed points of f, +where the index is defined as if(x) = ω(x)sign(f|x) with sign(f|x) is the signature of the +permutation which f induces on the simplex x, and where χf(G) is the super trace on +cohomology, which is defined as � +k≥0(−1)ktr(Uf|ker(Lk)), where Lk is the Hodge Laplacian +L = dd∗ + d ∗ d restricted to the linear subspace of functions on k-dimensional simplices, +where df(x) = � +y,|y|=|x|−1 sign(y|x)f(y) is the exterior derivative and Ufg(x) = g(f(x)) +is the linear Koopman map that f induces on functions. +1.10. +Note that the Lefschetz fixed point theorem [22] holds for all simplicial complexes +and all continuous functions. It generalized the theorem [51], which is a result in one +dimensions. In the continuum, one needs assumptions, like that there are only finitely many +fixed points. The discrete theorem had been formulated for graph endomorphisms [22] which +produce continuous maps on the corresponding simplicial complex but the proof works also +for continuous maps meaning for example that the map can contract. The Lefschetz fixed +point theorem has two special cases: the first case is if f is the identity, where it becomes +the Euler-Poincar´e formula � +x ω(x) = � +k(−1)kbk, where bk are the Betti numbers, +the dimensions of the kernels of Lk. An other special case is if the cohomology is trivial, +meaning that only constant functions are in the kernel of L. This applies for example if G is +an arbitrary contractible complex and among manifolds with boundaries if G is a k-ball. This +is the discrete Brouwer fixed point theorem: every continuous map on a finite abstract +simplicial complex that is a k-ball has at least one fixed point. An other example where +we always have fixed points is if G is an even-dimensional sphere and if f is continuous but +preserves the orientation of the maximal simplices. These are results in finite mathematics. +At no point, the concept of infinity is used. +1.11. +One of the main points was to have a clear definition what we mean with homeomor- +phism in finite topological spaces. Once one has such a notion, one can see what topolog- +ical invariants are. [4] defined combinatorial invariants as properties invariant under +Barycentric refinement. We are especially interested in numerical quantities, that are not +homotopy invariants. An example in the continuum is the analytic torsion [40], adapted +from the continuum [53]. There are other properties we believe to be topological like being +a Dehn-Sommerville space [27]. For all higher characteristics [25], which are only defined in +the discrete so far, there are topological expressions which could be used to compute them +in the continuum as we write down expressions which hold for the smallest open sets which +exist in the topology. For the second characteristic, the Wu characteristic ω = ω2, we +know omega(B) = (−1)k, where k is the dimension of a ball B. One way to see this is that +5 + +FINITE TOPOLOGY +for d-manifolds with boundary ω(B) = χ(B) − χ(δB) which in the case of a d-ball is by the +Euler gem formula 1−(1+(−1)d−1) = (−1)d. To compute the Wu characteristic of a space, +cover it by balls (which can have different dimensions but should match the dimension of the +covered part. Within part of a ball not covered by different balls, the dimension of the un- +derlying space should be the same than the ball. What makes Wu characteristic compatible +with topology is that we have the valuation formula ω(U ∪V ) = ω(U)+ω(V )−ω(U ∩V ). +This allows us to glue different parts together. Note that this valuation formula does not +work for closed sets which together with a formula � +x,y ω(x)ω(y)ω(U(x)∩U(y)) is the major +reason why Wu characteristic is a topological invariant. +Summary: A finite abstract simplicial complex G is a finite set of sets x such +that G is closed under the operation of taking finite subsets of G. The cardinality +|x| of x defines its dimension dim(x) = |x| − 1. The star U(x) = {y, x ⊂ y} of +x ∈ G is the set of simplices containing x. It is declared to be open. The set B +of stars U(x) together with ∅ is a topological basis for a finite topology on G. A +complex H is called a d-ball, if it is of the form G \ U(x), where G is a d-sphere. +A complex G is contractible if there exists x ∈ G such that both the unit sphere +S(x) = δU(x) = U(x) \ U(x) and G \ U(x) are contractible. An arbitrary set is called +contractible if its closure is contractible. A complex is called a d-sphere if it is a +d-manifold and G \U(x) is contractible for some x; a complex is a d-manifold if every +S(x) is a (d − 1)-sphere. The complex G1 of all vertex sets of complete sub-graphs of +the graph G1 = (V1, E1) with V1 = G and E1 = {(x, y), x ⊂ y, or y ⊂ x} is called +the Barycentric refinement of G. Define Gn = (Gn−1)1. A simplex x ∈ G is locally +maximal if x ⊂ y implies y = x. +A complex H is declared to be a continuous +image of G if there exists a continuous surjective map f : Gn → H for some n such +that (i) if U(x) ⊂ H for any locally maximal k-simplex x ∈ H has a pre-image whose +closure is a k-ball and (ii) that every f −1S(x) ⊂ Gn is homeomorphic to the unit sphere +S(x) ⊂ H. Two complexes are homeomorphic if each is a continuous image of the +other. These definitions are inductive either with respect to number of elements or +dimension: the empty complex = void 0 = {} is the (−1) sphere. The complex +1 = {{1}} is contractible and the 0-ball. +2. Topology +2.1. +An abstract finite simplicial complex is a finite set G of non-empty sets that is +closed under the operation of taking finite non-empty subsets. +A finite simple graph +G = (V, E) carries the Whitney complex G of G which is the set of vertex sets of complete +subgraphs of G. While not all simplicial complexes come from graphs in such a way, 2 every +simplicial complex G defines a finite simple graph G in which the sets x of the complex +are the vertices and where two sets are connected by an edge if one is contained in the other. +If G came from a graph G as a Whitney complex, the graph G1 obtained from G is the +Barycentric refinement of G. Similarly, the Whitney complex G1 of G1 is a simplicial +complex, called the Barycentric refinement of the simplicial complex G. Since we can +switch between graphs and complexes while doing Barycentric refinements, the two concepts +“graphs” and “complexes” can be interchanged. We like to keep both graphs and complexes +2Examples are the (k − 1)-dimensional skeleton complex of the complete graph Kk which is a (k − 1)- +sphere, also known as the boundary sphere of the simplex. +6 + +OLIVER KNILL +and see them equipped with finite topological spaces. Simplicial complexes are attractive +mathematical objects because they have the simplest axiom system imaginable in geom- +etry: there is only one single axiom. Graphs on the other hand are unmatched in providing +geometric intuition, featuring accessibility, and being supported by computer algebra +systems, much more than sets of sets. 3 +2.2. +As part of the definition, Barycentric refined objects are all homeomorphic so that +from a topological point of view, they are identified. The goal is to use standard notions +of topology. We can not use the classical notion of homeomorphism for finite topological +spaces because this forces the finite topologies to be identical. We see all cyclic graphs to +be homeomorphic for example or to see the icosahedron isomorphic to the octahedron as +both are 2-dimensional complexes which approximate under Barycentric refinements more +and more spheres. We will call a complex H a continuous image of G if there exists a +continuous surjective map f : Gn → H such that for every x, the boundary S(x) of U(x) +is homeomorphic to the boundary of f −1(U(x)). This allows to use induction with respect +to dimension. We also require that for locally maximal simplices x which have the property +that the closure of the open set f −1(U(x)) is a ball, a simplicial complex which is obtained +from a sphere by removing an open set U(z). If G is a continuous image of H and G is a +continuous image of H, the two spaces are considered homeomorphic. +2.3. +A finite abstract simplicial complex G is so always equipped with a finite topology O +on G. This is understood in the classical sense: a topology contains the empty set and G, +it is closed under finite intersections and closed under arbitrary unions. In the finite case, +we of courses can avoid the “finite” word but all we do here can be generalized to infinite +but locally finite simplicial complexes. We stick with the finite, because the text should be +seen as part of a larger and more ambitious project investigating the question: which parts +of geometry can be replaced with finite combinatorial notions? We hope to be able to define +within finite sets of sets whether two simplicial complexes are homeomorphic or not and point +out that this is sharp than the softer equivalence relation given by homeomorphic geometric +realizations. The classical “homeomorphic notion” is too rigid for finite topological spaces +as it forces a bijection between the atoms U(x) making up the basis. Applied to graphs it +would require the graphs to be isomorphic. We want cyclic graphs Cn with n ≥ 4 to be all +homeomorphic for example. We want an edge refinement of a graph to be homeomorphic +deformations and capture the notion of homeomorphism which is used in graph theory when +graphs are considered one-dimensional simplicial complexes. +2.4. +If G is a finite abstract simplicial complex and x ∈ G is given, it defines the star +U(x) = {w ∈ G, v ⊂ w}. The collection B of all these stars together with the empty set ∅ +is a set of sets B that covers G. The collection B is also closed under intersections because +U(x) ∩ U(y) = U(x ∩ y) if x ∩ y is not empty and U(x) ∩ U(y) = ∅ else. It therefore defines +a base for a topology O on G. By definition in point set topology, a topology O is a +set of subsets of G which (i) contains ∅, (ii) contains G and which is (iii) closed under finite +intersections and (iv) closed under arbitrary unions. A sub-base of the topology is the set +3Part of graph theory is accessible in secondary school education. +Simplicial complexes on the other +hand tend to appear first in college topology or algebraic topology courses. The abstract version is more +accessible because higher dimensional Euclidean spaces, usually only introduced in linear algebra courses, +are not invoked. +7 + +FINITE TOPOLOGY +of sets U({v}), with v ∈ V = � +x x. It is a sub-base because every base element U(x) is an +intersection of such sets U(x) = � +v∈x U({v}) and so generates the base from intersections. +2.5. +A suspension of graph G is the Zykov join of G with the zero sphere S0 (the +graph with two vertices and no edges). +Doing this twice is a double suspension. +A +rational homology 3-sphere is a 3-manifold that has the same cohomology than a 3- +sphere. 4 The double suspension G of a rational homology 3-sphere is a concrete example of +a simplicial complex that is not a discrete 5-sphere because not all unit spheres are spheres. +But G has a geometric realization that is a 5-sphere by the double suspension theorem +of Edwards and Cannon. +The finite topology can distinguish complexes in the discrete, +which are indistinguishable using the tool of topological realizations. +5 Edward [9] works +with the Mazur homology 3-sphere and shows that the double suspension is S5. +A +consequence of the double suspension theorem is the existence of “exotic triangulations”: +there are topological manifolds which are not equivalent to a piecewise linearly homogeneous +polyhedron P meaning that for any x, y ∈ P, there exists a piecewise linear homeomorphisms +h : P → P such that h(x) = y. A triangulated topological manifold M on the other hand +only requires h(x) = y which is a local homeomorphisms. Every smooth manifold has a +PL structure. But a general topological manifold does not need to be homeomorphic to +polyhedra: Casson gave 4-manifold counterexamples. Examples in higher dimensions have +appeared more recently [42]. +2.6. +We start by defining some subsets in a finite abstract simplicial complex G. Every +x ∈ G defines the star U(x) := {y ∈ G, x ⊂ y} which is an open set and the core K(x) = +W −(x) := {y ∈ G, y ⊂ x} which, unlike U(x) in general, is always a sub-simplicial complex +of G and so a closed set. The closure B(x) of U(x) of U(x) contains {x} = W −(x) and is +called the unit ball of a point x ∈ G. Its boundary S(x) = δB(x) = B(x) \ U(x) is a closed +set called the unit sphere of x. By definition, this is a closed set. In general, the boundary +of any open set δU = U \ U = U ∩ U c is closed because it is the intersection of two closed +sets. 6 +2.7. +The unit sphere S(x) = δU(x) is in the language of simplicial complexes, also known as +the link of x, (but it is usually only defined for 0-dimensional x so that we avoid the term). +The open set U(x) = W +(x) = {y, x ⊂ y} and closed set K(x) = W −(x) = {x} = {y, y ⊂ x} +are somehow dual to each other. The closed set W −(x) is contained within S(x). In the case +when x is locally maximal meaning that it is not contained in a strictly larger simplex, +then S(x) is the boundary complex of the simplex x and so a sphere. A general complex G +is a sphere: there exists y ∈ G such that G \ U(y) is contractible and also for every y ∈ S(x), +the unit sphere S(y) within S(x) is a co-dimension-one sphere again. +2.8. +The terminology for graphs is similar. In a finite simple graph G and a vertex v, we +call the graph generated by the vertices w adjacent to v the unit sphere of v. If G comes +from a simplicial complex G, then each vertex has a dimension and adjacent vertices are +4There are implementations of the homology 3-sphere with 16 maximal simplices, leading to a complex +with 392 simplices. The corresponding graph has 2552 edges. The double suspension has 394 vertices. +5PL geometry in the continuum would capture the finite topology too but it also would use the continuum +and as Euclidean spaces are used the concept of infinity. +6The graph that can be constructed from the complex S(x) is isomorphic to the subgraph of all points in +distance 1 to the vertex x in the Barycentric graph G1, the graph which is constructed from the complex G. +8 + +OLIVER KNILL +ordered. The sphere S(x) is now the join of S+(v) generated by the vertices w for which w +contains v in G and S−(v) is generated by all w which are subsets of v. The sphere S(v) of +the graph is then the Zykov join [64] of S−(x) and S+(x) because the vertex sets are the +disjoint union and every element in S−(x) is connected to every element in S+(x). We like to +think also of S±(x) as the stable and unstable manifolds in the “hyperbolic structure” +defined by the Morse function f(x) = dim(x) of Morse index dim(S−(x)) + 1. Let us +define this more generally. The join can also be defined directly for simplicial complexes by +G + H = G ∪ H ∪ {x + y, x ∈ G, y ∈ H}. +2.9. +Let R be an ordered ring like Z, Q, Z. A function f : G → R is called locally injective +if f(x) ̸= f(y) for every y ∈ S(x). A Morse function on a complex G is defined as a locally +injective function function G which has the property that S− +f (x) = {y ∈ S(x), f(y) < f(x)} +is a (k − 1)-sphere. Its Morse index is k. If we do not want to refer to the graph and so to +the Barycentric refined topology, we would require that S− +f (x) is a simplicial complex which +is a sphere. The function f(x) = dim(x) is a special Morse function. If G is a d-manifold, +then the graph S(x) in the graph G1 is a (d − 1)-sphere and agrees with the topological join +of the two spheres S±(x). The geometric realization of a Zykov join of two graphs agrees +with the topological join of the geometric realizations +Summary: A finite abstract simplicial complex G carries a finite topology O in which +the stars B form a basis. A finite simple graph carries so a natural topology on its +Whitney complex. The topology is finite and so Alexandroff: every point x has a +smallest neighborhood U(x), the star of the simplex x. We like to think of them as +atoms of space. The closure B(x) of a star U(x) is is called the unit ball. Its boundary +S(x) is called unit sphere of x. +3. Continuity +3.1. +The classical definition of continuity can be applied immediately to functions f between +simplicial complexes f : G → H if we just silently assume the topology generated by stars +as the natural topology on the complex. If we talk about continuous maps f : G → H of +graphs G, H, then rather looking at maps on the vertex sets V (G), V (H), we look at maps +from its simplicial Whitney complex G of G to the simplicial Whitney complex H of H. As +usual in point-set topology, a map f is called continuous if the inverse f −1(A) of an open +set A in H is an open set in G. +3.2. +A map is continuous if and only if the inverse image of closed sets is closed. The +standard definition of continuity works well, but it was necessary to modify the notion of +“homeomorphismI” in the finite as classically, homeomorphic finite topological spaces are +identical. It would be unacceptable for example to consider an icosahedron and octahedron as +being topologically different, or to consider a cyclic graph with 6 elements to be topologically +different than a cyclic graph with 5 elements. Even Barycentric refined complexes would +not be homeomorphic with the narrow definition from point set topology, requiring the two +continuous maps which are inverses of each other. +3.3. +In the context of simplicial complexes, a continuous map is a bit more general than +a simplicial map. The later is a map from G to H that preserves order and does not +increase dimension if x ⊂ y then f(x) ⊂ f(y) and therefore satisfies dim(f(x)) ≤ dim(x). A +simplicial map must map 0-dimensional simplices to 0-dimensional simplices. A continuous +9 + +FINITE TOPOLOGY +map f : G → H does not need to do that. A constant map which has as an image a positive +dimensional simplex is continuous but not a simplicial map. It does not necessarily map +simplicial complexes into simplicial complexes. Most permutations f : GG of a simplicial +complex are not continuous. They scramble around the simplices without preserving the +order. But simplicial maps are always continuous: if f is a simplicial map, then the inverse +image of any simplicial complex {y}) consists of unions of simplicial complexes {xk}, which +is closed. Because the inverse image of any closed set is a closed set, the map is continuous. +3.4. +Any map from a 0-dimensional complex G to a complex H is always continuous because +every set in G is both open and closed. Such a map neither does have to be injective, nor does +it have to be surjective. An other extreme case is a constant map f(x) = c from a complex +G to a complex H. It is always continuous and a simplicial map if c is zero dimensional: the +set f −1(A) is either empty (if c /∈ A) or then the entire space G. The image {c} is however +not open. So, even for finite topologies, a continuous map does not need to be an open +map, a map that transports open sets into open sets. +3.5. +Sometimes it is good to look at maps defined within graphs G = (V, E), H = (W, F) +alone and not directly look at the simplicial complex. A map f : V (G) → V (H) is contin- +uous graph map f : G → H if e = (a, b) in G then either (f(a), f(b)) in E(H) or then +that f collapses e collapses to vertex a = (a, a) in V (H). Such a map f lifts to a continuous +map on the corresponding Whitney simplicial complexes G → H. It actually even lifts to +a simplicial map because zero dimensional parts get mapped into zero dimensional parts. +Every continuous map between graphs as just defined leads so to a continuous map on the +Whitney complex: given x ∈ G(G), define f(x) = � +v∈x f(v). This is an element in the +complex H of H. If it was not, then there would exist a, b ∈ x such that (f(a), f(b)) /∈ E +nor f(a) = f(b), contradicting the assumption that f is a continuous graph map. Again, +we should tell that there are continuous maps between simplicial complexes G(G), H(H) +of graphs G, H which do not come from continuous graphs maps, the constant map to a +positive dimensional simplex is an example of a continuous map that does not come from a +graph map because a continuous graph map necessarily maps zero dimensional parts to zero +dimensional parts. +3.6. +For a continuous map between graphs f : G → H, the maximal dimension of the +image graph f(G) ⊂ H is always smaller or equal than the maximal dimension of G. More +generally, any simplicial map from a simplicial complex to an other simplicial complex does +not increase dimension on the image: if f : G → H is a continuous map between simplicial +complexes then dim(f(x)) ≤ dim(x). The dimension of the image of a continuous map can +be strictly smaller of course: the constant map mapping every simplex in G to a single fixed +vertex v (zero-dimensional simplex) in H is continuous and a simplicial map. The constant +map to a positive dimensional simplex however is continuous but not a simplicial map. +3.7. +A graph homomorphism 7 is a map between graphs G = (V, E) and H = (W, F) +such that it maps V into W and E into F. A graph homomorphism therefore maps simplices +into simplices on the simplicial complex level and so defines a simplicial map and therefore a +continuous map on its Whitney simplicial complexes: the inverse image of an open set U(x) +in G(H) is open in G(G) because it is a disjoint union of sets U(yj) where yj are the set of +7To have “homomorphism” and “homeomorphism” so close in the landscape of words is unfortunate but +it is very much entrenched. The two terms rarely appear in the same context, but here they do. +10 + +OLIVER KNILL +simplices which are mapped into x. On the other hand, not every continuous map is a graph +homomorphism because a graph homomorphisms by definition is not allowed to collapse an +edge to a vertex. +Summary: Continuity of maps between simplicial complexes is defined as usual in +topology: the inverse image of an open set is open. For simplicial complexes, simplicial +maps are continuous but the converse is not necessarily true. There are continuous +maps between simplicial complexes which are not simplicial maps. Graph homomor- +phisms define simplicial maps on their complexes and so are continuous too, but also +here, the converse is not always true. In general, simplicial maps or graph maps only +can lower the maximal dimension: the dimension of G is larger or equal than the di- +mension of f(G) ⊂ H. But continuous maps between simplicial complexes do not need +to lower the dimension as the constant map f(x) = c from a zero dimensional complex +G to a positive dimensional complex H with a fixed positive dimensional c ∈ H shows. +4. Homeomorphism +Figure 3. +Two homeomorphic finite abstract simplicial complexes G, H are +displayed, with vertices, edges and triangles filled out. Both are a wedge sum +of a 2-sphere, a 3-sphere and a 2-ball. In order to show that the two spaces are +homeomorphic, one can first show that all d-spheres are homeomorphic and all +d-balls are homeomorphic and that if two pointed spaces are homeomorphic +then their wedge sums are homeomorphic. The wedge sum of two path graphs +can be a star graph or a path graph and they are not homeomorphic. One- +dimensional complexes are homeomorphic if and only if they are classically +homeomorphic, that is if one can get from one to the other by a sequence of +edge refinements or edge collapses that come from edge refinements. +4.1. +As pointed out in the introduction, the classical notion of homeomorphism is too +rigid for finite topologies. Already pioneers like Poincar´e used in a combinatorial set-up +at equivalence classes of finite geometries and considered Barycentric refinement complexes +11 + +FINITE TOPOLOGY +equivalent. +8 +So, to start with, we assume that two geometries which are Barycentric +refinement of the other are homeomorphic. The Barycentric refinement G1 = (V1, E1) +of a graph G = G0 is a new finite simple graph defined as follows: the vertex set of G1 is +V1 = G. The Barycentric refinement of a simplicial complex G is the Whitney complex of +the graph defined by G. The edge set is the set of pairs (x, y) for which either x ⊂ y or +y ⊂ x. We can iterate the Barycentric construction and look at the Barycentric refinements +Gn = (Gn−1)1 for every n ≥ 0. We have just seen that a continuous map G → H can be +lifted to a continuous map G1 → H1. This means now that every continuous map G → H +between two graphs be lifted to a continuous map on the Barycentric refinements Gn → Hn. +4.2. +We repeat the new definition: if there is a continuous map f : Gn → H such that for +every y ∈ H, the boundary of f −1(U(y)) is homeomorphic to the boundary S(y) of U(y) and +for all locally maximal simplices y, the ball B(y) = U(y) in H has a pre-image f −1(B(y)) +which is a ball, we say H is a continuous image of G. We say G and H are homeomorphic +if H is a continuous image of G and G is a continuous image of H. A d-ball is a complex that +is of the form S − U(z), where S is a d-sphere and z ∈ S. Remember that a d-sphere S is a +d-manifold which when punctured (S \U(z)) becomes contractible and that a d-manifold is a +complex for which all unit-sphere is a (d−1)-sphere. The empty complex is the (−1)-sphere +and the 1 point complex contractible. 9 +4.3. +An immediate consequence of the definition is that 0-dimensional complexes are home- +omorphic if and only if they have the same number of elements. The reason is that we require +the inverse image of every B(x) = {x} to be a 0-ball which is a one-point complex K1. This +implies that f must be injective. The map f therefore has to be a bijection. This defines +now both an equivalence relation between simplicial complexes as well as for graphs: it is +reflexive and symmetric. To see transitivity of the relation, note that we have now a chain +of maps Gn → Hm → Kl showing that K is a continuous image of G. Since the reverse holds +also, the complexes K and G are homeomorphic, if G and H are homeomorphic and H and +K are homeomorphic. +4.4. +The notion of homeomorphism goes over to graphs G if we look at the Whitney sim- +plicial complex G attached to the graph G. The topology of G is then taken. By definition +then, the Barycentric refinements Gn of a graph G are all homeomorphic to each other. We +could also restate this by noting that G and H are homeomorphic if and only if Gn and Hn +are homeomorphic for some n. As an example: two cyclic graphs G = Cn and H = Cm +with n, m ≥ 4 are homeomorphic. If G is a triangulation of a compact manifold M and +H is a triangulation of a compact manifold N such that G, H have a common Barycentric +refinement or common edge refinement, then G, H are homeomorphic and M, N are com- +binatorially equivalent. Any combinatorial invariants, a term coined in [4] (meaning a +property that does not change under Barycentric refinements) must also be a topological +invariant (meaning a property that is the same for homeomorphic objects). +8As pointed out earlier, PL-geometry [54] would do the job in the continuum. But we do not want to use +Euclidean spaces, nor use of infinity. +9This is a recursive definition as it refer to homeomorphism smaller dimensions. In zero dimensions, +homeomorphic means equal cardinality. +12 + +OLIVER KNILL +4.5. +It is useful to reformulate the notion of homeomorphism as the property that there +exists maps f : Gn → Hm and g : Hm → G which are both continuous and such that the +homeomorphism works also locally in that the smallest spheres have pre-images which are +homeomorphic and that the inverse of locally maximal unit balls are actual balls. We still +wonder whether the assumption on the locally maximal balls can be avoided. The definition +includes it so that we can prove things, like that if two complexes are homeomorphic and +one of them is a manifold, then the other must also be a manifold. +4.6. +Let us dwell on this a bit more: if a map f : G → H is continuous, we could try to ask +that all unit spheres S(x) = δU(x) are homeomorphic to δf −1U(x) and use this alone as a +recursive definition for homeomorphism. We would then start with the assumption that zero- +dimensional complexes are homeomorphic if they have the same cardinality, the property +δf −1U(x) homeomorphic δU(x) would then recursively lift the notion of homeomorphisms +dimension by dimension. It is still not clear whether this alternative definition alone would +work. +4.7. +While the definition of homeomorphism is by design symmetric, we could also ex- +plore and try assuming only direction: checking this for f : Gn → H in one direction only +might already determine that G and H are homeomorphic. This works if G and H are one- +dimensional. In that case, we need a map from Gn to H such that the vertex degrees of +S(x) = δU(x) is the same than the vertex degrees of δf −1U(x) for every x. We come back +to this question again at the end. +Summary: The complex H is declared to be a continuous image of G, if the natural +surjective map from some Barycentric refinement Gn → G factors as Gn → Hm → G +and the map induces homeomorphisms on unit spheres and maximal unit balls have +pre-images that are actual balls, punctured spheres. If two complexes are continuous +images of each other, we call them homeomorphic. This notion defines an equivalence +relation on complexes as well as an equivalence relation on graphs. +5. Closed +5.1. +Closed sets of a graph G are subsets of G which themselves form simplicial complexes. +Not all closed sets in the Whitney complex of a graph do have to be induced from a sub- +graph of G. The boundary S(x) of a maximal simplex for example is always closed but it is +only a skeleton complex of a Whitney complex of a graph. 10 To see the correspondence +between open and closed sets: note that any simplicial complex K in G has as a complement +the union of all stars U(x) with V (x)∩V (K) = ∅ and this union is an open set. For a graph +G, a closed set K contains all the simplicial complexes of subgraphs of G. +5.2. +In the case G = K3, where we have G = {{1, 2, 3}, {1, 2}, {2, 3}, {1, 3}, {1}, {2}, {3}}. The topology +O = {U1, · · · U19} has 19 elements: +U1 = ∅, U2 = {(1, 2, 3)}, U3 = {(1, 2), (1, 2, 3)}, U4 = {(1, 3), (1, 2, 3)}, U5 = {(2, 3), (1, 2, 3)}, +U = {(1, 2), (1, 3), (1, 2, 3)}, U7 = {(1, 2), (2, 3), (1, 2, 3)}, U8 = {(1, 3), (2, 3), (1, 2, 3)}, U9 = +{(1), (1, 2), (1, 3), (1, 2, 3)}, +U10 = {(2), (1, 2), (2, 3), (1, 2, 3)}, +U11 = {(3), (1, 3), (2, 3), (1, 2, 3)}, +U12 += +{(1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U13 += +{(1), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U14 += +{(2), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U15 += +{(3), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U16 += +{(1), (2), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U17 += +{(1), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, +U18 += +{(2), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3)}, G = U19 = {(1), (2), (3), (1, 2), (1, 3), (2, 3), (1, 2, 3)}. +10Taking subgraphs as closed sets is motivated by the Zariski topology, where closed sets are algebraic +subsets H of an algebraic variety G. +13 + +FINITE TOPOLOGY +The number of closed sets {U c, U ∈ O} is of course the same than the number of open sets. +The sub-graphs: there is the empty graph of dimension −1, 7 graphs of dimension 0, 3 graphs +of dimension 1 with one edge, 3 graphs withe one edge and one vertex, 3 graphs of dimension +1 with 2 edges and then the complete graph. This gives 1+7+3+3+3+1 = 18 sub-graphs. +The 1-dimensional skeleton complex C3 of K3 is a closed set in the topology which does no +correspond to a sub-graph. Its complement is the open set U((1, 2, 3)) = {(1, 2, 3)}. +5.3. +Let A be an arbitrary set of sets in G. The closure of A is the smallest closed set +(simplicial complex) which contains A. +11 For example, if A = {x} consists of a single +simplex, then its closure A is the simplicial complex {y ⊂ x} generated by A. If G is the +Whitney complex of a graph, the closure of a set of simplices is often (but not always) the +subgraph of the Whitney complex of the smallest sub-graph of G which contains all simplices +x. The example of the closed set C3 = {(1), (2), (3), (1, 2), (2, 3), (3, 1)} in K3 which is the +skeleton complex of the Whitney complex of K3 shows that not all closed subsets in the +topology of G correspond to sub-graphs of G. 12 +5.4. +If X is a topological space and Y a subset, then Y is called locally closed if it is an +intersection of an open set A and closed subset K. We think then of the induced topology of +X on K by taking all intersections U ∩K as open sets in K. By definition, every U ∩K is an +intersection of a closed and an open set. Locally closed sets are not necessarily closed in the +topology X but part of the Borel σ-algebra of the topology. Locally closed sets are sets which +are open in some closed subset K with induced topology. In our context, where a closed set +is a simplicial complex, a locally closed set is a set which is an open set in that simplicial +complex. It does not need to be open in the original topology O. Take for example the set +which consists of a single simplex A = {x} which is not maximal, nor zero-dimensional. An +example is if x is a boundary edge in a Wheel graph G. This set is neither open nor closed. +But the set is locally closed because we can write it as an open set in the boundary sub +complex K which is a circular graph complex K. The set A is an open set in K but not +open in G. The complement of A in K is closed in K as well as closed in G. Similarly, look +at the closure of A which is the complete complex with 2 elements. Its complement in K is +open in K but not open in G. +5.5. +Let us look at some examples: in a finite abstract simplicial complex, every single +simplex {x} is locally closed because it is the intersection of the open set U(x) and the +closed set W −(x) = {x}}. In the simplicial complex of K3, the set {{1}, {1, 2}, {1, 2, 3}} is +not locally closed. From the 128 possible subsets of G, there are 64 which are not locally +closed and 64 which are. In a one-dimensional simplicial complex, all subsets are locally +closed. In K4, where we have 215 = 32768 possible set of subsets in G, where only 167 sets +are open and 167 are closed, there are 3605 locally closed sets. +11This corresponds to a classical notion in the theory of simplicial complexes when they are considered +as geometric realizations in Euclidean space. The closure contains all boundary simplices, meaning to look +at all subsets of x and so also in the continuum it means to look at the smallest simplicial complex which +contains K. +12One could define an other topology, where the closed sets are simplicial complexes of sub-graphs +but this is a rougher topology and closer to the Zariski topology. +14 + +OLIVER KNILL +Summary: The finite topology on a complex is of Zariski type: sub simplicial com- +plexes are the closed sets. Sub-graphs of a graph define closed sets of a graph but not +all closed sets come from subgraphs. On complexes coming from graphs, we could get +a slightly rougher topology by declaring sets to be closed if their simplicial complexes +are Whitney complexes coming from closed sub-graphs. Finally, we have looked at +locally closed sets, sets which are intersections of open and closed sets. +6. Compact +6.1. +Traditionally, a topological space is declared to be compact if every open cover has +a finite sub-cover. Compactness in this strict sense is not a very useful in finite topological +spaces: using the definition, every set is compact (whether it is open, closed and even +if it is neither). It makes therefore much sense in a finite topological space to to identify +compact sets with closed sets. If topologies are considered for non-finite graphs, a set then +would be compact in a complex or graph, if it is a closed and finite set defined by a finite +abstract simplicial complex. Again, it would make sense to consider being compact as a +synonym for being finite and closed. In an infinite graph G, many (but not all) compact +sets are represented by finite subgraphs of G. Their Whitney complexes are then simplicial +complexes and also closed. On a general graph (V, E) with no restrictions on V, E, one could +also look at the slightly rougher topology in which finite subgraphs define closed sets. There +are less closed sets then and therefore also less open sets. In zero dimensions, we would get +the co-finite topology, which in the finite case agrees with the discrete topology and +where compactness is equivalent to being finite. +6.2. +We have just seen that in general, like if one looks at topological spaces which are not +so commonly used - finite topological spaces are examples - one has to be a bit more careful +when using notions which involve compactness. The property that every open cover has a +finite subcover is really not a good notion for compactness in the case of finite topological +spaces. For example, a map is called proper if the inverse image of a compact set is compact. +In the case of finite topologies with the standard definition of compactness, every map (even +a not continuous one) would be proper, simply because the inverse image of any finite set +is a finite set, which by the definition of having a finite cover, would be declared to be +compact. So, the notion of “proper” does not really say much. If one requires a compact set +to be closed too, then on finite topologies, the notion of proper is the same than continuous +because there, every closed set is compact and the fact that all inverse images of closed sets +is closed is equivalent to continuity. +6.3. +Looking art finite topological spaces could be named “radically elementary topology” +similarly as radically elementary probability theory covers a lot of traditional probability +theory [49]. Finite structures are not that limiting, especially if one considers them in a +non-standard frame work. In internal set theory IST for example, [48, 50] compact sets +can be modeled by finite sets. Compact simplicial complexes X therefore can be modeled +by finite abstract simplicial complexes. Of course, the number of elements is non-standard +if the set X is infinite. If one looks at a continuous map from a compact topological space +to itself, then in general there are infinitely many fixed points. In a situation like in the +context of the Lefschetz fixed point theorem, one traditionally assumes that there are +finitely many fixed points. In the non-standard frame work, one would just assume that the +15 + +FINITE TOPOLOGY +number of fixed points is standard (an axiomatically defined term). This will then assure +that also the sum of the Lefschetz numbers is standard. +Summary: In a finite topological space, every set is compact so that the classical +notion is not very useful. Every map would be proper for example. It makes sense +therefore to consider all closed finite sets as compact instead. If one would consider +infinite complexes, every finite sub-simplicial complex would be considered compact. +Especially, every finite subgraph graph defines a closed compact set. In non-standard +analysis frame-works, ”compact topological spaces” can be treated like ”finite topolog- +ical spaces”. +7. Connectivity +7.1. +A graph G is called path connected if for every two vertices a, b, there is a path +e1 = (a, v1), . . . , en = (vn−1, b) (a finite collection of edges) connecting a = v0 with b = vn. +A simplicial complex G is path connected if the graph G1 defined by the complex is path +connected. A graph is path connected if and only its topology O(G) is connected: here is +the proof: if G = U ∪ V is the disjoint union of two open sets U, V , then there can not be +simplex which contains x ∈ U and y ∈ V . This means that there is no path connecting +x with y in the graph G1 and G is not path connected. On the other hand, if K1, K2 are +not path connected components in G1, then their smallest open neighborhoods U1, U2 are +disjoint and G is not connected. +7.2. +In graph theory, connectivity is also called 0-connected. A graph is called 1-connected, +if it is connected but there exists a vertex v which when removed, renders G \ v discon- +nected. The graph K2 is 1-connected but its Barycentric refinement, the path graph P3 is +not 1-connected. A simplicial complex can be defined to be 1-connected, if is graph G1 +is 1-connected. If f : G → H is a continuous surjective map between graphs and G is +1-connected, then also H is 1-connected: to prove this, note first that H must be connected +because the continuous image of a connected topological space is always connected. If H +was not 1-connected, all unit spheres S(x) in H would be connected. So, also f −1(S(x)) is +connected. This does not generalize to higher connectivity in graph theory: it is possible +that the continuous image of a 2-connected graph can be only 1-connected. As an exam- +ple, take the kite graph G in which one edge has been removed from K4. This graph is +2-connected, but a continuous map from G to K2 has an image that is only 1-connected but +not 2-connected. +Summary: The topology of a complex is connected if and only if the complex is +classically connected in the sense that the graph of the complex is path connected. +The topology of a graph on its simplicial complex is connected if and only if it is path +connected. A complex is connected if and only if its graph is connected. A continuous +image of a connected complex is a connected complex. If a graph is 1 connected, then +a continuous image is still 1 connected. +8. Separation +8.1. +The topology is called a Kolmogorov space or T0, if for any pair of distinct points +x, y, there is at least one point who has a neighborhood not containing the other. +The +topology O of a simplicial complex G is always a Kolmogorov space: this is clear in the case +16 + +OLIVER KNILL +x ⊂ y or y ⊂ x. In the case x ⊂ y, then U(y) does not contain x. If x ∩ y is not empty, then +U(x) does not contain y and U(y) does not contain x. +8.2. +However, the topology O is not Fr´echet = T1 if the dimension is positive. Fr´echet +means means that for every two points x, y, there exists a neighborhood of x which does +not contain y and a neighborhood of y that does not contain x. As an example, take x ̸= y +with x ⊂ y. Now, only y can be separated from x but x can not be separated from y. Every +neighborhood of x contains y. If the complex has positive dimension, the it is never Fr´echet. +Of course, all zero-dimensional complexes are Fr´echet because the topology is the discrete +topology. +8.3. +The topology of a simplicial complex G is also not Hausdorff or T2 if the complex +has positive dimension. In particular, the topology of a graph G is not Hausdorff if G has +positive dimension. Two vertices x, y in a graph which are connected by an edge can not be +separated by open sets in the complex: as x must contain U(x) and y must contain U(y) +and these two open sets have an intersection U(x) ∩ U(y) which contains the edge {e}. +8.4. +The topology is also not normal = T4: two closed sets can in general not be separated +by open sets. Examples are the closures x, y of two simplices x, y ∈ G that have a non- +empty intersection. They are closed sets but every neighborhood of one intersects with any +neighborhood of the other. One could think that the non-Hausdorff property is a handicap. +However, it can be a blessing in the context of connection calculus, when we consider +higher order characteristics generalizing the Euler characteristic χ(G) = � +x∈G ω(x) with +ω(x) = (−1)dim(x). +While Euler characteristic satisfies for every subset A, B ⊂ G the +valuation property χ(A ∪ B) = χ(A) + χ(B) − χ(A ∩ B), this property does no more hold +in the case of Wu characteristic ω(G) = � +x,y,x∩y∈G ω(x)ω(y). But the valuation property +will hold for open sets. +8.5. +A topology is called Alexandroff if every point x has a smallest non-empty open +neighborhood U(x). In an Alexandroff topology, space has smallest atoms. Equivalently, +an Alexandroff topological space has the property that arbitrary intersections of open sets +are open. Most topological spaces we are familiar with are not Alexandroff. If a metric space +is Alexandroff, it must be a discrete topological space as then, every single point needs to +be open. In general, any discrete topology and the indiscrete topology O = {∅, X} is +Alexandroff. Any finite topology must be Alexandroff because the intersection of all open +sets containing x is an open set U(x), containing the point. It is the star. Calling a finite +topology a “finite Alexandroff topology” would be a pleonasm. Still, it is good to use the +name as Alexandroff was one of the firs who seriously considered finite topological spaces. +8.6. +To compare, note that the geometric realization |G| (in some Euclidean space) of a +finite abstract simplicial complex G (or finite simple graph with the Whitney complex) is +always a Hausdorff, because |G| is a closed subset of a Hausdorff topological space. We +will see that looking at the topological realization of a complex loses some information like +the topological nature of spheres in the space. Topological manifolds can in general not be +described by one simplicial complex (or equivalence class of Barycentric refinements) alone. +This is not a surprise. Most topological spaces we look at, even compact ones like Cantor +type sets can not be described by one single finite standard topological space. 13 +13Nonstandard analysis teaches us however that we can describe it by a non-standard finite topological +space. Standard finite topological spaces are then the spaces we look at here. The axiom system however +17 + +FINITE TOPOLOGY +Summary: The topology of a complex or graph is Kolmogorov (T0) not Fr´echet (not +T1), not Hausdorff (not T2) and not normal (not T4), but (like all finite topologi- +cal spaces) is Alexandroff. The non-Hausdorff property is in sharp contrast with the +topology given by geometric realizations which are Hausdorff. +9. Dimension +9.1. +The maximal dimension of a simplicial complex is defined as maxx∈Gdim(x). The +maximal dimension of a graph G is the maximal dimension of its Whitney complex G. +We have already seen that a continuous map f : G to H between simplicial complexes has +the property that dim(f(x)) ≤ dim(x). The maximal dimension of a continuous image of a +complex G is therefore smaller or equal than the dimension of G. An open cover {Uj} of +G is a set of open sets such that � +j Uj = G. A cover defines a ˇCech nerve graph, in which +the sets Uj are the vertices and where two are connected if they simultaneously intersect in a +non-empty set. The maximal dimension of this graph is called the dimension of the cover. +The minimum over all dimensions of covers of G is called the topological dimension of G. +9.2. +The following fact is an other reason why the topology associated to a simplicial +complex or graph is the right one. The topological dimension and the maximal dimension +d are the same for every complex and every graph. The reason is that we can cover the +space with sets U({v}) with v ∈ V = � +x∈G x. This cover can not be refined further because +removing one would keep some vertex v uncovered. The dimension of this cover is equal +to the maximal dimension of G because if x is a simplex of dimension d, then all the open +sets {U(v)}v∈G intersect. Therefore, the topological dimension of G is at at most d. Every +cover of G must contain all sets U({v}) with v ∈ V because otherwise {v} ∈ G would not +be covered. So, the dimension is also at least d. +9.3. +The ˇCech graph of an open cover U of a graph or simplicial complex is defined as the +graph in which the open sets U are the vertices and where two vertices are connected if they +have a non-empty intersection. In general, the ˇCech graph of the cover {U({v}), v ∈ V } +is the graph G itself and the ˇCech cover of the cover {U(x), x ∈ G} is the Barycentric +refinement G1. One usually looks at covers for which every of the open sets are contractible +(every star U(x) is considered contractible because its closure B(x) is contractible.) +Summary: The topological dimension of a topology on a complex as defined by +Lebesgue, agrees with the maximal dimension of the complex. The ˇCech graph of the +base cover of a complex is the graph G1. If G is the Whitney complex of a graph G, +then the ˇCech graph of the cover {U(v)}v∈V (where V is the set of zero dimensional +simplices) is the graph G itself. +10. Product +10.1. +Every data structure, whether we deal with graphs, with simplicial complexes or with +topological spaces has notions of products. The Shannon product G∗H [56] of two graphs +G, H is the graph for which V (G∗H) = V (G)×V (H) is the Cartesian product of sets and +where E(G ∗ H) = {((a, b), (c, d)), (a, c) ∈ E(G) ∪ V (G) or (b, d) ∈ E(H) ∪ V (H)}, meaning +does not allow us to define the intersection of all standard open sets so that compact topological spaces do +not have atoms U(x). +18 + +OLIVER KNILL +that two points are connected if both projections have the property that they project onto +a vertex or edge. The Shannon product does not go over naturally to complexes. We can +of course look at the complex of the Shannon product. What is nice about the Shannon +product is that it allows to see graphs as a ring. We have explored this a bit in [28, 39, 34]. +10.2. +The box product topology 14 on G ∗ H is the finest topology on the simplicial +complex of G ∗ H such that both the projections on G and H are both continuous. The +graph topology of the product G ∗ H is in general much finer than the product topology. +One can see this already for G = H = K2, where G ∗ H = K4. The topologies of G and H +have only 5 elements, while the topology of G ∗ H has 167 elements. +10.3. +The graph topology on U ∗ V is in general finer than the topology generated by the +“cubes” U ∗ V where U, V are basis elements of the factors. Some simplices in G ∗ H are of +the form x∗y which is a (k+1)∗(l+1)−1-simplex if x was a k simplex and l was a l simplex. +But not every simplex in G ∗ H is of the form x ∗ y. For example, for G = H = K2, only the +0, 1 and 3-dimensional simplices in G ∗ Y are products. The 2-simplices (the triangles) in +G ∗ H are not products because 2 = (k + 1) ∗ (l + 1) − 1 implies either k + 1 = 3 or l + 1 = 3 +but there are no 2-simplices in neither G nor H. One can see this also from the fact that +there are 22 − 1 = 3 simplices in G and H and 24 − 1 = 15 simplices in G ∗ H. Only 9 of +them are of the form x ∗ y with x a simplex in G and y a simplex in H. +10.4. +The Shannon product does not preserve manifolds. The Stanley-Reisner product +of two simpicial complexes G and H is defined as the Whitney complex of the graph in which +the Cartesian product G × ×H are the vertices and where two different vertices (x, y), (u, v) +are connected by an edge if either x ⊂ u, y ⊂ v or u ⊂ x, v ⊂ y. The Stanley-Reisner +product of a p-manifold with a q-manifold is a (p + q)-manifold. As the graph defining +G × H is homotopic to the Shannon product, it inherits properties of the former like the +K¨unneth formula or the compatibility with the Euler characteristic. We will write more +about the compatibility with higher characteristics elsewhere and especially show that they +are topological invariants. +10.5. +One can now ask whether there is not a natural ring structure on simplicial complexes +which corresponds to the Shannon ring, or whether there is even a ring structure which +preserves manifolds. One problem is that if we take the Cartesian product of two simplicial +complexes, we don’t have a closed set. We would have to close it but it would not have the +properties we like. Much more elegant is to expand the class from simplicial complexes to +delta-sets. This structure is more general than simplicial sets, a popular construct which +has more structure than δ-sets. Every simplicial set of course is also a delta set by just +forgetting the degeneracy maps si maps and only keep the face maps di. The disadvantage +of working with δ sets is that we have to carry around not only sets of sets but also keep +track of the maps di. And the entire elegance of having a simple topology etc is gone. δ +sets also are useful when describing quivers, which generalize finite simple graphs. There is +a Whitney functor from quivers to δ sets generalizing the functor from graphs to simplicial +complexes. 15 +14For finitely many products, the box product topology agrees with the product topology. +15In order to talk about funtors one needs to adapt the morphisms on graphs and simplicial complexes +and continuous maps are the most natural common denominator as both simplicial maps as well as graph +homomorphisms are continuous maps. +19 + +FINITE TOPOLOGY +Summary: The projections from the Shannon product G∗H of two graphs to one of its +factors is a continuous map. The graph topology of the product graph is in general much +finer than the product topology in general. The Shannon product does not preserve +topological quantities like higher characteristic or manifold properties. The Stanley- +Reisner product however does. The Stanley-Reisner product is more compatible with +topology but does not provide a ring structure as associativity fails. The Shannon +product on the other hand defines a ring and so an arithmetic. +11. Join +11.1. +The join G ⊕ H of two graphs G, H has as vertex set V (G) ∪ V (H) and as the edge +set E(G) ∪ E(H) ∪ {(a, b), a ∈ V (G), b ∈ V (H)}. The join operation in graphs theory was +first defined by Zykov [64] and does exactly what the join does for geometric realizations of +the Whitney complex. The join with a 0-sphere is a suspension. In general, the join of two +spheres is a sphere. The proof follows from the sphere formula SG⊕H(x) = SG(x) ⊕ H and +SG⊕H(y) = G ⊕ SH(y). so that by induction if each G, H, SG(x), SH(y) are spheres, then the +unit sphere of any point in G ⊕ H is a sphere. The join of a graph with the 0-sphere S0 is +called the suspension. Since S0 ⊕ S0 = C4 is a cyclic graph and so a discrete sphere, the +join of a graph with a cyclic graph is a double suspension. +11.2. +The join of two simplicial complexes G, H can be defined also without referring +to their graphs. The simplices in G⊕H are the union of G, H and G⊕H where x⊕y = x∪y is +a k+l+1-dimensional simplex obtained by taking the disjoint union of the two sets x, y. If x +had k+1 elements and y had l+1 elements, then x⊕y has k+1+l+1 = k+1+2 = (k+l+1)+1 +elements so that k + l + 1 is the dimension of x ⊕ y. So, for example, if G = {a, b} is the +zero sphere and H = {1, 2, 3, 4, (12), (23), (34), (41)} is a discrete circle, then G ⊕ H =} = +{a, b, 1, 2, 3, 4, (12), (23), (34), (41), (a1), (a2), (a3), (a4), (a12), (a23), (a34), (a41) +(b1), (b2), (b3), (b4), (b12), (b23), (b34), (b41)} is the suspension of a circle and the octahe- +dron complex. +11.3. +The join operation is dual to the disjoint union G+H as addition: if G′ denotes the +graph complement of G in which edges and non-edges are switched, then (G⊕H)′ = G′+H′, +where G + H is the disjoint union of the graphs. If G1 is the Barycentric refinement of G, +then the unit sphere S(x) of a point x ∈ G = V (G1) is of the form S+(x) ⊕ S−(x), where +S+(x) = {y, x ⊂ y} is the unstable sphere and S −(x) = {y, y ⊂ x} is the stable sphere. +In a discrete manifold, where every unit sphere is a sphere, both the stable and unstable +spheres are spheres. +11.4. +If G, H are graphs and if x is a simplex in G and y is a simplex in H, then x ⊕ y is a +simplex in G⊕H. So, the Whitney complex of the Zykov join G(G⊕H) = G ⊕H is the join +of the simplicial complexes which is the G ∪ H ∪ {x ⊕ y, x ∈ G, y ∈ H}. The topology of the +join by definition has as a basis the sets UG⊕H(z), where z = x ⊕ y is a simplex in G ⊕ H. +This is UG⊕H(x ⊕ y) = UG⊕H(x) ⊕ H ∪ G ⊕ UG⊕H(y). The total set of stars U(x ⊕ y) is a +basis of G ⊕ H and generates the topology of G ⊕ H. +20 + +OLIVER KNILL +Summary: The embedding of G in G ⊕ H with the induced topology is a classical +homeomorphism onto the image, as it is a bijection onto the image. The join of the +topological base in G and H defines a base for the join G ⊕ H. The smallest atomic +open sets U(x⊕y) in G⊕H is a basis: the base of the join contains open sets U(x)⊕H +as well as open sets G ⊕ U(y). +12. Subgraph +12.1. +We have seen that a subcomplex H of a complex G is a closed set. The topology +induced from G on a subcomplex H is the topology of H itself. A general subset H of G +can still be given a topology by taking as open sets U ∩ H with U in the topology of G. +This topological space agrees with the closure of H. Similar standard consequences hold for +graphs. Any subgraph H of G has a topology which agrees with the induced topology from +G. A subset A of G(H) is open if and only if it is of the form O ∩ G(H). The simplicial +complex of a subgraph A of G is a closed set. As in general when we take the relative +topology on a closed subset K of a topological space X, the relative topology has the sets +U ∩ K as open sets, where U ranges over the open sets in G. All the axioms for a topology +are satisfied. The relative topology is the finest topology on K which has the property that +the inclusion i : K → G is continuous. +12.2. +To reformulate this, the relative topology on a subgraph H of G agrees with the +graph topology of H. We only have to look at a basis to see this. If x is a simplex in H, +then UH(x) is the intersection UG(x) ∩ H. So, the basis for the topology on H is the same +than the restriction of the basis of the topology on G to H. This proves the statement. +The subgraph H can be generalized. As for any topological space on some set X we can +restrict the topology on any subset Y of X. So, we can build a topology on any subset of +the simplicial complex G of a graph G. +12.3. +If a subset W ⊂ V of the vertex set of a finite simple graph G = (V, E) is given, one +can look at the subgraph H generated by W. This means to take the largest subgraph of +G which contains the vertex set W. One could look at the closure of a subgraph H as the +subgraph generated by the vertex set of W. This is in general a much larger graph. For +a Hamiltonian subgraph of G (a graph which passes through all vertices) for example +this “closure” would be the graph itself. The topology defined by the simplicial complex +however make a subgraph naturally closed already as the simplicial complex H generated by +the simplices in H is a sub-simplicial complex of the complex G of G and so closed. +Summary: Sub simplicial complexes correspond to closed sets in the topology. Sub- +graphs of a graph define a subclass of closed sets in the topology on the simplicial +complex defined by the graph. The relative topology on a subgraph of a graph is the +graph topology of H itself and does not use the topology of the host graph G. The +relative topology on a subcomplex of a complex is the topology of the subcomplex itself +which is the same without looking at the ambient space G. +13. Quotient +13.1. +If G is a graph and ∼ is an equivalence relation on vertices honoring the edges, +then the set of equivalence classes H = G/ ∼ can carry a topology. +First of all, the +21 + +FINITE TOPOLOGY +equivalence relation induces an equivalence relation on complete subgraphs and x/ ∼ is +the complete graph on the set of equivalence classes of V (x)/ ∼. Define G(H) as the set +{x/ ∼, x ∈ G(G)}. If G = K2 with G = {{1, 2}, {1}, {2}} for example and ∼ identifies +the two points 1, 2, we get G(H) = {{1}}. Assume G is a cover of H, meaning that there +is a surjective graph homomorphism f : G → H, then we can see H as a quotient G/ ∼, +where v ∼ w if f(v) = f(w). An example is the cover C8 → C4 with f(v) = v mod 4 if +V (C8) = {0, 1, 2, 3, 4, 5, 6, 7}. An other example is the cover S2 → P 2 of a sufficiently large +2-sphere for which the equivalence relation defines a manifold. Some 2-spheres are too small. +For an octahedron O for example, a graph with 6 elements, identifying opposite vertices +produces not no projective plane but O/ ∼= K3. +13.2. +In order that an equivalence relation on the sets of simplicial complex G produces a a +quotient topology, one needs to make some assumptions. In a Barycentric refinement, things +are easier. Having the complex G too small can make things weird. Lets look for example +the cycle complex C4 = {1, 2, 3, 4, (12), (23), (34), (41)} and impose the equivalence relation +where we identify the vertices 1 and 3. The quotient is no more a simple graph, but a quiver +because multiple connections appear. We have now a graph with three vertices 1, 3, 4 and +double bond connections (13) and double bond connections (34). We can however look at +the situation in the Barycentric refinement, where the identification becomes now a figure +8 graph. This situation matters if we look at Riemann-Hurwitz formulas which relate +the Euler characteristic of the quotient with the Euler characteristic of the complex itself as +well as using ramification points. In a case of a covering and having a group A of order +|A| acting on G so that G/A is again a complex, then The Riemann-Hurwitz formula tells +χ(G/A) = χ(G)/|A|. For example, if A = Z2 acts on a sphere G and H = G/A is a projective +space, then χ(H) = χ(G)/2. But if the complex is too small like for the Octahedron complex +considered above, then G/A is not a complex any more. For the Barycentric refinement +however it works and we get like that complexes representing a projective plane. +Summary: The topology on a quotient H = G/ ∼ can be defined as usual in topology +as the finest topology which makes the projection from G to the space of equivalence +classes continuous. +If H is a simplicial complex, then its topology is the quotient +topology. If quotients come from covers, then we have completely analogue situations +like in the continuum. We can for example look at the quotient of an antipodal map on +a Barycentric refined sphere and get a finite topological space representing a projective +space. +14. Manifold +14.1. +A graph G is called a d-manifold if every unit sphere S(x) in G is a (d − 1)-sphere. +A d-sphere is a d-manifold such that for some vertex v, the graph G − v without v is +contractible. A graph is contractible if there exists v such that S(v) and G − v are both +contractible. These notions can be defined also for complexes without referring to graphs. +14.2. +A complex G is a d-manifold if every unit sphere S(x) = B(x) \ U(x) is a (d − 1)- +sphere, where B(x) = U(x) is the unit ball the closure of U(x). The unit S(x) is always +closed and so carries a simplicial complex structure. A d-sphere is a d-manifold G such that +G \ U(x) is contractible for some x. A complex is contractible if there exists x such that +S(x) and G \ U(x) are both contractible. One can extend contractibility to non-closed sets +22 + +OLIVER KNILL +by defining for example an open set to be contractible if its closure is contractible. Every +star U(x) is contractible with this definition. 16 +14.3. +All these inductive definitions are primed by the assumption that the empty graph 0 +or the empty complex 0 is the (−1)-sphere and that the one point graph 1 = K1 or 1-point +complex 1 is contractible. If G is a graph that is a d-manifold, then all its Barycentric +refinements Gn are d-manifolds too. +If G is a complex which is a d-manifold, then all +their Barycentric refinements are manifolds. Examples of discrete manifolds in the sense +just defined are combinatorial triangulations of a manifold. But not all triangulations +are manifolds. +The tetrahedron K4 for example is contractible and so not a manifold. +We could look at the 2-skeleton complex of K4 however and get a sphere complex. The +Barycentric refinement of K4 is a 3-ball with 2-dimensional boundary which corresponds to +the Barycentric refinement of the 2-skeleton complex. +14.4. +Let us add a remark coming from the continuum: +every PL manifold (a mani- +fold equipped with a PL-structure) admits a combinatorial triangulation. +The question +of Poincar´e from 1899, whether every smooth manifold admits a triangulation has been +answered positively in the 1930ies: every smooth manifold has an essentially unique PL- +Structure. (The converse is not true. There are PL-manifolds which can not be smoothed or +admit different smooth structures.) The question shifted then to the topological situation. +See [16] and especially [55] for history or [42] for more recent developments. Also PL struc- +tures do not exist in general on topological manifolds in dimensions 4 or larger. 4-manifolds +examples were given using the Kirby-Siebenmann invariant. Non-PL triangulations of man- +ifolds were constructed using the Edwards-Cannon double suspension theorem. +14.5. +The fact that the Barycentric refined graph G1 obtained from the Whitney complex +G of a d-manifold graph G is a d-manifold can be proven by induction with respect to +dimension. Indeed, G is a manifold if and only if G1 is a manifold. If G was not a manifold, +then some unit sphere S(v) in G would not be a sphere. But then also SG1(v) which is +the Barycentric refinement of S(v) would not be a sphere. By induction assumption (unit +spheres are one dimension smaller) this is a contradiction. +14.6. +If G and H are homeomorphic and G is a manifold and H is not, then every Gn is +a manifold and non of the Hm are. Take a unit sphere S(v) in H. Since its inverse image +is homeomorphic to S(v) it is a sphere. By definition, it is the boundary of a ball. All unit +spheres of interior points in this ball are by definition spheres. Now, every vertex in G\ is in +the interior of the inverse of a ball B(x) in H. The following statement in the summary is not +true if “homeomorphic” would be replaced by “has a homeomorphic geometric realizations”. +The notion of homeomorphism proposed here is probably is equivalent to PL-equivalent but +we do not prove this because we don’t deal with infinity here: +Summary: If G, H are homeomorphic and G is a d-manifold then H is a d-manifold. +16We identify collapsible and contractible and use homotopic to 1 if we mean that a complex can be +deformed to 1 = K1 by both expansions and contraction steps. While homotopic to 1 is a computationally +difficult equivalence relation, contractibility is easy to check. +23 + +FINITE TOPOLOGY +15. Contractible +15.1. +The concept of “contractible” entered in a crucial way in the definition of “sphere” +and so in the definition of manifold. +The notion 17 makes sense for general graphs and +general simplicial complexes. It is different from homotopic to 1, where one can do both +homotopy extensions and contractions. The dunce hat graph is a concrete example of a +finite simple graph which is homotopic to a point which is not contractible. 18 The graph G +is contractible if and only if G1 is contractible. A complex G is contractible if and only if G1 +is contractible. Contractibility can be extended to non-closed sets by assuming the closure +to be contractible. +15.2. +The continuous image of a contractible graph does not need to be contractible: an +example is the map from the linear graph Ln+1 to the graph Cn mapping the initial and +end point to the same point. While Ln+1 is contractible, the graph Cn is not. It is also +not true that if H = f(G) is contractible then G is contractible. Let f map a 0-sphere +G = {V = {a, b}, {}} to K1 = {{a}, {}}. This is continuous because both graphs have the +discrete topology but the 0-sphere is not contractible. +Summary: As in the continuum, homotopy transformations are not continuous in +general. But contractions = homotopy reductions of graphs f : G → G−v or complexes +G → G \ U(x) are continuous. In the Unlike the unit spheres S(x), the balls B(x) are +always contractible. +16. Boundary +16.1. +A d-manifold with boundary is a graph or complex with the property that every +unit sphere is either a (d − 1)-sphere or is a (d − 1) ball. The same definition applies for +simplicial complexes. If S(x) is a (d − 1) sphere, we have an interior point x, if S(x) is +a (d − 1)-ball, x is a boundary point. For a manifold with boundary, the boundary is a +(d − 1) manifold without boundary. An example of a manifold with boundary is a d-ball +which by definition is a d-sphere with a point removed. An other example is a complete +graph Kd+1, where all points are boundary points. We can include Kd+1 into the class of +manifolds with boundary just in order to have Barycentric invariance. We want a graph to be +homeomorphic to its Barycentric refinement and in general to have a graph G homeomorphic +to H if there exist continuous maps Gn → Hm → G. We can say: +Summary: If G is a manifold with boundary and G is homeomorphic to H, then H is +a manifold with boundary. The same holds for graphs. +17. Duality +17.1. +Continuity is not compatible with some duality notions in graph theory or the theory +of simplicial complexes. The operation of mapping a graph to its graph complement is +in general not continuous. Already the dimensions do not work. The graph complement of +a cyclic graph Cn is always homotopic to a sphere or then to a wedge sum of two spheres +17Again: we avoid the term collapsible used often in the literature. +18The Dunce hat can be realized as a graph G with 17 vertices, 52 edges and 36 triangles. Its unit spheres +are all either 1-spheres or homeomorphic to figure eight graphs (wedge sums of two 1-spheres). There are +homotopy expansions which make it contractible. +24 + +OLIVER KNILL +[38]. Only for special cases like C5, where the graph complement is the same graph, the +complement operation can be made to be a homeomorphism. +17.2. +One can ask however whether the graph complement operation maps homeomorphic +graphs to homeomorphic graphs. But also here the answer is no: let f : G → H be a +continuous map between finite graphs like for example f(x) = x mod 4 from G = C8 to +H = C4. Does there exist a continuous map from Gc to Hc? The graph complement of +C8 is homotopic to a 2-sphere. The graph complement of C4 is a disconnected union of +two graphs K2. Since Hc is disconnected and Gc is connected and a continuous map can +not map a connected space to a disconnected space (as continuity preserves the property of +being connected), we can also not see the graph complement operation as a map preserving +homeomorphic graphs. +17.3. +Let us reformulate this a bit differently. We have identified simplicial complexes or +graphs which are Barycentric refinements of each other. The notion of graph complement is +not at all compatible with the Barycentric refinement notion. The Barycentric refinement of +C4 is C8. But there is no topological similarity between Cc +4 which is homotopic to a 0-sphere +and Cc +8 which is homotopic to a 2-sphere. +17.4. +For simplicial complexes, there is the Alexander duality operation: if G is a complex +and V � +x∈G x is the set of 0-dimensional simplices, then the Alexander dual of G is the +complex {y ⊂ V, (V \ y) /∈ G}. +17.5. +one can ask whether there is a duality notion which corresponds to the graph com- +plement. If G is a graph, we can look at the simplicial complex G of Gc but that does have +little to do with the simplicial complex G. One can experiment with other notions like if G +is an arbitrary simplicial complex and V = � +x x is the vertex set of G. We can look at the +complement Gc of G in the complete complex on V . This is a duality notion, but of course, +Gc is almost never a simplicial complex and the closure Gc of G is a complex but G → Gc +is not a duality notion. We see that graphs have their purpose especially with respect to +arithmetic. +Summary: While interesting for other reasons, duality notions like graph comple- +ment or Alexander duality are not compatible with homeomorphisms. The topic of +duality also shows that having different data structures for finite geometries is useful. +The graph complement duality for graphs for example works well with arithmetic. It +provides an isomorphism between the Sabidussy ring with join and large product as +operations, and the Shannon ring with disjoint union and Shannon product as opera- +tions. +18. Edge refinement +18.1. +Edge refinement are topological transformations of graphs. They induce topological +modifications of simplicial complexes. It is best described on graphs. Given a graph G and +an edge e = (a, b), we can refine the graph by adding a new vertex v, remove e and connect v +to the intersection of S(a) and S(b). When applied to the cyclic graph Cn, it produces Cn+1. +When applied more generally to a discrete d-manifold, we get a new d-manifold. The reason +is that the new unit sphere S(v) is the join of the 0-sphere {a, b} and the (d − 2)-sphere +S(a) ∩ S(b) and so again a (d − 1)-sphere. The unit spheres S(a) and S(b) are not changed. +The unit spheres of vertices w with the edge e = (a, b) in the unit sphere are themselves +25 + +FINITE TOPOLOGY +edge refined. Using induction with respect to dimension, one has now verified that edge +refinements preserve basic invariants like Euler characteristic. +18.2. +In full generality, an edge refined graph Ge is homeomorphic to G. First we can define +a surjective continuous map from Ge to G induced from the rule that every vertex goes itself +and that the new vertex goes to a. In order to see that this map is continuous, check the +properties. The inverse of every star in G is homeomorphic to a star in Ge. The star U(x) +of x = {a} has as as an inverse image the union of the star of a and star of v. The star of +the edge (a, v) has the empty set as an inverse image, The star of the edge (v, b) has as an +inverse image the star of (a, b). Similarly, the star of any simplex containing (a, v) has an +empty inverse image while the star of any simplex containing (v, b) has as the inverse image +the star of the corresponding simplex containing (a, b). +18.3. +To check the other direction, we have to construct a continuous map from the Barycen- +tric refinement G1 of G to Ge. We can take the canonical homeomorphism projection map +from G1 to G and modify it so that it becomes a map from G1 to Ge. +18.4. +We should mention that also the Dehn-Sommerville property is preserved by edge +refinement and Barycentric refinements. Dehn Sommerville spaces generalize spheres and +like spheres produce a monoid under the join operation. They therefore can be used to +generate spaces more general than manifolds but still have many properties of manifolds +like that odd-dimensional manifolds have zero Euler characteristic. Dehn-Sommerville d- +spaces Xd [27] are inductively defined. They must have the property that χ(G) = 1+(−1)d +and that all their unit spheres satisfy S(x) ∈ Xd−1. The induction starts with X−1 = {}. +Having the class Xd−1 of (d−1)-dimensional Dehn-Sommerville spaces topologically invariant +immediately bootstrap to see that also d-dimensional Dehn-Sommerville spaces have the +property that they are invariant under homeomorphisms. +Summary: The edge refinement operation produces a homeomorphic graph and so +of its Whitney complex. If a graph is a d-manifold, then the edge refined graph is a +d-manifold. One can look at edge refinements as local Barycentric refinements. We +also mentioned that Dehn-Sommerville spaces, a class of graphs generalizing spheres +and like spheres forming a submonoid of all complexes, are topological in nature. A +homeomorphic sibling of a Dehn-Sommerville space is Dehn-Sommerville. +19. Fundamental group +19.1. +A closed curve in a graph G can be defined as a continuous map from a circular +graph Cn to G. This means that the vertices x0, x1, . . . , xn = x0 are mapped into vertices +y0, y1, . . . , yn = y0 such that either yi = yi+1 or (yi, yi+1) ∈ E. The fundamental group +of a graph equipped with a reference point v is defined as the equivalence classes of closed +curves in G starting at v modulo curve homotopy deformations, where two curves are +called curve homotopic, if they can be morphed into each other by homotopy steps. +19.2. +A homotopy step is an operation, where an edge of the path attached to a triangle +t (an embedded complete graph K3) is replaced with the two other sides or then reverses +such a homotopy deformation and replaces two edges of the path in a triangle with the other +edge. If f : G → H is a continuous map, then a closed path maps either into a closed path or +26 + +OLIVER KNILL +a point. The following result mirrors corresponding results in the continuum. It is however +a statement in finite topological spaces. +19.3. +Replacing 1-spheres by a d-sphere S equipped with a base point, one can look at sphere +embeddings continuous images of S attached to a base point in G and so look at homotopy +groups πn(G). The sum of two such embeddings S1, S2 can be obtained by embedding a wedge +sum. We were once interested in graph complements of circular graphs [38] because there, +all higher dimensional wedge sums of spheres appear (at least homotopically equivalent) as +graphs complements of cyclic graphs. It would be nice if one could use this to compute +higher homotopy groups better but this has not worked yet. +Summary: If f : G → H is a continuous map on graphs, it induces a group homo- +morphism f∗ : π1(G) → π1(H) on the fundamental groups. +20. Euler characteristic +20.1. +The Euler characteristic of a finite abstract simplicial complex G is defined as +χ(G) = � +x∈G ω(x), where ω(x) = (−1)dim(x). The quantity can be seen in different ways. +It is first of all a valuation, meaning that it satisfies χ(G ∪ H) = χ(G) + χ(H) − G ∩ H. +If fk(G) counts the number of k-dimensional simplices then also χ(G) = �∞ +k=0(−1)kfk(G). +If G = (V, E) is a graph, its Euler characteristic χ(G) is defined as the Euler characteristic +of its Whitney complex G. If f : V → R is a locally injective function on vertices, then +if(v) = 1 − χ(S−(v)) is the PoincarHopf index of f at the vertex v. By induction, one can +check the Euler-Poincar´e formula χ(G) = � +v if(v). Applying this to the graph G1 of +the Whitney simplicial complex G of a graph G and using the function f(x) = dim(x) which +is locally injective, one immediately can see that the Euler characteristic of G and G1 are +the same. The reason is that if(v) = 1 − χ(S−(v)) = 1 + (−1)k = ω(v) because in the +case of the dimension functional, S−(x) is the boundary sphere complex of x which has by +the Euler-Gem formula the Euler characteristic 1 + (−1)k if x has dimension k. The Euler +characteristic therefore preserves Barycentric refinements. One can see this also by explicitly +writing down the linear map transforming the f vector (f0, f1, . . . , fd) of G to the f vector +of its Barycentric refinement. There is only one eigenvalue 1 of this linear operator T and +the corresponding eigenvector of T ∗ defines the Euler characteristic. +20.2. +One can see from the Poincar´e-Hopf formula immediately that homotopy extensions +and homotopy reductions preserve the Euler characteristic: choosing a function f which has +the property that it is maximal on the added vertex, we get if(v) = 1 − χ(S−(v)) = 0 +because S−(v) is contractible and because recursively one sees that contractible graphs have +Euler characteristic 1. One can also see that edge refinements in general preserve the Euler +characteristic: if e = (a, b) is an edge, then the edge refinement replaces the join of K2 +with of S(a) ∩ S(b) with the join of the path graph P3 with S(a) ∩ S(b). The operation +just replaces a contractible part with an other contractible part meaning that the Euler +characteristic of that part does not change. Similarly, one can show other operations like +flip diagonal operations on embedded kite graphs do not change the Euler characteristic. +But flip diagonal operations does not preserve d-manifolds in general. +27 + +FINITE TOPOLOGY +20.3. +If simplices in G are equipped with an orientation 19 one can interpret an arbitrary +function f : G → R as a differential form. For y ⊂ x, define sign(y, x) = 1 if the orientation +of y matches the orientation of x restricted to y, and −1 else. The exterior derivative +df(x) = � +y⊂x,|x|−|y|=1 sign(y, x)f(y) is a n × n matrix if G has n elements. +Because if +z ⊂ y ⊂ x with |z| = |y|−1 satisfies � +y sign(z, y)sign(y, x) = 0, the matrix d satisfies d2 = 0 +so that the Hodge Laplacian L = dd∗ + d∗d is block diagonal with fk × fk block matrices +Lk leaving invariant the class of k-forms, functions on k-dimensional simplices. The kernel +of Lk is called the k’th Betti number of G. By using the McKean-Singer symmetry that +the non-zero eigenvalues of L on even forms agrees with the non-zero eigenvalues of L on +odd forms, one can see that the super trace str(A) = � +k(−1)kAkk has the property that +χ(G) = str(e−tL) for any t. For t = 0, one has str(1) = � +k(−1)kfk(G) and in the limit +t → ∞, where only the kernels of Lk survives, one gets χ(G) = � +k(−1)kbk. The identity +� +k(−1)kfk(G) = � +k(−1)kbk(G) is called the Euler-Poincar´e formula. +20.4. +The Betti numbers of the Barycentric refinement G1 are the same than the Betti +numbers of G. This can be seen as a consequence of the K¨unneth formula which relates +the Betti numbers of H · G with the Betti numbers of H and the Betti numbers of G. +The Stanley-Reisner product H · G is homotop to the Shannon product H ∗ G for which +one can show the K¨unneth formula by taking the product of Harmonic functions d∗fg. +Homotopy deformations preserve the Betti numbers. If H = f(G) is a continuous image of +G then bk(H) ≤ bk(G). From these statements one can get immediately that homeomorphic +geometries have the same Betti numbers and so the same Euler characteristic. +Summary: +Betti numbers, cohomology group, Euler characteristic are topological +invariants. Also the sphere spectrum � +x∈G χ(S(x)) is a topological invariant. The +valuation property χ(U∪V ) = χ(U)+χ(V )−χ(U∩V ) holds for Euler characteristic and +all subsets U, V of G. The Euler-Poincar´e identity � +k(−1)kfk(G) = � +k(−1)kbk(G) +can be seen by heat deformation and using McKean-Singer symmetry. +21. Characteristics +21.1. +Euler characteristic is the first of many higher characteristics. +The next after +Euler characteristic is Wu characteristic. It is defined as ω(G) = � +x∩y∈G ω(x)ω(y). Also +all higher characteristics are invariant under Barycentric refinements. For manifolds with +boundary, it is χ(G) − χ(δG) (see [25]). +It is a multi-linear valuation but not a valua- +tion. It had been puzzling to us how the Wu characteristic behaves, even when looking at +wedge sums. While the Euler characteristic is invariant under homotopy and so also under +homeomorphisms the Wu characteristic is only invariant under homeomorphisms. +21.2. +To see why topology is involved, we have to restate that the energy theorem tells +χ(G) = � +x,y∈G g(x, y). There is a quadratic identity to that ω(G) = � +x,y∈G ω(x)ω(y)g(x, y)2 +(see [37]). Because g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) is expressed in terms of the topology. +What happens is that f −1(U(x) ∩ U(y)) is an open set with the same Euler character- +istic. +What happens is that if U, V are arbitrary open sets in the topology of G, then +ω(U ∪ V ) = ω(U) + ω(V ) − ω(U ∩ V ). +19There does not need to be compatibility with intersecting simplices +28 + +OLIVER KNILL +21.3. +The Euler characteristic χ(G) as a linear combination of basic valuations fk(G) count- +ing simplices. It satisfies the valuation formula χ(G ∪ H) = χ(G) + χ(H) − χ(G ∩ H) if +G and H are arbitrary simplicial complexes. This formula does not hold for the Wu charac- +teristic. For example, if G is the Octahedron complex and H is the circle complex C4, +then ω(G) = χ(G) = 2, ω(H) = χ(H) = 0. If we look at the wedge sum G ∧ H which is +G ∪ H with a common 1 point complex K1. Now ω(K1) = 1. We compute ω(G ∪ H) = 3 so +that obviously, the valuation formula does work as in the case of Euler characteristic, where +χ(G ∪ H) = 1. Now we know the solution to the puzzle: while G, H, G ∩ H are all open +sets by themselves, in the topology of G ∪ H, the complexes G, H are only closed in G ∪ H +and no more open. We have however the identity ω(U ∩ V ) = ω(U) + ω(V ) − ω(U ∩ V ) +for open sets within the topological space O in X = G ∪ H. If we look at the open balls +A = G \ {x}, B = H \ {x} (they are open as an open set intersected with the complement +of a closed set), and the open set C = U(x) in X. Now A, B, C are open sets in X and we +have ω(A ∪ B ∪ C) = ω(A) + ω(B) + ω(C) − ω(A ∩ B) − ω(B ∩ C) + ω(A ∩ B ∩ C). +21.4. +Consider the figure 8 graph X which is the wedge sum of two circular graphs X = +G sup H = C4 ∧ C4. We can explain ω(X) = 7 by putting things together. The valuation +formula does not work for closed sets. For example, the following formula does not work: +7 = ω(X) = ω(G) + ω(H) − ω(G ∩ H) = 0 + 0 − 1. However, we can write G as a union +of three open sets U, V, W. Both U, W are linear graph without boundary which have Wu +characteristic 1. The star graph without boundary has Wu characteristic 9. The intersection +between U and V has Wu characteristic 2. So, we have ω(X) = ω(U) + ω(V ) + ω(W) − +ω(U ∩ V ) − ω(V ∩ W) = 1 + 9 + 1 − 2 − 2 = 7. +21.5. +We first used the old definition ω(U) = � +x,y,x∩y̸=∅ ω(x)ω(y) for Wu characteristic +and not the correct definition ω(U) = � +x,y,x∩y∈U ω(x)ω(y). There is no difference between +the two definitions if we deal with simplicial complexes which are closed sets. It matters +however if we deal with open sets for example, take the two open sets U = {(1, 2, 3), (1, 2)} +and V = {(1, 2, 3), (2, 3)}, which are stars in the complete complex K3. Now W = U ∩ V = +{(1, 2, 3)} is the star of the facet (1, 2, 3) which has ω(W) = 1. We have X = U ∪ V = +{(1, 2), (2, 3), (1, 2, 3)} with ω(x) = −1. We have ω(U) = ω(V ) = 0 and ω(W) = 1, ωX = +−1. The identity ω(U)+ω(V )−ω(U ∩V ) = ω(U ∪V ) is valid but only because the simplices +x = (1, 2) and y = (2, 3) were not allowed to “interact”. Their intersection was not in U ∪V . +21.6. +A convenient way to compute Wu characteristic therefore is to write the complex as a +union � +j Uj of open sets, then use the inclusion exclusion property ω(G) = � +j ω(Uj) − +� +i∩j ω(Ui ∩ Uj) + � +i∩j∩k ω(Ui ∩ Uj ∩ Uk). +This explains again the known fact that for +d-manifolds M we have ω(M) = χ(M). Lets assume now that G is a manifold graph with +vertex set V . We can cover G with the open sets U(v), v ∈ V . Now use that ω(U(x)) = 1 +for any simplicial complex. So we have ω(G) = � +x=(v)∈G ω(U(x))−� +x=(v,w)∈G ω(U(x))+... +which is � +x(−1)dim(x)ω(x) = χ(G). For manifolds ω(G) = χ(G). +21.7. +In general we have the star formula: +ω(G) = +� +x∈G +ω(x)ω(U(x)) +using the stars U(x) of the simplex x. +We had previously proven the formula ω(G) = +� +x,y∈G ω(x)ω(y)χ(U(x) ∩ U(y))2 in [37] we have here a sum over G and not a more costly +29 + +FINITE TOPOLOGY +sum over pairs in G. We also see the ball formula for the unit balls B(x) = U(x) which is +remarkable because there is no direct relation between ω(B(x)) and ω(U(x)). The relation +ω(G) = +� +x∈G +ω(x)ω(B(x)) +follows from � +x∈G ω(x)ω(S(x)) = 0. (See [36] Corollary 6). +Summary: +Wu characteristic is no homotopy invariant but a topological invariants. +The valuation property χ(U∪V ) = χ(U)+χ(V )−χ(U∩V ) holds for all open sets, where +ω(U) = � +x,y,x∩y∈U ω(x)ω(y) for a set of sets U and not ω(U) = � +x,y,x∩y̸=∅ ω(x)ω(y). +There is the Gauss-Bonnet type formula ω(G) = � +x∈G ω(x)ω(U(x)) complement- +ing χ(G) = � +x∈G ω(x). +This allows to compute the Wu characteristic for larger +spaces. +There is also an energy theorem ω(G) = � +x,y g(x, y), where g(x, y) = +ω(x)ω(y)ω(U(x) ∩ U(y)). +22. Dynamics +22.1. +If T is a continuous map from a finite topological space O into itself then every point +is eventually periodic. Similarly, a simplicial map T on a simplicial complex. The attractor +of T is a finite set and on every connected component of the attractor, one just cyclically +permutes points. One calls the forward attractor also the ω-limit set. In the case of a +homeomorphism, there is also the α limit set which is the ω-limit set of the inverse map. +22.2. +The Lefschetz fixed point theorem for graphs [22] tells that if T is a graph +endomorphism T : G → G then the sum of the indices of the fixed points agrees with the +Lefschetz number χT(mathcalG), the super trace � +k(−1)ktr(L|Hk) of T induced each +space of harmonic forms Hk = ker(Lk). This result formulated for graphs [22] obviously +works for arbitrary simplicial complexes and a continuous map T : G → G. If F is the +set of fixed points of T and index iT(x) = ω(x)sign(T|x) with sign(T|x) being signature of +the permutation of T induced on x, then the Lefschetz formula tells � +x∈F iT(x) = χT(G). +The formula is easy to prove using the heat flow. The Koopman operator U : f → f(T) has +as the super trace � +x∈F iT(x). Applying the heat flow does not change the super trace of +e−tLU as non-zero eigenvalues in the odd forms and even forms agree. In the limit t → ∞, +only the map induced on the kernel survives and this is the Lefschetz number. A special +case of the Lefschetz fixed point theorem is the Brouwer fixed point theorem which applies +in the case when the complex has trivial cohomology. A special case is if G is contractible. +An even more special case is if G is a d-ball. +22.3. +We can also start with an arbitrary finite topological space O and fix a pre-basis B +which generates the topology. This defines a nerve simplicial complex G on B, where the +complex consists of all subset of B which have a non-empty intersection. A continuous map +f on O now defines a continuous map on the simplicial complex G so that the Lefschetz +fixed point theorem applies. We can now define the cohomology of the topological space +(equipped with the base) as the cohomology of G. The Lefschetz number of the super trace +of the from T induced map on the cohomology is then equal to the sum of the indices of +fixed points of f on G. This means that there is an open set in O which is fixed. +30 + +OLIVER KNILL +22.4. +A homeomorphism T : G → G of a finite geometry can be enhanced to a sequence +of homeomorphisms Tn : Gn → Gn. As more iterations are needed, as more Barycentric +refinements are required. +For a homeomorphism T this means specifying a sequence of +permutations Tn : On → On of the topologies of Gn and then require some compatibility. +How well the map Tn on On approximates the dynamics of Tm on Om determines the amount +of regularity of smoothness. +22.5. +The upgrade of a homeomorphism T : G → G to a stratified sequence of home- +omorphisms Tn : Gn → Gn is motivated by various similar constructions in mathematics, +like computing with sequences of rational numbers with a larger and larger number of digits +in order to approximate real numbers, to do numerical computations of partial differential +equations on sequences of meshes or the concept of inverse limit in constructions like p-adic +integers or then martingales, where a stochastic process is observed on a sequence of adapted +σ-algebras. Since every stochastic process given in the form of a sequence of IID random +variables Xn can be assigned a compact topological space Ω a continuous function f and a +homeomorphism T such that Xn = f(T n). The sigma-algebra An generated by the random +variables X1, . . . , Xn is the Borel σ algebra of a topological space On which is the product +space Ωn. If X has a finite set as range, then On is a finite topological space and An is the +Borel σ algebra generated by On. +Summary: The Lefschetz fixed point theorem and so the Brouwer fixed point the- +orem naturally work for continuous maps on simplicial complexes. It even works for +a finite topological space when applied to the nerve of a pre-basis. In order to study +the dynamics of a homeomorphism, one has to pick a choice of concrete homeomor- +phisms on refinements. The length of the orbit which one wants to compute accurately +determines how many Barycentric refinement lifts are needed. +23. Categorical +23.1. +Here are some general contemplations about the various categories: complexes, graphs +and topologies involved. Finite simplicial complexes form a category Sim with simplicial +maps as morphisms. +Finite graphs form a category Gra with graph homomorphisms as +morphisms. Finite topological spaces Top form a category too where continuous maps are +the morphisms. +We have the Whitney map from graphs to complexes, the Alexandroff +map from complexes to topological spaces and the ˇCech map from topological spaces to +the nerve graph. These maps are not functors because the morphisms do not correspond +directly. We can however enlarge the class of morphisms both on Sim as well as on Gra +to have morphisms. For example, in order to see the Whitney map Gra → Sim as a +functor between categories one has to expand the possible morphisms on graphs allowing +not only graph homomorphisms but maps from one graph to an other in which edges can +collapse to vertices. In order to have a functor between Sim and Top, we enlarge the class +of simplicial maps and allow also continuous maps, still order preserving but not mapping +simplicial subcomplexes to simplicial subcomplexes necessarily (these are the open maps). +Also the map assigning to a simplicial complex a graph is a functor again if one uses the +larger class of morphisms. The composition of the functors from Graphs to Complexes back +to Graphs is the Barycentric refinement. If Sim/ be the equivalence classes of complexes +under Barycentric refinement and Gra/ the equivalence classes of graphs under Barycentric +refinement. The Whitney map now identifies these two categories. A continuous map from +31 + +FINITE TOPOLOGY +some Gn → H could serve as the morphism. We chose however to make more assumptions +and use the notion of having H a continuous image of G to define homeomorphisms. +23.2. +Some of the graph theory literature assumes graphs are one-dimensional simplicial +skeleton complexes. While useful for some set-up’s it is rather limiting as graphs are so +much more than one dimensional objects. +The Whitney complex reflects rather general +topological spaces. There are other simplicial complexes associated to graphs of course, like +the graphical matroid other skeleton complexes or the neighborhood complex. +23.3. +Topological graph theory looks at graphs embedded in two-dimensional manifolds. In +that case, a graph naturally naturally defines a cell complex in which the two-dimensional +faces are the connected components of the complement of the embedded graph. This uses +infinity but it allows to deal with two-dimensional complexes which have the discrete topol- +ogy of the underlying surface. On a surface of degree g for example, the number of vertices +v, the number of edges e and the number of faces f satisfies v − e + f = 2 − 2g. Also in +topological graph theory one can sometimes avoid infinity. The notion of being planar for +example is settled with Kuratowski’s theorem completely within finite mathematics. That +theorem uses homeomorphism in the narrow sense as homeomorphic as one-dimensional +simplicial complexes. +23.4. +Discrete CW complexes extend simplicial complexes. A discrete CW structure +can be introduced within combinatorics, once one has defined what a k-sphere is: Start +building up the geometry G0 = {} and successively attach k-balls (called cells or handles) +to already existing (k − 1)-spheres. We can for example attach a 0-ball (called a vertex) to +a −1 sphere (the empty graph). Once the 0-dimensional part is built, we can attach 1-balls +(called edges) to 0-spheres (2 disjoint points). Then one can start adding 2-dimensional balls +(faces) to 1-spheres. For example, one can add triangles 2-simplices to the a triangular circle. +Obviously, every finite abstract simplicial complex is also an abstract CW complex. While +simplicial complexes are natural and given just as they are, a CW complex comes with a +“timeline” of how the structure has been built up. We can so build also multi-graphs or +quivers or more general complexes called δ-sets, which are simplicial complexes in which +simplices can occur with multiplicities. Adding a bit more structure produces a subclass +of δ-sets called simplicial sets. In category theory this is known as a pre-sheaf on the +simplex category. If on a δ-set boundary maps are defined, one has a cohomology like on +simplicial complexes. +23.5. +We have more recently also looked at quivers, graphs where self-loops and multiple +connections can happen. In that case, one can naturally attach δ-sets to a quiver. δ-sets +generalize simplicial complexes in that different sets can appear multiple times and where +boundary maps are specified. The category of δ sets generalize the category of simplicial +sets. The later are δ sets with more structure attached. δ sets (and so also simplicial sets) +have a cohomology attached. One can now ask, what natural topologies can be associated to +a quiver. We have not yet investigated this. One possibility would be to do this on the sets +x in the δ-complex and take the basis U(x). But now, the space is not even T0 any more as +different points can have the same minimal open sets. One can no more distinguish points +by their minimal open neighborhoods. +32 + +OLIVER KNILL +Summary: +Simplicial complexes, finite simple graphs and topological spaces can not +be directly linked with their traditional morphisms. But topology glues them together +if we look at continuous maps as morphism. Whether we talk about a graph with +continuous maps on them, simplicial complexes with continuous maps on them or finite +topological spaces with continuous maps on them, we always can switch to the other two +pictures. It requires however to change already what we mean by morphisms. While +traditionally, these three categories use different notation and jargon, topology unifies +them nicely and allow us to work on finite geometries using a trinity of interpretations. +Non-standard analysis links this radically finite geometry with rather arbitrary compact +topological spaces. +24. Ringed spaces +24.1. +As in commutative algebra approaches to geometry, one can use the notion of +ringed space. Attach a ring F(U) to ever open set and call it a section of U. Given +restriction maps produces a pre-sheaf. +The usual way to rephrase this is that this is +a contra-variant functor from the category of open sets with inclusion morphisms to the +category of rings. +To get a sheaf, we need existence (gluing) and uniqueness (locality) +properties: Gluing is related to existence h(x)|U(x) ∩ U(y) = h(y)|U(x) ∩ U(y) then there +exists a h with h|U(x) = h(x). Locality relates to uniqueness because h ∈ R(x) = k ∈ R(x) +for all x, then we have the same h = k. +24.2. +A simple case is to the sheaf of ring-valued continuous functions on open sets such +that if V ⊂ U, the restriction of F(U) to F(V ) is a ring homomorphism. In our case, where +the topology of a simplicial complex with the topology, we deal with a ringed simplicial +complex. The section F(x) = F(U(x)) is in this context the stalk of x and its elements +are the germs. Given a commutative local ring R and any function h : G → R defines +already a locally ringed sheaf. But things can be much more general. The restriction +maps from U(x) → U(y) if x ⊂ y do not have to be the obvious ones. Actually, any ring +valued matrix r(x, y) can serve as a transition map F(x) → F(y) if x ⊂ y. The pre-sheave +condition now means for x ⊂ y ⊂ z that the cocycle condition r(y, z)r(x, y) = r(x, z) holds +and especially that r(y, x)r(x, y) = r(x, x). +24.3. +Classically, when looking at general topological spaces, the stalk F(U(x)) at some +point x is the direct limit F(U) over all the open sets U containing x. In the finite topology +case, the stalk at x is F(U(x)), which is just a ring attached to the star U(x). A locally +ringed space is a ringed space in which every stalk is a local ring, meaning that it has a +unique maximal ideal. An example of a ringed space are differential forms. If an orientation +is fixed on each simplex x ∈ G, then these are just the functions from G to the ring R. +24.4. +Finite geometries also allow to use the frame work of schemes in an elementary frame +work. A ringed space by definition attaches to every open set a ring and its spectrum, the set +of prime ideals in the ring. If U is an open finite set, then the space OX(U) of functions +on U have as prime ideals the functions which vanish at some simplex x. The spectrum +therefore is just the set of simplices in U. The theory as developed for general ringed spaces +can be taken over word for word. The frame work can be useful also in combinatorics. For +example, we can take C-valued functions and require that the restriction maps from ring +F(U) to the ring F(V ) if V ⊂ U is not the obvious ones. Locally ringed topological space +33 + +FINITE TOPOLOGY +can have global properties are not necessarily the obvious ones: going around a closed loop +for example can produce a non-trivial map. +24.5. +The lack of linear structures prevents having constructs like tangent spaces in the +discrete. However, we have attached to each simplex a unit sphere S(x) and so a sphere +bundle. What we can do in general is to have transition maps on spheres S(x) coming from +positive dimensional simplices. These transitions tell what happens if one looks at S(x) as +part of S(v) or S(w) if v, w ⊂ x. Going from a pre-sheaf to a sheaf means to have transition +maps on positive dimensional simplices. +Summary: +Having a topology on a complex allows to use sheaf theoretical concepts +on finite spaces. In finite topological spaces, the ring R(x) attached to a star is called +a stalk and its elements are the germs. +25. Morse extensions +25.1. +A function f : G → R to a totally ordered space R like R of Z is called a Morse +function if it is locally injective, meaning that f(x) ̸= f(y) if x ⊂ y or y ⊂ x and +S−(x) = {y ∈ S(x), f(y) < f(x)} is a (k − 1)-sphere for some k ≥ 0 or then contractible. In +the former case, we have added a critical point, in the later case a regular point. If x is +a critical point, the integer k ≥ 0 is called the Morse index of the point x. If xk is a fixed +enumeration of points such that f(xk) ≥ f(xl) if k ≥ l, then Gn = {y, f(y) < f(xn)} is a +Morse build-up an χ(Gn+1) = χ(Gn) + χ(B+(xn)) − χ(S−(xn)) = χ(Gn) + if(xn) so that the +Poincar´e-Hopf formula χ(G) = � +x if(x) [19] holds. This formula holds for any locally +injective function f but for Morse functions the Poincar´e-Hopf index if(x) ∈ {−1, 1}. We +see that the existence of a Morse function implies that G can be seen as a CW complex in +which successively k-balls 20 are attached to (k − 1)-spheres in the previous step. +25.2. +The Morse build-up of a graph are not homeomorphism even if we look at the step +when a regular point is added. One can see this already from the fact that the dimension can +increase without adding a critical point. A continuous map can not increase dimension. The +contraction process is however a continuous process. The stable unit sphere S− +f (x) is then a +subgraph of S(x) and so a closed set. We can state this all in other words and say that a real +valued function on a simplicial complex needs not to be continuous but that every function +on the vertex set of a graph is continuous in the topology of the Barycentric refinement. +25.3. +It would not be useful to enforce continuity because {f(x) < c} can be a single point +which is neither open nor closed in the topology of G. When we look however at the situation +on the graph level with a function f : V (G) → R, then the sets Gn = {v, f(v) ≤ vn} generate +graphs which are closed sets in G1. In this sense any function f : V → R on a graph G is +continuous in the topology of G while a function on a-priori given simplicial complex G is +hardly ever continuous in that topology. A function G → R becomes only continuous if we +look at it as a function of the graph G1 and so using the topology of G1. +20also called handles +34 + +OLIVER KNILL +Summary: +Morse theory is a concrete way to build up an abstract finite CW- +complex using a Morse function as a guidance. While the sets Gn in a Morse build- +up of a complex are neither open or closed, the topology of their graphs make them +topological spaces in a refinement. Functions on simplicial complexes become naturally +continuous when considered in the topology of the Barycentric refinement. +26. Calculus +26.1. +A function f : G → R can be interpreted as a differential form. Similarly, if G is a +graph, we look at functions f : G → R. When restricted to k-dimensional simplices, one gets +k-form. Calculus can be studied on arbitrary Barycentric refinement levels. Provided that +orientations are fixed on G any scalar function is just a differential form. In order to define +level surfaces, we need only the very mild assumption that functions are locally injective, +meaning that adjacent vertices take different values. Lest look in this section at graphs. In +case of a simplicial complex G, look at the graph G1 in which the vertex set is G and where +two are connected if one is included in the other. +26.2. +A locally injective function f : G → R can be lift to a function on G1 by assigning to a +simplex x the average of f over the vertices in x. We can also take a function f : V (G) → R +and distribute its values f(v) equally to all points x in U(v). This produces a new function +G1 → R. What might happen under such a refinement of a locally injective function, that +it is no more locally injective. We can fix this by looking at the lexicographic order of the +pair (f, dim). Because in a Barycentric refinement, dim(x) ̸= dim(y) if x, y are connected in +G1, this is a well defined ordering. We can now use f on G1 to build again a level surface. +26.3. +Given a function on the vertex set of G1 we can move the content from vertices in G1 +which are sets of vertices in G to vertices of G by equally distributing the value f(x) to to all +vertices v ∈ x. With f(x) = ω(x), this produces f(v) = κ(v), where κ(x) is the curvature +[41]. The consequence � +x ω(x) = � +v κ(v) is the Gauss-Bonnet theorem. See [18, 21]. +26.4. +If f is a function on a discrete d-manifold G which is locally injective. Then the level +surface U = {f = c} generated by the set of simplices on which f changes sign. See [24]. +In the topology of the complex, this is an open set. However, its graph defines a discrete +(d − 1) manifold if it is not empty. This manifold now carries a topology again. Note that +as a set M in G, the set U is always open because if x is in M then every set y containing +x is in M. We can still make f locally injective by replacing f with f(x) + ϵdim(x) for ϵ +small. This allows us to extend f to G1 in a determined way. We can now look at a variety +{f1 = 0, . . . , fm = 0} for m locally injective functions f1, . . . , fm in the n’th Barycentric +refinement Gn as the set of simplices, where all the lifted functions of fj change sign. +26.5. +We have seen that using the “dimension trick” providing a lexicographic order of +the function on a higher level, we can lift any function on a graph uniquely to Barycentric +refinement where we can again define level sets. This level set is a graph where two simplices +are connected if one is contained in the other. Remarkably, by the discrete Sard theorem, +we never run into singularities. We expect if the functions fk are lifted nicely to a Barycentric +refinement, then the corresponding manifold is homeomorphic to S. We expect that there +could be surprises if we take a situation from the continuum, where {f1 = 0, . . . , fm = 0} +is a classical variety which is not a manifold. In that case, there could be surprises near +singularities. The topology depends on the choice of the Barycentric refinement. +35 + +FINITE TOPOLOGY +Summary: If G is a n-manifold and m ≤ n locally injective functions are given on +G, then the “variety” {f1 = 0, . . . , fm = 0} is a well defined graph again. It is either +empty or a (n − k)-manifold S. +27. Interaction energy +27.1. +Given a simplicial complex G with n elements x and any n × n matrix taking values +in some ring R, we can define the internal energy of a subset A ⊂ G as +ω(A) = +� +x,y,x∩y∈A +h(x, y) . +The matrix h does not have to be symmetric. We can think of h(x, y) also as a current from +x to y and ω(A) as the total current or traffic flowing overall through A. Now look at the +matrix +g(x, y) = +� +x,y +ω(x)ω(y)ω(U(x) ∩ U(y)) . +This matrix gives a potential energy between the simplices x and y. Unlike h, the matrix +g is always symmetric. The energy theorem tells ω(G) = � +x,y g(x, y). This result assures +that the total potential energy of G is the total internal energy. This theorem generalizes +an energy theorem proven before, see [36, 37, 35]. For example, if h(x, y) is diagonal with +h(x, x) = ω(x), then ω(A) = χ(A) is the Euler characteristic. In that case the matrix g is the +inverse of the operator L(x, y) = χ(x∩y), where x is the closure of {x}, a simplicial complex. +The frame work also captures energized simplicial complexes where h(x, x) = h(x) and +h(x, y) = 0 for x ̸= y. +27.2. +To prove the more general energy theorem, note that the map h → g is linear and +that both the energy and the total sum are both linear expressions. We only need to verify +the statement therefore for the matrix h satisfying h(x0, y0) = 1 and h(x, y) = 0 else for some +fixed simplices x0, y0. These n2 basis elements are fixed-points of the linear map T(h) = g +and the energy relation � +x,y,x=x∩y∈G h(x, y) = � +x,y g(x, y) holds. By linearity, the relation +holds then for all h. To verify the statement for a basis element, note that the left hand side +is 1 if x0 and y0 intersect and 0 else. The right hand side is ω(x)ω(y)ω(U(x) ∩ U(y)) which +is non-zero only if x0 and y0 intersect and both x0, y0 are contained in U(x) and U(y). This +means that the union x0 ∪ y0 is contained in U(x) ∩ U(y). This means that both x and y +have to contain x0 ∪y0 . This means that the simplex x∩y has to contain the simplex x0 ∪y0 +and so x0 ∩ y0. But � +x0⊂x,y0⊂y,x0∩y0⊂x∩y ω(x)ω(y) = � +x0⊂x ω(x) � y0 ⊂ yω(y) = 1 ∗ 1 = 1 +because the Euler characteristic of a simplex is 1. +27.3. +We also have as before a curvature relation κ(x) = � +y g(x, y) = ω(x)g(x, x) = +ω(x)χ(U(x)) and so � +x κ(x) = χ(G) which is a Gauss-Bonnet relation. It can also be +thought of as a Poincar´e-Hopf theorem for the locally injective function f(x) = −dim(x) +because then, the atom U(x) is the stable sphere S−(x) = {y, f(y) < f(x)}. Still, since the +dimension function f is not so well visible, it is good to think of κ not as an index but as +a curvature. The internal energy of the “atom” U(x) is up to a sign a curvature. Summing +up the curvature gives the total energy. We can also think of the relation � +x ω(x)g(x, x) as +the super trace str(g) of the matrix g. In the context of simplicial complexes, the notion +of super trace is natural since str(1) = χ(G) is the Euler characteristic and because of the +McKean-Singer formula str(e−tL) = χ(G). [20, 45]. +36 + +OLIVER KNILL +27.4. +Actually, we would like to announce here already that arbitrary tensor energy +theorems hold. Let h(x1, . . . , xm) be arbitrary ring-valued functions of m variables. We +already had for +ω2(A) = +� +x,y,x∩y∈A +h(x, y) +that g2(x, y) = ω(x)ω(y)ω2(U(x) ∩ U(y)) satisfies � +x,y g2(x, y) = ω2(G). If G is a finite +abstract simplicial complex and h(x, y, z) arbitrary R valued function. For a subset A ⊂ G, +define the internal cubic energy +ω3(A) = +� +x,y,z,x∩y∩z∈A +h(x, y, z) . +Now define g3(x, y, z) = ω(x)ω(y)ω(z)ω3(U(x) ∩ U(y) ∩ U(z)) and think about it as the +potential energy of the triple. +Then the total potential energy agrees with the total +internal energy +� +x,y,z +g3(x, y, z) = ω3(G) . +This works also with more interaction like quartic +ω4(A) = +� +x,y,z,w,x∩y∩z∩w∈A +h(x, y, z, w) . +For g4(x, y, z, w) = ω(x)ω(y)ω(z)ω(w)ω(U(x) ∩ U(y) ∩ U(z) ∩ U(w)), the total potential +energy agrees with the total internal energy +� +x,y,z,z +g4(x, y, z, w) = ω4(G) . +Summary: +Given an interaction transfer rule between m intersecting simplices in a +simplicial complex G, we can assign internal energies k tuples of sets. The total energy +can be expressed also as the sum of all potential energies. The internal m-energy of a +set ω(A) = � +� +j xj∈A h(x1, . . . , xm) is associated to closed sets, the potential energy uses +open sets gm(x1, . . . , xk) = � +j ω(xj)ω(�k +j=1 U(xj)). The energy theorem assures that +the total internal energy is the total potential energy ωm(G) = � +x1,...,xk g(x1, . . . , xk). +In the case m = 1 and h(x) = ω(x), where the total energy is the Euler characteristic, +this is a Gauss-Bonnet theorem χ(G) = � +x ω(x)ω(U(x)), where U(x) is the smallest +open set containing x. In the case h(x1, . . . , xm) = �m +k=1 ω(xk), the energy ω(G) is the +m-th characteristic, a topological invariant for the complex. Unlike for m = 1, which +gave the Euler characteristic, we have no homotopy invariant however for m > 1. +28. Remarks +28.1. +Inspired by Alexandroff and Zariski, we have revisited here at the finite topology on +a simplicial complex G defined by stars and especially for simplicial complexes coming from +a finite simple graph G = (V, E). +21 The analogy to algebra is that the vertices of the +graph play the role of the maximal ideals and that the simplices play the role of prime +ideals. A basis for the topology is the set of stars of a simplex, the set of simplices which +21Since open sets are still quite local, the drawbacks of Zariski topology appear not really relevant. +37 + +FINITE TOPOLOGY +contain x, as well as an added empty set. Unlike any topology on the vertex set V like the +one given by distance which would render the graph completely disconnected, our topology +honors connectivity and dimension. It shares the non-Hausdorff property with the Zariski +topology on prime ideals of a commutative ring. The closed sets in our graph topology are +exactly the simplicial complexes of subgraphs. They play the role of algebraic subsets of a +variety in algebraic geometry. The closure of a set A of simplices is the smallest abstract +simplicial complex which contains the set A. +28.2. +One of the motivations to look at finite topologies is that there are finite simple spaces +G, H for which the geometric realization |G|, |H| are homeomorphic but which are not home- +omorphic in the finite topology. Open sets U(x), U(y) be entangled in a complicated way in +the finite topology if the dimension is large. But these entanglements are not always visible +when looking at the topology induced from Euclidean distance in geometric realizations. The +topology of the Euclidean realization is not sophisticated enough. In other words, there are +triangulations of topological manifolds which have the manifold as a geometric realizations +but which are not discrete manifolds as defined here. The definition of homeomorphism is +motivated by the notion of piecewise linear map in topology. A map f : |G| → |H| be- +tween geometric realizations of simplicial complexes is PL, if there is a piecewise linear map +between Barycentric refinements |Gn| → |Hm|. A PL homeomorphism is then a simplicial +map such that there is a homeomorphism |Gn| → |Hm| for some refinement. This looks +equivalent to what we do here in the finite but leaves finite mathematics. As we are not +interested in infinity here, we do not bother showing the equivalence. +28.3. +An other motivation has been to answer the question why higher characteristics +like Wu characteristic ω(G) = � +x,y,x∩y∈G ω(x)ω(y) is a topological notion while Euler +characteristic χ(G) = � +x ω(x) is more. We will write about this more in the future but +one of the key facts is that ω(U ∪V ) = ω(U)+ω(V )−ω(U ∩V ) holds for open sets but not for +closed sets or sets which are neither closed nor open in general. For Euler characteristic this +valuation formula holds for all subsets U, V of G. Euler characteristic does super count +simplices, while Wu characteristic does super count intersecting simplices. This interacting +points should not be separable. In indeed, if an intersecting pair x, y is in the intersection +U ∩ V of two open sets it must be in both sets U and V . An other mystery which still needs +more investigation is the notion of analytic torsion A(G) = � +k Det(Lk)k(−1)k+1, where Lk +are the blocks of the Hodge Laplacian L = D2 = (d+d∗)2 of the Whitney complex and Det is +the pseudo determinant. We wrote this as a super determinant SDet(D) = � +k Det(Dk)(−1)k +of the Dirac operator D with Dirac blocks Dk = d∗ +kdk. Analytic torsion can be defined +for any simplicial complex but so far it has been accessible only in 2 cases: the first is +when G is homotopic to 1. In that case A(G) = |V | where V = � +x∈G x. The second case +was when G = |V ||V ′| for even dimensional spheres and G = |V |/|V ′| for odd dimensional +spheres, where V ′ is the number of maximal simplices in G. But analytic torsion is not +a topological invariant. Even for manifolds like a torus, the analytic torsion changes if we +make a Barycentric refinement. +As already in the etymology of the name, a non-trivial +fundamental group makes the functional A(G) more complicated as we can get torsion terms +along non-contractible closed loops. Still it is not only that. If we make homotopy extensions +of a sphere which are not homeomorphisms, the torsion formula for the sphere in terms of +|V | and |V ′| disappears. +38 + +OLIVER KNILL +28.4. +Finite and so Alexandroff topologies can be interesting from a physics point of view. +First of all, there is local interaction of simplices which are contained in each other. We +can not separate such points using open sets. The fact that we have smallest non-empty +open Planck units U(x) is some sort a space quantization or atoms of space. If we +look at a manifold with a very fine triangulation, both the lack of the Hausdorff topology +and the Alexandroff feature are hardly visible. +The situation is also present in floating +point arithmetic, when a computer deals with small numbers. Every point has a smallest +neighborhood which can no more be resolved. We also can not separate two points which are +too close from each other even so they are different. Points which are identified by the equal- +tolerance parameter define the machine graph. With a machine precision log10(252) ∼ +15.65 the distance below which two numbers are identified is about 2−46 = 2−52−1+7 = +1.42109 · 10−14. +28.5. +The definition of homeomorphism is motivated by the fact that if we have two finite +topological spaces coming from a finite abstract simplicial complex and a continuous sur- +jective map f : X → Y and a continuous surjective map g : Y → X and the unit spheres +S(x) have homeomorphic pre-images and unit balls of locally maximal simplices have balls +as images, then X, Y are homeomorphic. 22 +28.6. +The given definition of homeomorphism within finite topology has shifted a bit +while writing down this text. We first tried to avoid unit spheres S(x) and balls but failed +to prove some results. Using unit sphere S(y) in the definition allows the use of induction. +For the unit balls of maximal simplices, there is no interesting topology as they are balls +and we just require that the inverse image of such a unit ball is a ball (which is precisely +defined). It is postulated that all d-dimensional balls are homeomorphic but it could also be +proven from the definition. It also seems to be necessary to ask such a requirement. We need +it for example that if G is a manifold and H is homeomorphic, then H is a manifold. We can +imagine continuous surjective maps going both ways which collapse substantial parts of the +topology somewhere in the interior of the manifold so that the inverse image of S(y) could +become a complicated object in G. +28.7. +Looking at stars U(x) = W +(x) and cores V (x) = {x} = W −(x) of simplices is also +motivated by connection calculus. We have seen for example that the Green function +matrix g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) is always the inverse to the matrix L(x, y) = +χ(V (x)∩V (y)), where χ(A) = � +x∈A ω(y) is the Euler characteristic of an arbitrary subset of +G and ω(x) = (−1)dim(x). While V (x)∩V (y) which is always a simplicial complex, are closed, +the sets U = U(x) ∩ U(y) are open. The Euler characteristic of the closure B(x) = U is in +general different from the Euler characteristic of U. Actually χ(B(x)) = χ(U(x)) + χ(S(x)) +where S(x) is the boundary of U. 23 +22We wonder whether it is true in general: does already the existence of two continuous surjective maps +f : X → Y, g : Y → X force X, Y to be homeomorphic. +Using the axiom of choice, one can invert +the surjections and have injections, showing with Cantor-Schroeder-Bernstein that the cardinalities are the +same so that there is a bijection between the topologies. In the non-Alexandroff case, where points can be +written as intersections of open sets, this should then give a homeomorphisms. +23This phenomenon prevented us for some time to find the Green star formula for the Green function. +39 + +FINITE TOPOLOGY +28.8. +Both the star U(x) and the core V (x) = {x} = W −(x) can be seen as measurable +sets in the graph. If we close a topology O under complements and countable intersections +and unions, we get the Borel σ algebra A as usual. This set A still does not cover all +subsets of G if G has dimension 2 or more. The reason is that if {e = (a, b)} is a single +non-maximal simplex, then it is neither open nor closed. The smallest open set containing it +is U(e), the smallest closed set containing it is the simplicial complex {e, {a}, {b}}. We can +look for probability theory on the graph and look for example at the measure fk(A)/fk(G) +counting the fraction of k-dimensional simplices in A. For probability theory on finite set, +see [49]. +28.9. +In the literature, one often a “graph” as a topological space that is obtained as a +geometric realization as a one-dimensional simplicial complex. A more topological ap- +proach is to look at the geometric realization of its Whitney complex in which all the complete +subgraphs Kn+1 are realized as simplices. One can then also look at other simplicial com- +plexes attached to a graph, similarly as one can attach other topologies to Rn. The notion of +homeomorphism could be extended to such cases too. There are simplicial complexes and so +graphs that are not homeomorphic but which have homeomorphic realizations. An example +is a double suspension of a rational homology sphere. It is not topologically equivalent to a +sphere in our sense but in a geometric realization, it is by the double suspension theorem. +While also in the discrete, any manifold that is a suspension of a manifold must be a sphere, +in the discrete, a suspension of a non-manifold is by definition not a discrete manifold. +28.10. +We have searched for notions of homeomorphism within finite combinatorics for a +while like [23], where we looked at ˇCech type notions like the nerve of an open cover and +asked that two homeomoprhic graphs have isomorphic nerves. In 2016 we experimented +(motivated by the Zarisiki topology) with the concept of having the closed subgraphs play +the role of closed sets. +We have not taken it too seriously: do we want to work with +topological spaces that are non-Hausdorff? We decided now it is better to work with the +topology generated by the star basis. When reviewing the Lusternik-Schnirelmann category, +where open covers play a role, the concept fits better with what one does in the continuum. +Since 2016, we have also realized more the importance of stars U(x) = U(x) and cores +x = W −(x) = {x} of simplices as they form a hyperbolic structure and because g(x, y) = +ω(x)ω(y)χ(U(x) ∩ U(y)) and L(x, y) = χ(W −(x) ∩ W −(y)). In the current notation, we +would write this Green-Star formula as g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)). +The fact +that U(x) ∩ U(y) can be topologically quite complicated even so both sets U(x), U(y) are +smallest open sets and have contractible closures, the intersection U(x)∩U(y) can be rather +complicated. The local smallest Planck units U(x) in a complex can be entangeled in a +complicated way. Also, the simplicial complex belonging to the closure of U(x) ∩ U(y) can +be topologically very different from U(x) ∩ U(y)! 24 +28.11. +There are many interesting open questions and many opportunities for experimen- +tation or further explorations. We can ask for example about the fraction |O(G)|/2|G(G)| +telling us in a graph what fraction of subsets of the simplicial complex are open sets. As the +number of open sets and closed sets agree, this is equivalent to count the number of sub- +graphs of a given graph. Numerical computations become quickly too hard to do. We have +24This was a reason to drove us almost insane in 2016 while looking for the Green star formula as the +formula with the closure U(x) ∩ U(y) worked in most cases and especially small complexes, but that it had +rare failures. +40 + +OLIVER KNILL +φ(C4) = 48/256 and φ(C5) = 124/1024 and φ(C6) = 323/4096 and φ(K1) = 1, φ(K2) = 5/8, +φ(K3) = 19/128 and φ(K4) = 167/32768. +28.12. +Let us add a comment on the literature. The paper [1], was dedicated to Emmy +Noether, assumes that the space is locally finite spaces which under a global compactness +assumption means finiteness. Alexandroff already identifies discrete T0 spaces with partially +ordered sets and identifies closes sets as simplicial complexes. He notes that if in an Alexan- +droff space, two smallest open sets U(x) = U(y) agree, then x = y. The reason is that +then x ∈ U(y) and so y ⊂ x and y ∈ U(x) and so x ⊂ y. He calls simplicial complexes +vollst¨andige Mengensystems = complete set systems. The completion of an arbitrary +set of sets is the closure. Interestingly, the void {} = empty set is not considered of this type, +even so today we consider this to be a simplicial complex. It is technically a finite set of sets +which is closed under the operation of taking finite non-empty subsets. Also in topology, the +empty set is a closed set as in any topological space we consider ∅, X to be clopen = closed +and open and define connectedness as the property that the only clopen sets are ∅ and X. +The modern point of view is to see the void 0 as the (−1) dimensional sphere. Alexandroff +also notes that pre-base of stars centered at 0-dimensional simplices define what we would +call today a ˇCech graph. In our terminology we would say that every locally finite simplicial +complex coming from a graph G has a ˇCech cover (the pre-base) whose graph is G. It has +also a ˇCech cover (coming from the base) which is the Barycentric refinement. Alexandroff +also notes that a continuous map can be lifted to Barycentric refinements. He however refers +to the geometric realization as a polyhedron in order to define something analog to homeo- +morphism. Alexandroff also reformulates the construction of the cohomology ring following +Alexander-ˇCech and Whitney. +28.13. +Finite topologies spaces were picked up again as such in the 1960ies like [60]. Strong +for example shows that for any finite topological space, there is a unique minimal base. +Connectedness and path connectedness are equivalent. That a continuous self-map f on +a finite topological space that is either injective or surjective must be a homeomorphism. +Strong shows already that on finite topological spaces, continuity is equivalent with simplicial +map: x ⊂ y if and only if f(x) ⊂ f(y). Strong equips the space HG of continuous maps G → +H with the compact-open topology. It is ordered with g ≤ f if for all points g(x) ≤ f(x). +If f ≤ g then f, g are homotopic. The connectivity components of HG are the homotopy +classes of maps from G to H. [44] starts with Finite topological spaces have more interesting +topological properties than one might suspect at first. Indeed, McCord points out that for +every finite topological space, there is a finite simplicial complex which is a weak homotopy +equivalent in the sense that the induced maps on all homotopy groups are isomorphisms +(meaning for π0 which is not equipped with a group structure, that the number of connected +components are the same). McCord is known also for a version of the nerve theorem stating +that the homotopy type of a nice topological space is encoded in the ˇCech nerve of a nice +open cover. This certainly applies for finite topological spaces and the cover coming from +minimal open sets. What is needed for example is that the intersection of two such sets is +either contractible or empty. The ˇCech nerve of a cover has been introduced by Alexandroff. +41 + +FINITE TOPOLOGY +29. Code +29.1. +The following few Mathematica lines allow to compute the topology of a complex or +the topology of the Whitney complex of a graph. We see that the number of topologies on +the cyclic graph Cn is the Lucas number L(2n). We then display the code for Figure 1. +� +Closure [ A ]:= If [A=={},{},Delete [ Union [ Sort [ Flatten [Map[ Subsets ,A ] , 1 ] ] ] , 1 ] ] ; +Whitney [ s +]:= If [ Length [ EdgeList [ s ]]==0 ,Map[{#}&, VertexList [ s ] ] , +Map[ Sort , Sort [ Closure [ FindClique [ s , Infinity , All ] ] ] ] ] ; +UU[ G , x ]:=Module[{U={}},Do[ If [ SubsetQ [G[ [ k ] ] , x ] , +U=Append[U,G[ [ k ] ] ] ] , { k , Length [G] } ] ;U] ; +Basis [ G ]:= Table [UU[G,G[ [ k ] ] ] , { k , Length [G] } ] ; +SubBasis [ G ]:=Module[{V=Union [ Flatten [G] ] } , +Table [UU[G,{V[ [ k ] ] } ] , { k , Length [V ] } ] ] ; +UnitSpheres [ G ]:=Module[{B=Basis [G] } , +Table [Complement[ Closure [B [ [ k ] ] ] , B [ [ k ] ] ] , { k , Length [B ] } ] ] ; +UnitBalls [ G ]:=Map[ Closure , Basis [G ] ] ; +Cl [ U , A ]:=Module[{V=U} ,Do[V=Union [Append[V, +Union [V [ [ k ] ] ,A[ [ l ] ] ] ] ] , { k , Length [V]} ,{ l , Length [A] } ] ;V] ; +Topology [ G ]:=Module[{V=B=Basis [G] } , +Do[V=Cl [V,B] , { Length [ Union [ Flatten [G ] ] ] } ] ; Append[V, { } ] ] ; +GraphBasis [ s +]:= Basis [ Whitney [ s ] ] ; +GraphTopology [ s +]:= Topology [ Whitney [ s ] ] ; +N u l l i t y [ Q ]:=Length [ NullSpace [Q ] ] ; +Fvector [ G ]:= Delete [ BinCounts [Map[ Length ,G] ] , 1 ] ; +ToGraph [ G ] +:=Module[{ n=Length [G] , v , e , s } , v=Range[ n ] ; e ={}; +Do[ If [ ( SubsetQ [G[ [ k ] ] ,G[ [ l ] ] ] | | +SubsetQ [G[ [ l ] ] ,G[ [ k ] ] ] ) && +Not[G[ [ k]]==G[ [ l ] ] ] , e=Append[ e , v [ [ k]]−>v [ [ l ] ] ] ] , +{k , n} ,{ l , k+1,n } ] ; +s=UndirectedGraph [ Graph [ v , e ] ] ] ; +BarycentricGraph [ s +]:=ToGraph [ Whitney [ s ] ] ; +BarycentricComplex [ G ]:= Whitney [ ToGraph [ s ] ] ; +w[ x ]:=−(−1)ˆLength [ x ] ; +Wu1[ A ]:= Total [Map[w,A ] ] ; +Chi=Wu1; +Wu2[ A ]:=Module[{ a=Length [A] } ,Sum[ x=A[ [ k ] ] ; Sum[ y=A[ [ l ] ] ; +If [MemberQ[A, Intersection [ x , y ] ] , 1 , 0 ] ∗w[ x ]∗w[ y ] , { l , a }] ,{ k , a } ] ] ; Wu=Wu2; +Wu3[ A ]:=Module[{ a=Length [A] } ,Sum[ x=A[ [ k ] ] ; Sum[ y=A[ [ l ] ] ; Sum[ z=A [ [ o ] ] ; +If [MemberQ[A, Intersection [ x , y , z ] ] , 1 , 0 ] ∗w[ x ]∗w[ y ]∗w[ z ] , { o , a }] ,{ l , a }] ,{ k , a } ] ] ; +FastChi [ A ]:=Module[{UU=Basis [A] } ,Sum[w[A[ [ k ] ] ] ∗ Chi [UU[ [ k ] ] ] , { k , Length [A ] } ] ] ; +FastWu [ A ]:=Module[{UU=Basis [A] } , Sum[w[A[ [ k ] ] ] ∗Wu[UU[ [ k ] ] ] , { k , Length [A ] } ] ] ; +FastWu3 [ A ]:=Module[{UU=Basis [A] } ,Sum[w[A[ [ k ] ] ] ∗ Wu3[UU[ [ k ] ] ] , { k , Length [A ] } ] ] ; +Suspension [ G ]:=Module[{ q=Max[ Flatten [G]]+1 , n=Length [G] } , +Closure [ Union [ Table [Append[G[ [ k ] ] , q ] , { k , n } ] , Table [Append[G[ [ k ] ] , q+1] ,{k , n } ] ] ] ] ; +JoinAddition [ A , B ]:=Module[{ q=Max[ Flatten [A] ] ,Q,G=A} ,Q=Table [B [ [ k ]]+ q , { k , Length [B ] } ] ; +Do[G=Append[G, Union [A [ [ a ] ] ,Q[ [ b ] ] ] ] , { a , Length [A]} ,{ b , Length [Q] } ] ;G=Union [G,Q] ; +If [A=={},G=B ] ; +If [B=={},G=A] ; G] ; +DoubleSuspension [ G ]:= Suspension [ Suspension [G ] ] ; +WuBetti [ G ]:=Module[{ Cohomology2 , n , n2 ,G2, l l , ln , dd1 , dd2 , LL2 , L2 , dd , br , D2,DD} , +n=Length [G] ; +length [ x ]:=Length [ x [ [ 1 ] ] ] + Length [ x [ [ 2 ] ] ] ; +G2={}; +IS=Intersection ; +Do[ If [ Length [ IS [G[ [ k ] ] ,G[ [ l ] ] ] ] > 0 ,G2=Append[G2,{G[ [ k ] ] ,G[ [ l ] ] } ] ] , { k , n} ,{ l , n } ] ; +n2=Length [G2 ] ; +G2=Sort [G2, length [#1]< length [#2] & ] ; +l l = Map[ length ,G2 ] ; +ln=Union [ l l ] ; br=Prepend [ Table [Max[ Flatten [ Position [ l l , ln [ [ k ] ] ] ] ] , { k , Length [ ln ] } ] , 0 ] ; +d e r i v a t i v e 1 [ { x +, y }]:=Table [{ Sort [ Delete [ x , k ] ] , y} ,{k , Length [ x ] } ] ; +dd1=Table [0 ,{ n2 } ,{ n2 } ] ; Do[ u=d e r i v a t i v e 1 [G2 [ [m ] ] ] ; +If [ Length [ u] >0 , +Do[ r=Position [G2, u [ [ k ] ] ] ; +If [ r !={} , dd1 [ [m, r [ [ 1 , 1 ] ] ] ] = ( − 1 ) ˆ k ] , { k , Length [ u ] } ] ] , {m, n2 } ] ; +d e r i v a t i v e 2 [ { x +, y }] +:=Table [{ x , Sort [ Delete [ y , k ] ] } , { k , Length [ y ] } ] ; +dd2 = Table [0 ,{ n2 } ,{ n2 } ] ; Do[ u = d e r i v a t i v e 2 [G2 [ [m ] ] ] ; +If [ Length [ u] >0 , +Do[ r=Position [G2, u [ [ k ] ] ] ; +If [ r !={} , dd2 [ [m, r [ [ 1 , 1 ] ] ] ] = ( − 1 ) ˆ ( Length [G2 [ [m, 1 ] ] ] + k ) ] , +{k , +Length [ u ] } ] ] , +{m, n2 } ] ; +dd = dd1 + dd2 ; D2=dd+Transpose [ dd ] ; +L2 =D2.D2; +LL2=Table [ Table [ L2 [ [ br [ [ k ]]+ i , br [ [ k ] ] + j ] ] , { i , br [ [ k+1]]−br [ [ k ] ] } , +{ j , +br [ [ k + 1]] − br [ [ k ] ] } ] , +{k , Length [ br ] −1}]; +Cohomology2=Map[ NullSpace , LL2 ] ; Map[ Length , Cohomology2 ] ] ; +� +� +42 + +OLIVER KNILL +29.2. +As example computations, we compute the number of elements in the topology of a +circle Cn where |G| = 2n. One can show by induction in n that the number of open sets in the +topology is L(2n), where L(n) is the Lucas number defined by L(0) = 2, L(1) = 1, L(2) = 3 +and the recursion L(n + 1) = L(n) + L(n − 1) is the Fibonacci recursion. (The only +difference is that for the Lucas numbers, the entry L(0) = 2, while for the Fibonacci numbers, +the entry F(0) = 1 is assumed.) +� +Table [ Length [ GraphTopology [ CycleGraph [ k ] ] ] , { k , 4 , 7 } ] +Table [ LucasL [2 n ] , { n , 4 , 7 } ] +� +� +29.3. +Here we compute the Euler characteristic and Wu characteristic of star graphs: +� +Table [ s=StarGraph [ k ] ; { Chi [ Whitney [ s ] ] ,Wu[ Whitney [ s ] ] } , { k , 3 , 1 0 } ] +� +� +29.4. +Here we compute the Wu characteristic of the basis of a random graph +� +s=RandomGraph [ { 1 5 , 5 4 } ] ; Map[Wu, GraphBasis [ s ] ] +� +� +29.5. +This is the code for Figure 1 +� +e={1−>2,2−>3,3−>1,3−>4,3−>6,3−>8,8−>9,8−>10}; V=ViewVertical ; +s=UndirectedGraph [ Graph [ e ] ] ; BG=BarycentricGraph ; +A=GraphPlot3D [ s , ViewPoint−>{1,−3,−1},V− >{1 , −1 , −0.3}]; +B=GraphPlot3D [BG[BG[ s ] ] , ViewPoint−>{0,−2.5,−2},V− >{1 ,0 ,0}]; +S=GraphicsRow [ {A,B} ] ; +Export [ ” f i g u r e 1 . pdf ” ,S , ”PDF” ] ; +Show[ S ] +� +� +29.6. +We illustrate Gauss-Bonnet +ω(G) = +� +x∈G +ω(x)ω(U(x)) = +� +x∈G +w(x)ω(B(x)) +and � +x∈G ω(x)ω(S(x)) = 0. These formulas hold for any simplicial complex G. We have +seen an analog formula χ(G) = � +x∈G ω(x)χ(U(x)) for Euler characteristic ω1(G) = χ(G) +before. But it holds for any higher characteristic ωm(G). +� +s=RandomGraph [{44 , +2 2 0 } ] ; G=Whitney [ s ] ; +{ Timing [ FastWu [G] ] , +Timing [Wu[G] ] } +U=Basis [G] ; +S=UnitSpheres [G] ; +B=UnitBalls [G] ; +{Wu[G] ,Sum[w[G[ [ k ] ] ] ∗Wu[U [ [ k ] ] ] , { k , Length [G] } ] , +Sum[w[G[ [ k ] ] ] ∗Wu[ S [ [ k ] ] ] , { k , Length [G] } ] , +Sum[w[G[ [ k ] ] ] ∗Wu[B [ [ k ] ] ] , { k , Length [G] } ] } +� +� +29.7. +For small complexes, the direct Wu computation is faster. But already if G has several +hundred entries, the fast Wu computation is faster. In the first example, where the complex +had 59 elements, the direct computation was 4 times faster. In the third of the following +cases the fast Wu computation took 23 seconds while the Wu computation took 90 sections. +The complex had 1355 elements. +� +s=RandomGraph [{14 , +3 0 } ] ; G=Whitney [ s ] ; +{ Timing [ FastWu [G] ] , +Timing [Wu[G] ] } +s=RandomGraph [{44 , +2 2 0 } ] ; G=Whitney [ s ] ; +{ Timing [ FastWu [G] ] , +Timing [Wu[G] ] } +s=RandomGraph [{54 , +4 2 0 } ] ; G=Whitney [ s ] ; +{ Timing [ FastWu [G] ] , +Timing [Wu[G] ] } +� +� +43 + +FINITE TOPOLOGY +29.8. +In general, we measure ω(U(x)) + χ(B(x)) − χ(S(x)) ≥ 0. This is equivalent to +ω(U(x)) ≥ χ(U(x)). This is something, we have not been able to explain yet: +� +s=RandomGraph [{14 , +3 0 } ] ; G=Whitney [ s ] ; +U=Basis [G] ; +S=UnitSpheres [G] ; +B=UnitBalls [G] ; +Map[ Chi , U] − Map[ Chi , B] + Map[ Chi , +S ] +Map[Wu, U] − Map[ Chi , U] +� +� +29.9. +We also have an energy theorem for Wu characteristic +ω(G) = +� +x,y +g2(x, y) , +where g2(x, y) = ω(x)ω(y)ω(U(x) ∩ U(y)) is a Green function matrix. Unlike in the case of +Euler characteristic, where g1(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) was unimodular, the matrix +g2 is no more unimodular in general. The determinant is in general not 1. For manifolds, +we compute it here for a 3-sphere, a double suspension of a cyclic graph C4. +� +s=RandomGraph [{20 , +4 0 } ] ; G=Whitney [ s ] ; +n=Length [G] ; U=Basis [G] ; +g2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗Wu[ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +Print [{Wu[G] , Total [ Flatten [ g2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g2 [ [ k , k ] ] , { k , n } ] } ] ; +Print [ Det [ g2 ] ] +� +� +29.10. +There would be a lot more to explore. We can look for example at the sphere +Green matrix +s2(x, y) = ω(x)ω(y)ω(S(x) ∩ S(y)) +and compare it with the ball Green matrix +b2(x, y) = ω(x)ω(y)ω(B(x) ∩ B(y)) +and the star Green matrix +g2(x, y) = ω(x)ω(y)ω(U(x) ∩ U(y)) +for which we see that the trace and the total sum of all entries and the determinant are all +zero. Is there some significance to the nullities we see in the ball or sphere Green function +entries. If Wu is replaced with Euler we get Green function matrices s1, b,g1. +� +s=RandomGraph [ { 2 0 , 4 0 } ] ; +G=Whitney [ s ] ; n=Length [G] ;U=Basis [G] ; S=UnitSpheres [G] ; B=UnitBalls [G] ; +s1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [ S [ [ k ] ] , S [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +s2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [ S [ [ k ] ] , S [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +Print [{ Chi [G] , Total [ Flatten [ s1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ s1 [ [ k , k ] ] , { k , n } ] ,Det [ s1 ] } ] ; +Print [{Wu[G] , +Total [ Flatten [ s2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ s2 [ [ k , k ] ] , { k , n } ] ,Det [ s2 ] } ] ; +b1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [B [ [ k ] ] ,B [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +b2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [B [ [ k ] ] ,B [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +Print [{ Chi [G] , Total [ Flatten [ b1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ b1 [ [ k , k ] ] , { k , n } ] ,Det [ b1 ] } ] ; +Print [{Wu[G] , +Total [ Flatten [ b2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ b2 [ [ k , k ] ] , { k , n } ] ,Det [ b2 ] } ] ; +g1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +g2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ; +Print [{ Chi [G] , Total [ Flatten [ g1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g1 [ [ k , k ] ] , { k , n } ] ,Det [ g1 ] } ] ; +Print [{Wu[G] , +Total [ Flatten [ g2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g2 [ [ k , k ] ] , { k , n } ] ,Det [ g2 ] } ] ; a +Print [{ N u l l i t y [ g1 ] , N u l l i t y [ b1 ] , N u l l i t y [ s1 ] , N u l l i t y [ g2 ] , N u l l i t y [ b2 ] , N u l l i t y [ s2 ] } ] +� +� +44 + +OLIVER KNILL +29.11. +Gauss-Bonnet formulas for higher characteristic allows to compute higher invariants +more quickly. It still needs time. The homology 3 sphere is implemented with a simplicial +complex of 392 simplices. [3]. It has Euler and Wu characteristic 0. Its suspension has Euler +and Wu characteristic 2. The double suspension again has Euler and Wu characteristic 0. +We know by the double suspension theorem that the double suspension of the homology +sphere has a geometric realization that is homeomorphic to a 5 sphere. We unfortunately +can not compute the Wu cohomology yet as the complex is too large. We suspect that +Wu cohomology can distinguish the double suspension of the homology sphere from the 5 +sphere. The following computation still needs a few minutes to compute, even so we use +the Gauss-Bonnet version for Wu characteristic. Already in the case of the Suspension of +the homology sphere, the fast Wu computation is faster. For small complexes, the direct +computation is faster because the fast Wu procedure requires to pre-compute a basis of the +topology. +� +onesphere=Whitney [ CycleGraph [ 4 ] ] ; +moebius = Whitney [ GraphComplement [ CycleGraph [ 7 ] ] ] ; +c y l i n d e r=Whitney [ UndirectedGraph [ Graph [ +{1−>2,2−>3,3−>4,4−>1,5−>6,6−>7,7−>8,8−>5,1−>5,5−>2,2−>6,6−>3,3−>7,7−>4,4−>8,8−>1}]]]; +twosphere=Suspension [ onesphere ] ; +threesphere=DoubleSuspension [ onesphere ] ; +homologyB ={{1 ,2 ,4 ,9} ,{1 ,2 ,4 ,15} ,{1 ,2 ,6 ,14} ,{1 ,2 ,6 ,15} ,{1 ,2 ,9 ,14} ,{1 ,3 ,4 ,12} , +{1 ,3 ,4 ,15} ,{1 ,3 ,7 ,10} ,{1 ,3 ,7 ,12} ,{1 ,3 ,10 ,15} ,{1 ,4 ,9 ,12} ,{1 ,5 ,6 ,13} ,{1 ,5 ,6 ,14} , +{1 ,5 ,8 ,11} ,{1 ,5 ,8 ,13} ,{1 ,5 ,11 ,14} ,{1 ,6 ,13 ,15} ,{1 ,7 ,8 ,10} ,{1 ,7 ,8 ,11} ,{1 ,7 ,11 ,12} , +{1 ,8 ,10 ,13} ,{1 ,9 ,11 ,12} ,{1 ,9 ,11 ,14} ,{1 ,10 ,13 ,15} ,{2 ,3 ,5 ,10} ,{2 ,3 ,5 ,11} ,{2 ,3 ,7 ,10} , +{2 ,3 ,7 ,13} ,{2 ,3 ,11 ,13} ,{2 ,4 ,9 ,13} ,{2 ,4 ,11 ,13} ,{2 ,4 ,11 ,15} ,{2 ,5 ,8 ,11} ,{2 ,5 ,8 ,12} , +{2 ,5 ,10 ,12} ,{2 ,6 ,10 ,12} ,{2 ,6 ,10 ,14} ,{2 ,6 ,12 ,15} ,{2 ,7 ,9 ,13} ,{2 ,7 ,9 ,14} ,{2 ,7 ,10 ,14} , +{2 ,8 ,11 ,15} ,{2 ,8 ,12 ,15} ,{3 ,4 ,5 ,14} ,{3 ,4 ,5 ,15} ,{3 ,4 ,12 ,14} ,{3 ,5 ,10 ,15} ,{3 ,5 ,11 ,14} , +{3 ,7 ,12 ,13} ,{3 ,11 ,13 ,14} ,{3 ,12 ,13 ,14} ,{4 ,5 ,6 ,7} ,{4 ,5 ,6 ,14} ,{4 ,5 ,7 ,15} ,{4 ,6 ,7 ,11} , +{4 ,6 ,10 ,11} ,{4 ,6 ,10 ,14} ,{4 ,7 ,11 ,15} ,{4 ,8 ,9 ,12} ,{4 ,8 ,9 ,13} ,{4 ,8 ,10 ,13} ,{4 ,8 ,10 ,14} , +{4 ,8 ,12 ,14} ,{4 ,10 ,11 ,13} ,{5 ,6 ,7 ,13} ,{5 ,7 ,9 ,13} ,{5 ,7 ,9 ,15} ,{5 ,8 ,9 ,12} ,{5 ,8 ,9 ,13} , +{5 ,9 ,10 ,12} ,{5 ,9 ,10 ,15} ,{6 ,7 ,11 ,12} ,{6 ,7 ,12 ,13} ,{6 ,10 ,11 ,12} ,{6 ,12 ,13 ,15} ,{7 ,8 ,10 ,14} , +{7 ,8 ,11 ,15} ,{7 ,8 ,14 ,15} ,{7 ,9 ,14 ,15} ,{8 ,12 ,14 ,15} ,{9 ,10 ,11 ,12} ,{9 ,10 ,11 ,16} ,{9 ,10 ,15 ,16} , +{9 ,11 ,14 ,16} ,{9 ,14 ,15 ,16} ,{10 ,11 ,13 ,16} ,{10 ,13 ,15 ,16} ,{11 ,13 ,14 ,16} ,{12 ,13 ,14 ,15} , +{13 ,14 ,15 ,16}}; +homologysphere=Closure [ homologyB ] ; +Print [ FastWu [ homologysphere ] ] ; +Print [ FastWu [ Suspension [ homologysphere ] ] ] ; +Print [ FastWu [ DoubleSuspension [ homologysphere ] ] ] ; +� +� +29.12. +Here are some computations of Wu Betti numbers. We could not yet complete the +computation of the Wu cohomology for the double suspension of the homnology sphere. +� +WuBetti [ onesphere ] +(∗ +{0 ,1 ,1} +∗) +WuBetti [ twosphere ] +(∗ +{0 ,0 ,1 ,0 ,1} +∗) +WuBetti [ threesphere ] +(∗ +{0 ,0 ,0 ,1 ,0 ,0 ,1} +∗) +WuBetti [ moebius ] +(∗ +{0 ,0 ,0 ,0 ,0} +∗) +WuBetti [ c y l i n d e r ] +(∗ +{0 ,0 ,1 ,1 ,0} +∗) +WuBetti [ homologysphere ] +(∗ +not +yet +able +to +compute +∗) +WuBetti [ DoubleSuspension [ homologysphere ] ] +(∗ +d i t o +∗) +� +� +29.13. +Example 1: For the smallest positive dimensional example G = K2, we have the +simplicial complex G = {(1), (2), (1, 2)}. The basis has three elements and consists of +{{(1), (1, 2)}, {(2), (1, 2)}, {(1, 2)}}. The corresponding unit spheres are +{{(2)}, {(1)}, {(1), (2)}}. The topology has 5 elements +{{}, {(1), (1, 2)}, {(2), (1, 2)}, {(1, 2)}, G}. Any path graph is homeomorphic to this graph. +A homotopy reduction f : K2 → K1 given by 1− > 1, 2− > 1 is continuous. The topology +in K1 is {{}, (1)} and the inverse of every of the open sets is an open set in K2. There +45 + +FINITE TOPOLOGY +is no continuous surjective map from K1 to K2. Any Barycentric refinement of K1 is K1 +and a map on finite spaces can not increase cardinality. Also, any continuous map can only +decrease or preserve dimension. +29.14. +Example 2: All cyclic graphs Cn with n ≥ 4 are homeomorphic but Cn is not +homeomorphic to a path graph Pm. There is no homeomorphism as there would have to be +a continuous surjective map f : Cn → Pm. This is not possible because there are two unit +spheres S(x) in Pm for which the inverse image has 2 elements (a 0-sphere). As a 0-sphere S0 +is not homeomorphic to a 1-point graph (there is not even a surjective map from K1 to S0). +One can also see that Cn is not homeomorphic to Pm because the Euler characteristic does +not match. One can also see it from the fact that the fundamental groups do not match. +29.15. +Example 3: the definition of homeomorphism in finite spaces can be used to pro- +duce constructive verifications that two spaces are homeomorphic or not. To do so, cover +both spaces with balls which intersect in balls then try to match the balls up. Obviously, if +the maximal dimension of the two spaces is different they can not be homeomorphic. Let +us assume that we have two complexes G and H which are homeomorphic and both have +maximal dimension d, then the number of connected components of d-dimensional maximal +balls must be the same in both. Two star graphs S(n) and S(m) with different number +of rays can not be homeomorphic for example. The star graph S(n) has n different open +one-dimensional balls, while S(m) has m different connectivity components. +30. Questions +30.1. +The definition of “homeomorphism” proposed here seems have all the properties we +want: it has invariants like Euler characteristic, Wu characteristic, Lusternik-Schnirelmann +category (the minimal number of contractible sets which cover the space), Betti numbers, +Wu Betti numbers, cup length, Lebesgue dimension, connectivity type, separation properties +or being a manifold are the same for homeomorphic graphs. +30.2. +There are other notions which are not topological invariants like the number of k- +dimensional simplices fk, the eigenvalues of some Hodge Laplacian, curvature, inductive +dimension, average simplex cardinality, Dehn-Sommerville invariants for non-manifolds, the +Fermi characteristic φ(G) = � +x ω(x) which agrees with the determinant of the connection +Laplacian det(L) [36]. For Dehn-Sommerville, especially related to Gauss-Bonnet curvatures +[18, 25, 30]. We have shown for example that the Dehn-Sommerville property is invariant +under edge refinement, the join operation and Barycentric refinements. Also Poincar´e-Hopf +[19, 32, 33, 31] can be reformulated more conveniently in a topological frame work. +30.3. +A) One thing we could not explore yet whether the relative Wu characteristic +[25, 26] ω(G, H) for a subcomplex H defined as � +x∈G,y∈H,x∩y̸=∅ ω(x)ω(y) is depends on the +topology of H and on the embedding in G. The Wu characteristic ω(G) = � +x∩y̸=∅ ω(x)ω(y) +itself is a topological invariant. We would also like to know to compute the Wu characteristic +in a classical manner without triangulation. Obviously, just looking at the structure of open +covers does not work. What matters also are the local dimensions, the dimensions of the +covers. For example, if we glue two manifolds along a k-dimensionial part, then the dimension +of this connection matters. If two graphs C4 are glued along a point (one calls this a wedge +sum), we get the figure 8 graph G with ω(G) = 7. If we glue it along an edge, we get a +digital figure 8 graph H and ω(H) = 5. These two graphs are not homeomorphic because +46 + +OLIVER KNILL +there are unit sphere which are not homeomorphic. The vertex degrees of G are 2 or 4 while +the vertex degree of H are 2 or 3. +30.4. +In the context of calculus, there are questions about the minimal number of critical +points of locally injective functions. +In general, a critical point is a point x for which +S− +f (x) = {y ∈ S(x), f(y) < f(x)} is non-contractible. Points for which the Poincar´e-Hopf +index if(x) = 1 − χ(S− +f (x)) ̸= 0 are critical points but there can also be critical points of +index 0. The Lusternik-Schnirelman inequality assures cup(G) + 1 ≤ cat(G) ≤ cri(G) where +cup is the cup length (a homotopy and so topological invariant) and where cri counts the +minimal number of critical points, which a locally injective function can have. The Morse +inequalities count the minimal number of critical points of a Morse function can have. Also +here, one can ask whether this number is a topological invariant. More generally one can ask +whether the numbers ck counting the minimal number of Morse critical points of index k is +topological. The Morse inequalities produce the general bound � +k(−1)kbk ≤ � +k(−1)kvk. +30.5. +B) As the main focus of this note was a definition of homeomorphism, it would be +good to know more about redundancies in the definition. +We have played with various +versions of the definition. +• We tried first not to make any requirement about the unit ball and only have the +condition for unit spheres. +• An other modification would be to avoid talking about unit spheres and balls and +ask that every unit ball B(x) is homeomorphic to f −1B(x). (We have asked this +only for locally maximal simplices where the unit ball is a ball) This implies that the +boundary S(x) is homeomorphic to the boundary of the inverse f −1B(x). It would +also imply the for a maximal simplex where B(x) is a ball, the inverse f −1B(x) is a +ball. We did not want to use this as a definition however because B(x) is the same +so that we have have no induction to work with. It still make the definition local. +One could then postulate that all balls of the same dimension are homeomorphic. +But that is less elegant. +• We also tried to play with the requirement that f −1(B(x)) is contractible (which is +weaker than requiring it to be a ball) but we had difficulty from this to establish for +example that if two spaces are homeomorphic and one is a manifold, the other must +be a manifold. +C) The property that H = f(Gn) is a continuous image was defined as the property that +any unit sphere S(x) is homeomorphic to f −1(S(x)) for all x ∈ H and that the inverse of +the unit ball B(x) is a ball in Gn. Is this already is enough to establish that also G is a +continuous image of some g(Hm)? In dimensions 0 and 1 it is. +D) The Green function matrix gm(x, y) is still a bit of an enigma in the case when it is +defined by a general function hm(x, y) defining the energy. We see for example that g2 is +identically zero if h2 is anti-symmetric. We also see that if h(x, y) = 1 everywhere, then g is +invertible and positive definite. +References +[1] P. Alexandroff. Diskrete R¨aume. Mat. Sb. 2, 2, 1937. +[2] P.S. Alexandrov. Combinatorial topology. Dover books on Mathematics. Dover Publications, Inc, 1956. +Three volumes bound as one. +47 + +FINITE TOPOLOGY +[3] A. Bj¨orner and F.H. Lutz. A 16-Vertex Triangulation of the Poincar´e Homology 3-Sphere and Non-PL +Spheres with Few Vertices. +http://www.eg-models.de/models/Simplicial Manifolds/2003.04.001, 2003. +[4] R. Bott. Two new combinatorial invariants for polyhedra. Portugaliae Math., 11:35–40, 1952. +[5] N. Bourbaki. Topologie G´enerale. Diffusion, Paris, 1971. +[6] G. Burde and H. Zieschang. Development of the concept of a complex. In History of Topology. Elsevier, +1999. +[7] B. Chen, S-T. Yau, and Y-N. Yeh. Graph homotopy and Graham homotopy. Discrete Math., 241(1- +3):153–170, 2001. Selected papers in honor of Helge Tverberg. +[8] R.A. Piccinini D.L. Ferrario. 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The cohomology for Wu characteristics. +http://arxiv.org/abs/1803.06788, 2017. +[27] O. Knill. On a Dehn-Sommerville functional for simplicial complexes. +https://arxiv.org/abs/1705.10439, 2017. +[28] O. Knill. On the arithmetic of graphs. +https://arxiv.org/abs/1706.05767, 2017. +[29] O. Knill. The amazing world of simplicial complexes. +https://arxiv.org/abs/1804.08211, 2018. +[30] O. Knill. Dehn-Sommerville from Gauss-Bonnet. +https://arxiv.org/abs/1905.04831, 2019. +[31] O. Knill. More on Poincar´e-Hopf and Gauss-Bonnet. https://arxiv.org/abs/1912.00577, 2019. +[32] O. Knill. A parametrized Poincare-Hopf theorem and clique cardinalities of graphs. +https://arxiv.org/abs/1906.06611, 2019. +[33] O. Knill. Poincar´e-Hopf for vector fields on graphs. +https://arxiv.org/abs/1911.04208, 2019. +48 + +OLIVER KNILL +[34] O. Knill. Complexes, Graphs, Homotopy, Products and Shannon Capacity. +https://arxiv.org/abs/2012.07247, 2020. +[35] O. Knill. 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Sbornik N.S., 24(66):163–188, 1949. +Department of Mathematics, Harvard University, Cambridge, MA, 02138 +49 + diff --git a/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/load_file.txt b/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..abd1084d407898a2661352004da6ace0a4a8d41c --- /dev/null +++ b/OdE1T4oBgHgl3EQfaAQ0/content/tmp_files/load_file.txt @@ -0,0 +1,2724 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf,len=2723 +page_content='FINITE TOPOLOGIES FOR FINITE GEOMETRIES OLIVER KNILL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Besides exploring the definition in various contexts, we add some remarks like that the general Lefschetz formula works for any continuous map on any finite topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also noted that any higher order Wu characteristic as well as their cohomology are topological invariants which are not homotopy invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Energy theorems allow to express these topological invariants in terms of interaction energies of local open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' About 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When exploring finite geometries using finite topological spaces, one is challenged with the fact that homeomorphic finite topological spaces have the same cardinality, which is too rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A finite metric space produces the discrete topology which is too fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For the geodesic metric on a graph for example, the topology is totally disconnected and so does not reflect at all the connectivity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As probably first realized in 1937 by Alexandroff [1], non-Hausdorff finite topologies still can capture essential parts of a topology that is usually only explored using geometric realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' How can one avoid geometric realizations and still have a workable definition of homeomorphism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have explored a related notion in [23] using covers but use now classical finite topologies, accepting the fact that all reasonable finite topological spaces are Alexandroff and naturally non-Hausdorff if they capture connectivity properties of the space under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This working document has grown a bit longer than anticipated but has been a seed for further results like Green function formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As a remedy, we added summaries at the end of each section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The write-up is a contribution to a program of replacing continuum geometries in a finite set-up but with as little changes in notation as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We want to avoid geometric realizations because using the continuum is a rather serious step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is not just a philosophical obsession: the mathematics of topological manifolds has told lessons like that there are finite geometries G - and example is the simplicial complex obtained by taking the join of a discrete circle with a homology 3-sphere - which produces a geometric realization which is classically homeomorphic to the standard 5-sphere H even so from any finite point of view they are not homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They are geometries which have homeomorphic geometric realizations but should not be considered homeomorphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In some sense the mathematics of the Hauptvermutung [52] has indicated that the continuum can lead to identifications which are not expeected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A finite topology more honestly preserves details which geometric realizations do not see any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One could of course use piecewise linear geometry to capture what finite topology does, but also from a computer science point of view, it is desirable to have finite objects and finite data to deal with only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' All geometric objects are Date: January 8, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Topology, Simplicial complexes, Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='03156v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='CO] 9 Jan 2023 FINITE TOPOLOGY faithfully implemented using a finite amount of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The axiom of infinity is never used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite topologies by definition are always Alexandroff spaces [1], meaning that every point x has a smallest neighborhood U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When working with a sheaf over such a topology, we do not need to conceptualize direct limit constructions like “stalks” or “germs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case of simplicial complexes, the smallest atomic “Planck units of space” containing x is known as the “star” U(x) of the simplex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As connection calculus [36] illustrates, the topology of intersections of stars U(x) ∩ U(y) can be complicated, even so each U(x), U(y) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Indeed, the Euler characteristic of U(x) ∩ U(y) agrees up to a sign with the matrix entries of the inverse g(x, y) of the connection matrix L(x, y), which is 1 if the simplices intersect and 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In analogy with the Green functions in classical frame works, these numbers g(x, y) must be thought of the potential energies between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They can be rather arbitrary for large dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One of the reasons why small dimensional topology is so much different from larger dimensional ones is that there can be surprises in the topology of local “atomic parts of space”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While simple in small dimensions, the intersection U(x)∩U(y) can be entangled in a rather complicated way in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The figure shows a finite simple graph G and its second Barycen- tric refinement G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Whitney simplicial complex G of G has 17 sets x, leading to 17 basis elements U(x) = U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These open sets in G are minimal and called the stars of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure U(x) is the unit ball and its boundary S(x) = B(x)\\U(x) is the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The basis B generates a finite topology O with 3032 open sets and 3032 closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Most of the 217 = 131072 possible subsets of G are neither open nor closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology is not Hausdorff: one can not separate points which intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As every finite topological space, it is Alexandroff: every point x has a smallest neighborhood U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2 OLIVER KNILL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Graphs and simplicial complexes and finite topological spaces all provide model frame works in finite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The categories are closely related: if one of them is given, one can construct relatives in the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can get from a graph to a simplicial complex with the Whitney functor by assigning to the graph the vertex sets of complete subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The ˇCech nerve construction produces from a topological space a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1 From a simplicial complex, one can then construct a graph in which vertices are the sets of the complex and where two sets are connected if one is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Switching forth and back between complexes and graphs produces a Barycentric refinement of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' After identifying Barycentric refined complexes, simplicial complexes or graphs can serve the same purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We always construct the topology O on the simplicial complex G and denote individual points in G with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sub-simplicial complexes of G are then the closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In algebraic geometry, a similar constructions has led to the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While we deal here only with finite sets, most could be generalized to locally finite complexes, meaning that there is an upper bound on the number of elements in an atom U(x) and again define the basis by the smallest neighborhoods U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This local finite as- sumption corresponds in the continuum to the step to restrict to paracompact topological spaces, spaces where every open cover has a locally finite refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As for references in topology, see [5, 14, 46, 58] for topology, and especially [2], a text already using abstract sim- plicial complexes (introduced 1907 by Dehn and Heegaard [6]) and not the more commonly used definition using geometric realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Veblen [61] in 1922 defined a neighborhood of a k-simplex but still used geometric realizations and cites Poincar´e for introducing the notion of homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Veblen also used already the terminology of “stars”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every topology on a finite set is always an Alexandroff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is a topology, where points have smallest neighborhoods or alternatively where arbitrary intersections of sets are open too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of “Stars” was established in combinatorial topology like [1] but also entered some calculus textbooks like Whitney [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Still, as most texts, even Alexandroff look at it primarily at geometric realizations of cell complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff topology generalizes the co-finite topology for 0-dimensional complexes or the order topology of a general partially ordered set with a basis U(x) = {y, y ≥ x} [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As for simplicial complexes, see [8] or [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Simplicial complexes appearing in graph theory are covered in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The most important link between graphs and complexes is definitely the Whitney functor which assigns to a graph a complex which exactly has the topological properties which the graph suggests without leading to ambiguities like what we consider to be a face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In algebraic topology, the actual topology generated by the star basis has not obtained the attention it deserves but it appears, for example in [58] on page 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology defined in [57] is defined on the vertex set and of different nature, as even connectivity properties are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' See [43] for a review on finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also [59] is completely unrelated because the star of a vertex is defined by Stallings as the set of edges attached to it and graphs considered one-dimensional objects, a common perspective in the 20th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are quite a few other discrete frame works in finite geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We should mention Ivashchenko [15, 10] who translated Whitehead’s homotopy notion into concrete procedures in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It has been simplified in [7] and crucial for defining what a “sphere” is combinatorially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Morse approach is used in Forman’s discrete Morse theory [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In discrete combinatorics, one sometimes also looks a abstract simplicial 1The Whitney complex is also known as the face complex, clique complex, flag complex, or face poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3 FINITE TOPOLOGY complexes, in which the ∅ is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We use the frame work, where ∅ is not considered to be a simplex but where it is considered to be a (−1)-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The empty set itself is a simplicial complex, as it fulfills the axiom, but it also does not contain the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' All these definitions are compatible with the continuum, where d-spheres also have Euler characteristic 1 + (−1)d even for d = −1 and where simplices all are contractible and have Euler characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have explored the problem of defining homeomorphism within finite mathematics for a few years already, the first time more seriously in 2014 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Stars came up for us especially in the context of the Green star identity, which explicitly gives the matrix entries g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) of the inverse of the connection matrix L(x, y) = χ({x ∩ y}) attached to a simplicial complex G with simplices x with ω(x) = (−1)dim(x) and Euler characteristic χ(A) = � x∈A ω(x) of a subset A of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The connection matrix L involves the Euler characteristic of closed sets like {x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its inverse matrix g involves the Euler characteristic of open sets U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' See [29, 37, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sets U(x) = W +(x) are open sets and {x} = W −(x) are closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An important role plays the closure B(x) = U(x) called the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its boundary S(x) = δU(x) = U(x)\\U(x) called unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They all are closed sets and so simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A visualization of the star U(x) in two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Unlike what the picture suggests, we do not look at geometric realizations however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star of a point x ∈ G consists of all the simplices which contain x: that is U(x) = {y, x ⊂ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case of a finite simple graph G, we could also look at the topology O on its Whitney complex G, where the closed sets are the simplicial complexes coming from subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is a slightly rougher finite topology, because not all simplicial complexes are Whitney complexes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We are currently under the impression that the notion of homeomorphism discussed here is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It has shifted quite a bit while we were writing this text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A useful definition needs to be simple and lead to all expected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The goal had been to get a definition of homeomorphism which generalizes the definition used for one-dimensional complexes in topological graph theory [13], and which has the ability to identify different triangulations of obvious manifolds like the icosahedron 4 OLIVER KNILL and octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In one dimensions, the notion of homeomorphism is old and enters for example the Kuratowski theorem: two graphs G, H are called graph homeomorphic if there exists a graph isomorphism between some edge refined versions of G and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The story relating the finite and infinite is a bit tricky: more than a hundred years of work in the context of the Hauptvermutung has lead to surprises in the relation between finite and infinite models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite topology can contain more information than the topology to geometric realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Using the topology from realizations in Euclidean spaces allows for some surprising homeomorphisms in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Having a topology on simplicial complexes is useful as it allows to reformulate classical results in a more familiar language but within a finite frame work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example: given any simplicial complex G and any continuous map f : G → G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' then the Lefschetz fixed point formula [22] � x∈F if(x) = χf(G) holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' where F is the set of fixed points of f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' where the index is defined as if(x) = ω(x)sign(f|x) with sign(f|x) is the signature of the permutation which f induces on the simplex x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' and where χf(G) is the super trace on cohomology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' which is defined as � k≥0(−1)ktr(Uf|ker(Lk)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' where Lk is the Hodge Laplacian L = dd∗ + d ∗ d restricted to the linear subspace of functions on k-dimensional simplices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' where df(x) = � y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='|y|=|x|−1 sign(y|x)f(y) is the exterior derivative and Ufg(x) = g(f(x)) is the linear Koopman map that f induces on functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Note that the Lefschetz fixed point theorem [22] holds for all simplicial complexes and all continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It generalized the theorem [51], which is a result in one dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the continuum, one needs assumptions, like that there are only finitely many fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The discrete theorem had been formulated for graph endomorphisms [22] which produce continuous maps on the corresponding simplicial complex but the proof works also for continuous maps meaning for example that the map can contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Lefschetz fixed point theorem has two special cases: the first case is if f is the identity, where it becomes the Euler-Poincar´e formula � x ω(x) = � k(−1)kbk, where bk are the Betti numbers, the dimensions of the kernels of Lk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other special case is if the cohomology is trivial, meaning that only constant functions are in the kernel of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This applies for example if G is an arbitrary contractible complex and among manifolds with boundaries if G is a k-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is the discrete Brouwer fixed point theorem: every continuous map on a finite abstract simplicial complex that is a k-ball has at least one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other example where we always have fixed points is if G is an even-dimensional sphere and if f is continuous but preserves the orientation of the maximal simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These are results in finite mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' At no point, the concept of infinity is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One of the main points was to have a clear definition what we mean with homeomor- phism in finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Once one has such a notion, one can see what topolog- ical invariants are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' [4] defined combinatorial invariants as properties invariant under Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We are especially interested in numerical quantities, that are not homotopy invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example in the continuum is the analytic torsion [40], adapted from the continuum [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are other properties we believe to be topological like being a Dehn-Sommerville space [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For all higher characteristics [25], which are only defined in the discrete so far, there are topological expressions which could be used to compute them in the continuum as we write down expressions which hold for the smallest open sets which exist in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For the second characteristic, the Wu characteristic ω = ω2, we know omega(B) = (−1)k, where k is the dimension of a ball B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One way to see this is that 5 FINITE TOPOLOGY for d-manifolds with boundary ω(B) = χ(B) − χ(δB) which in the case of a d-ball is by the Euler gem formula 1−(1+(−1)d−1) = (−1)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To compute the Wu characteristic of a space, cover it by balls (which can have different dimensions but should match the dimension of the covered part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Within part of a ball not covered by different balls, the dimension of the un- derlying space should be the same than the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What makes Wu characteristic compatible with topology is that we have the valuation formula ω(U ∪V ) = ω(U)+ω(V )−ω(U ∩V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This allows us to glue different parts together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Note that this valuation formula does not work for closed sets which together with a formula � x,y ω(x)ω(y)ω(U(x)∩U(y)) is the major reason why Wu characteristic is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: A finite abstract simplicial complex G is a finite set of sets x such that G is closed under the operation of taking finite subsets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The cardinality |x| of x defines its dimension dim(x) = |x| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star U(x) = {y, x ⊂ y} of x ∈ G is the set of simplices containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is declared to be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The set B of stars U(x) together with ∅ is a topological basis for a finite topology on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex H is called a d-ball, if it is of the form G \\ U(x), where G is a d-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex G is contractible if there exists x ∈ G such that both the unit sphere S(x) = δU(x) = U(x) \\ U(x) and G \\ U(x) are contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An arbitrary set is called contractible if its closure is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex is called a d-sphere if it is a d-manifold and G \\U(x) is contractible for some x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' a complex is a d-manifold if every S(x) is a (d − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The complex G1 of all vertex sets of complete sub-graphs of the graph G1 = (V1, E1) with V1 = G and E1 = {(x, y), x ⊂ y, or y ⊂ x} is called the Barycentric refinement of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Define Gn = (Gn−1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A simplex x ∈ G is locally maximal if x ⊂ y implies y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex H is declared to be a continuous image of G if there exists a continuous surjective map f : Gn → H for some n such that (i) if U(x) ⊂ H for any locally maximal k-simplex x ∈ H has a pre-image whose closure is a k-ball and (ii) that every f −1S(x) ⊂ Gn is homeomorphic to the unit sphere S(x) ⊂ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Two complexes are homeomorphic if each is a continuous image of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These definitions are inductive either with respect to number of elements or dimension: the empty complex = void 0 = {} is the (−1) sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The complex 1 = {{1}} is contractible and the 0-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Topology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An abstract finite simplicial complex is a finite set G of non-empty sets that is closed under the operation of taking finite non-empty subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A finite simple graph G = (V, E) carries the Whitney complex G of G which is the set of vertex sets of complete subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While not all simplicial complexes come from graphs in such a way, 2 every simplicial complex G defines a finite simple graph G in which the sets x of the complex are the vertices and where two sets are connected by an edge if one is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G came from a graph G as a Whitney complex, the graph G1 obtained from G is the Barycentric refinement of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, the Whitney complex G1 of G1 is a simplicial complex, called the Barycentric refinement of the simplicial complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since we can switch between graphs and complexes while doing Barycentric refinements, the two concepts “graphs” and “complexes” can be interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We like to keep both graphs and complexes 2Examples are the (k − 1)-dimensional skeleton complex of the complete graph Kk which is a (k − 1)- sphere, also known as the boundary sphere of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6 OLIVER KNILL and see them equipped with finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Simplicial complexes are attractive mathematical objects because they have the simplest axiom system imaginable in geom- etry: there is only one single axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Graphs on the other hand are unmatched in providing geometric intuition, featuring accessibility, and being supported by computer algebra systems, much more than sets of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As part of the definition, Barycentric refined objects are all homeomorphic so that from a topological point of view, they are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The goal is to use standard notions of topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can not use the classical notion of homeomorphism for finite topological spaces because this forces the finite topologies to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We see all cyclic graphs to be homeomorphic for example or to see the icosahedron isomorphic to the octahedron as both are 2-dimensional complexes which approximate under Barycentric refinements more and more spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We will call a complex H a continuous image of G if there exists a continuous surjective map f : Gn → H such that for every x, the boundary S(x) of U(x) is homeomorphic to the boundary of f −1(U(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This allows to use induction with respect to dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also require that for locally maximal simplices x which have the property that the closure of the open set f −1(U(x)) is a ball, a simplicial complex which is obtained from a sphere by removing an open set U(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a continuous image of H and G is a continuous image of H, the two spaces are considered homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A finite abstract simplicial complex G is so always equipped with a finite topology O on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is understood in the classical sense: a topology contains the empty set and G, it is closed under finite intersections and closed under arbitrary unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the finite case, we of courses can avoid the “finite” word but all we do here can be generalized to infinite but locally finite simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We stick with the finite, because the text should be seen as part of a larger and more ambitious project investigating the question: which parts of geometry can be replaced with finite combinatorial notions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We hope to be able to define within finite sets of sets whether two simplicial complexes are homeomorphic or not and point out that this is sharp than the softer equivalence relation given by homeomorphic geometric realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The classical “homeomorphic notion” is too rigid for finite topological spaces as it forces a bijection between the atoms U(x) making up the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Applied to graphs it would require the graphs to be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We want cyclic graphs Cn with n ≥ 4 to be all homeomorphic for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We want an edge refinement of a graph to be homeomorphic deformations and capture the notion of homeomorphism which is used in graph theory when graphs are considered one-dimensional simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a finite abstract simplicial complex and x ∈ G is given, it defines the star U(x) = {w ∈ G, v ⊂ w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The collection B of all these stars together with the empty set ∅ is a set of sets B that covers G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The collection B is also closed under intersections because U(x) ∩ U(y) = U(x ∩ y) if x ∩ y is not empty and U(x) ∩ U(y) = ∅ else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It therefore defines a base for a topology O on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By definition in point set topology, a topology O is a set of subsets of G which (i) contains ∅, (ii) contains G and which is (iii) closed under finite intersections and (iv) closed under arbitrary unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A sub-base of the topology is the set 3Part of graph theory is accessible in secondary school education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Simplicial complexes on the other hand tend to appear first in college topology or algebraic topology courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The abstract version is more accessible because higher dimensional Euclidean spaces, usually only introduced in linear algebra courses, are not invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 7 FINITE TOPOLOGY of sets U({v}), with v ∈ V = � x x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is a sub-base because every base element U(x) is an intersection of such sets U(x) = � v∈x U({v}) and so generates the base from intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A suspension of graph G is the Zykov join of G with the zero sphere S0 (the graph with two vertices and no edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Doing this twice is a double suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A rational homology 3-sphere is a 3-manifold that has the same cohomology than a 3- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4 The double suspension G of a rational homology 3-sphere is a concrete example of a simplicial complex that is not a discrete 5-sphere because not all unit spheres are spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But G has a geometric realization that is a 5-sphere by the double suspension theorem of Edwards and Cannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The finite topology can distinguish complexes in the discrete, which are indistinguishable using the tool of topological realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5 Edward [9] works with the Mazur homology 3-sphere and shows that the double suspension is S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A consequence of the double suspension theorem is the existence of “exotic triangulations”: there are topological manifolds which are not equivalent to a piecewise linearly homogeneous polyhedron P meaning that for any x, y ∈ P, there exists a piecewise linear homeomorphisms h : P → P such that h(x) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A triangulated topological manifold M on the other hand only requires h(x) = y which is a local homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every smooth manifold has a PL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But a general topological manifold does not need to be homeomorphic to polyhedra: Casson gave 4-manifold counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Examples in higher dimensions have appeared more recently [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We start by defining some subsets in a finite abstract simplicial complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every x ∈ G defines the star U(x) := {y ∈ G, x ⊂ y} which is an open set and the core K(x) = W −(x) := {y ∈ G, y ⊂ x} which, unlike U(x) in general, is always a sub-simplicial complex of G and so a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure B(x) of U(x) of U(x) contains {x} = W −(x) and is called the unit ball of a point x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its boundary S(x) = δB(x) = B(x) \\ U(x) is a closed set called the unit sphere of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By definition, this is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, the boundary of any open set δU = U \\ U = U ∩ U c is closed because it is the intersection of two closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The unit sphere S(x) = δU(x) is in the language of simplicial complexes, also known as the link of x, (but it is usually only defined for 0-dimensional x so that we avoid the term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The open set U(x) = W +(x) = {y, x ⊂ y} and closed set K(x) = W −(x) = {x} = {y, y ⊂ x} are somehow dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closed set W −(x) is contained within S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case when x is locally maximal meaning that it is not contained in a strictly larger simplex, then S(x) is the boundary complex of the simplex x and so a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A general complex G is a sphere: there exists y ∈ G such that G \\ U(y) is contractible and also for every y ∈ S(x), the unit sphere S(y) within S(x) is a co-dimension-one sphere again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The terminology for graphs is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a finite simple graph G and a vertex v, we call the graph generated by the vertices w adjacent to v the unit sphere of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G comes from a simplicial complex G, then each vertex has a dimension and adjacent vertices are 4There are implementations of the homology 3-sphere with 16 maximal simplices, leading to a complex with 392 simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The corresponding graph has 2552 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The double suspension has 394 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5PL geometry in the continuum would capture the finite topology too but it also would use the continuum and as Euclidean spaces are used the concept of infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6The graph that can be constructed from the complex S(x) is isomorphic to the subgraph of all points in distance 1 to the vertex x in the Barycentric graph G1, the graph which is constructed from the complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8 OLIVER KNILL ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sphere S(x) is now the join of S+(v) generated by the vertices w for which w contains v in G and S−(v) is generated by all w which are subsets of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sphere S(v) of the graph is then the Zykov join [64] of S−(x) and S+(x) because the vertex sets are the disjoint union and every element in S−(x) is connected to every element in S+(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We like to think also of S±(x) as the stable and unstable manifolds in the “hyperbolic structure” defined by the Morse function f(x) = dim(x) of Morse index dim(S−(x)) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us define this more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join can also be defined directly for simplicial complexes by G + H = G ∪ H ∪ {x + y, x ∈ G, y ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let R be an ordered ring like Z, Q, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A function f : G → R is called locally injective if f(x) ̸= f(y) for every y ∈ S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A Morse function on a complex G is defined as a locally injective function function G which has the property that S− f (x) = {y ∈ S(x), f(y) < f(x)} is a (k − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its Morse index is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we do not want to refer to the graph and so to the Barycentric refined topology, we would require that S− f (x) is a simplicial complex which is a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The function f(x) = dim(x) is a special Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a d-manifold, then the graph S(x) in the graph G1 is a (d − 1)-sphere and agrees with the topological join of the two spheres S±(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The geometric realization of a Zykov join of two graphs agrees with the topological join of the geometric realizations Summary: A finite abstract simplicial complex G carries a finite topology O in which the stars B form a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A finite simple graph carries so a natural topology on its Whitney complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology is finite and so Alexandroff: every point x has a smallest neighborhood U(x), the star of the simplex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We like to think of them as atoms of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure B(x) of a star U(x) is is called the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its boundary S(x) is called unit sphere of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Continuity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The classical definition of continuity can be applied immediately to functions f between simplicial complexes f : G → H if we just silently assume the topology generated by stars as the natural topology on the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we talk about continuous maps f : G → H of graphs G, H, then rather looking at maps on the vertex sets V (G), V (H), we look at maps from its simplicial Whitney complex G of G to the simplicial Whitney complex H of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As usual in point-set topology, a map f is called continuous if the inverse f −1(A) of an open set A in H is an open set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A map is continuous if and only if the inverse image of closed sets is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The standard definition of continuity works well, but it was necessary to modify the notion of “homeomorphismI” in the finite as classically, homeomorphic finite topological spaces are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It would be unacceptable for example to consider an icosahedron and octahedron as being topologically different, or to consider a cyclic graph with 6 elements to be topologically different than a cyclic graph with 5 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Even Barycentric refined complexes would not be homeomorphic with the narrow definition from point set topology, requiring the two continuous maps which are inverses of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the context of simplicial complexes, a continuous map is a bit more general than a simplicial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The later is a map from G to H that preserves order and does not increase dimension if x ⊂ y then f(x) ⊂ f(y) and therefore satisfies dim(f(x)) ≤ dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A simplicial map must map 0-dimensional simplices to 0-dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A continuous 9 FINITE TOPOLOGY map f : G → H does not need to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A constant map which has as an image a positive dimensional simplex is continuous but not a simplicial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It does not necessarily map simplicial complexes into simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Most permutations f : GG of a simplicial complex are not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They scramble around the simplices without preserving the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But simplicial maps are always continuous: if f is a simplicial map, then the inverse image of any simplicial complex {y}) consists of unions of simplicial complexes {xk}, which is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Because the inverse image of any closed set is a closed set, the map is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any map from a 0-dimensional complex G to a complex H is always continuous because every set in G is both open and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Such a map neither does have to be injective, nor does it have to be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other extreme case is a constant map f(x) = c from a complex G to a complex H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is always continuous and a simplicial map if c is zero dimensional: the set f −1(A) is either empty (if c /∈ A) or then the entire space G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The image {c} is however not open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, even for finite topologies, a continuous map does not need to be an open map, a map that transports open sets into open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sometimes it is good to look at maps defined within graphs G = (V, E), H = (W, F) alone and not directly look at the simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A map f : V (G) → V (H) is contin- uous graph map f : G → H if e = (a, b) in G then either (f(a), f(b)) in E(H) or then that f collapses e collapses to vertex a = (a, a) in V (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Such a map f lifts to a continuous map on the corresponding Whitney simplicial complexes G → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It actually even lifts to a simplicial map because zero dimensional parts get mapped into zero dimensional parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every continuous map between graphs as just defined leads so to a continuous map on the Whitney complex: given x ∈ G(G), define f(x) = � v∈x f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is an element in the complex H of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If it was not, then there would exist a, b ∈ x such that (f(a), f(b)) /∈ E nor f(a) = f(b), contradicting the assumption that f is a continuous graph map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Again, we should tell that there are continuous maps between simplicial complexes G(G), H(H) of graphs G, H which do not come from continuous graphs maps, the constant map to a positive dimensional simplex is an example of a continuous map that does not come from a graph map because a continuous graph map necessarily maps zero dimensional parts to zero dimensional parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a continuous map between graphs f : G → H, the maximal dimension of the image graph f(G) ⊂ H is always smaller or equal than the maximal dimension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' More generally, any simplicial map from a simplicial complex to an other simplicial complex does not increase dimension on the image: if f : G → H is a continuous map between simplicial complexes then dim(f(x)) ≤ dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The dimension of the image of a continuous map can be strictly smaller of course: the constant map mapping every simplex in G to a single fixed vertex v (zero-dimensional simplex) in H is continuous and a simplicial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The constant map to a positive dimensional simplex however is continuous but not a simplicial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph homomorphism 7 is a map between graphs G = (V, E) and H = (W, F) such that it maps V into W and E into F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph homomorphism therefore maps simplices into simplices on the simplicial complex level and so defines a simplicial map and therefore a continuous map on its Whitney simplicial complexes: the inverse image of an open set U(x) in G(H) is open in G(G) because it is a disjoint union of sets U(yj) where yj are the set of 7To have “homomorphism” and “homeomorphism” so close in the landscape of words is unfortunate but it is very much entrenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The two terms rarely appear in the same context, but here they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10 OLIVER KNILL simplices which are mapped into x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' On the other hand, not every continuous map is a graph homomorphism because a graph homomorphisms by definition is not allowed to collapse an edge to a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Continuity of maps between simplicial complexes is defined as usual in topology: the inverse image of an open set is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For simplicial complexes, simplicial maps are continuous but the converse is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are continuous maps between simplicial complexes which are not simplicial maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Graph homomor- phisms define simplicial maps on their complexes and so are continuous too, but also here, the converse is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, simplicial maps or graph maps only can lower the maximal dimension: the dimension of G is larger or equal than the di- mension of f(G) ⊂ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But continuous maps between simplicial complexes do not need to lower the dimension as the constant map f(x) = c from a zero dimensional complex G to a positive dimensional complex H with a fixed positive dimensional c ∈ H shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Homeomorphism Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Two homeomorphic finite abstract simplicial complexes G, H are displayed, with vertices, edges and triangles filled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Both are a wedge sum of a 2-sphere, a 3-sphere and a 2-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order to show that the two spaces are homeomorphic, one can first show that all d-spheres are homeomorphic and all d-balls are homeomorphic and that if two pointed spaces are homeomorphic then their wedge sums are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The wedge sum of two path graphs can be a star graph or a path graph and they are not homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One- dimensional complexes are homeomorphic if and only if they are classically homeomorphic, that is if one can get from one to the other by a sequence of edge refinements or edge collapses that come from edge refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As pointed out in the introduction, the classical notion of homeomorphism is too rigid for finite topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Already pioneers like Poincar´e used in a combinatorial set-up at equivalence classes of finite geometries and considered Barycentric refinement complexes 11 FINITE TOPOLOGY equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8 So, to start with, we assume that two geometries which are Barycentric refinement of the other are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Barycentric refinement G1 = (V1, E1) of a graph G = G0 is a new finite simple graph defined as follows: the vertex set of G1 is V1 = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Barycentric refinement of a simplicial complex G is the Whitney complex of the graph defined by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The edge set is the set of pairs (x, y) for which either x ⊂ y or y ⊂ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can iterate the Barycentric construction and look at the Barycentric refinements Gn = (Gn−1)1 for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have just seen that a continuous map G → H can be lifted to a continuous map G1 → H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means now that every continuous map G → H between two graphs be lifted to a continuous map on the Barycentric refinements Gn → Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We repeat the new definition: if there is a continuous map f : Gn → H such that for every y ∈ H, the boundary of f −1(U(y)) is homeomorphic to the boundary S(y) of U(y) and for all locally maximal simplices y, the ball B(y) = U(y) in H has a pre-image f −1(B(y)) which is a ball, we say H is a continuous image of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We say G and H are homeomorphic if H is a continuous image of G and G is a continuous image of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A d-ball is a complex that is of the form S − U(z), where S is a d-sphere and z ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Remember that a d-sphere S is a d-manifold which when punctured (S \\U(z)) becomes contractible and that a d-manifold is a complex for which all unit-sphere is a (d−1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The empty complex is the (−1)-sphere and the 1 point complex contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An immediate consequence of the definition is that 0-dimensional complexes are home- omorphic if and only if they have the same number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that we require the inverse image of every B(x) = {x} to be a 0-ball which is a one-point complex K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This implies that f must be injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The map f therefore has to be a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This defines now both an equivalence relation between simplicial complexes as well as for graphs: it is reflexive and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To see transitivity of the relation, note that we have now a chain of maps Gn → Hm → Kl showing that K is a continuous image of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since the reverse holds also, the complexes K and G are homeomorphic, if G and H are homeomorphic and H and K are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of homeomorphism goes over to graphs G if we look at the Whitney sim- plicial complex G attached to the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology of G is then taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By definition then, the Barycentric refinements Gn of a graph G are all homeomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We could also restate this by noting that G and H are homeomorphic if and only if Gn and Hn are homeomorphic for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As an example: two cyclic graphs G = Cn and H = Cm with n, m ≥ 4 are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a triangulation of a compact manifold M and H is a triangulation of a compact manifold N such that G, H have a common Barycentric refinement or common edge refinement, then G, H are homeomorphic and M, N are com- binatorially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any combinatorial invariants, a term coined in [4] (meaning a property that does not change under Barycentric refinements) must also be a topological invariant (meaning a property that is the same for homeomorphic objects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8As pointed out earlier, PL-geometry [54] would do the job in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But we do not want to use Euclidean spaces, nor use of infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 9This is a recursive definition as it refer to homeomorphism smaller dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In zero dimensions, homeomorphic means equal cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12 OLIVER KNILL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is useful to reformulate the notion of homeomorphism as the property that there exists maps f : Gn → Hm and g : Hm → G which are both continuous and such that the homeomorphism works also locally in that the smallest spheres have pre-images which are homeomorphic and that the inverse of locally maximal unit balls are actual balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We still wonder whether the assumption on the locally maximal balls can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The definition includes it so that we can prove things, like that if two complexes are homeomorphic and one of them is a manifold, then the other must also be a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us dwell on this a bit more: if a map f : G → H is continuous, we could try to ask that all unit spheres S(x) = δU(x) are homeomorphic to δf −1U(x) and use this alone as a recursive definition for homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We would then start with the assumption that zero- dimensional complexes are homeomorphic if they have the same cardinality, the property δf −1U(x) homeomorphic δU(x) would then recursively lift the notion of homeomorphisms dimension by dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is still not clear whether this alternative definition alone would work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While the definition of homeomorphism is by design symmetric, we could also ex- plore and try assuming only direction: checking this for f : Gn → H in one direction only might already determine that G and H are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This works if G and H are one- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case, we need a map from Gn to H such that the vertex degrees of S(x) = δU(x) is the same than the vertex degrees of δf −1U(x) for every x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We come back to this question again at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: The complex H is declared to be a continuous image of G, if the natural surjective map from some Barycentric refinement Gn → G factors as Gn → Hm → G and the map induces homeomorphisms on unit spheres and maximal unit balls have pre-images that are actual balls, punctured spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If two complexes are continuous images of each other, we call them homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This notion defines an equivalence relation on complexes as well as an equivalence relation on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Closed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Closed sets of a graph G are subsets of G which themselves form simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Not all closed sets in the Whitney complex of a graph do have to be induced from a sub- graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The boundary S(x) of a maximal simplex for example is always closed but it is only a skeleton complex of a Whitney complex of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10 To see the correspondence between open and closed sets: note that any simplicial complex K in G has as a complement the union of all stars U(x) with V (x)∩V (K) = ∅ and this union is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a graph G, a closed set K contains all the simplicial complexes of subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case G = K3, where we have G = {{1, 2, 3}, {1, 2}, {2, 3}, {1, 3}, {1}, {2}, {3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology O = {U1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' · · · U19} has 19 elements: U1 = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U2 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U3 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U4 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U5 = {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U7 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U8 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U9 = {(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U10 = {(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U11 = {(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U12 = {(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U13 = {(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U14 = {(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U15 = {(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U16 = {(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U17 = {(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U18 = {(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G = U19 = {(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10Taking subgraphs as closed sets is motivated by the Zariski topology, where closed sets are algebraic subsets H of an algebraic variety G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 13 FINITE TOPOLOGY The number of closed sets {U c, U ∈ O} is of course the same than the number of open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sub-graphs: there is the empty graph of dimension −1, 7 graphs of dimension 0, 3 graphs of dimension 1 with one edge, 3 graphs withe one edge and one vertex, 3 graphs of dimension 1 with 2 edges and then the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This gives 1+7+3+3+3+1 = 18 sub-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The 1-dimensional skeleton complex C3 of K3 is a closed set in the topology which does no correspond to a sub-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its complement is the open set U((1, 2, 3)) = {(1, 2, 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let A be an arbitrary set of sets in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure of A is the smallest closed set (simplicial complex) which contains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11 For example, if A = {x} consists of a single simplex, then its closure A is the simplicial complex {y ⊂ x} generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is the Whitney complex of a graph, the closure of a set of simplices is often (but not always) the subgraph of the Whitney complex of the smallest sub-graph of G which contains all simplices x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The example of the closed set C3 = {(1), (2), (3), (1, 2), (2, 3), (3, 1)} in K3 which is the skeleton complex of the Whitney complex of K3 shows that not all closed subsets in the topology of G correspond to sub-graphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If X is a topological space and Y a subset, then Y is called locally closed if it is an intersection of an open set A and closed subset K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We think then of the induced topology of X on K by taking all intersections U ∩K as open sets in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By definition, every U ∩K is an intersection of a closed and an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Locally closed sets are not necessarily closed in the topology X but part of the Borel σ-algebra of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Locally closed sets are sets which are open in some closed subset K with induced topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In our context, where a closed set is a simplicial complex, a locally closed set is a set which is an open set in that simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It does not need to be open in the original topology O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Take for example the set which consists of a single simplex A = {x} which is not maximal, nor zero-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example is if x is a boundary edge in a Wheel graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This set is neither open nor closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But the set is locally closed because we can write it as an open set in the boundary sub complex K which is a circular graph complex K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The set A is an open set in K but not open in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The complement of A in K is closed in K as well as closed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, look at the closure of A which is the complete complex with 2 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its complement in K is open in K but not open in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us look at some examples: in a finite abstract simplicial complex, every single simplex {x} is locally closed because it is the intersection of the open set U(x) and the closed set W −(x) = {x}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the simplicial complex of K3, the set {{1}, {1, 2}, {1, 2, 3}} is not locally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' From the 128 possible subsets of G, there are 64 which are not locally closed and 64 which are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a one-dimensional simplicial complex, all subsets are locally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In K4, where we have 215 = 32768 possible set of subsets in G, where only 167 sets are open and 167 are closed, there are 3605 locally closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11This corresponds to a classical notion in the theory of simplicial complexes when they are considered as geometric realizations in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure contains all boundary simplices, meaning to look at all subsets of x and so also in the continuum it means to look at the smallest simplicial complex which contains K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12One could define an other topology, where the closed sets are simplicial complexes of sub-graphs but this is a rougher topology and closer to the Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14 OLIVER KNILL Summary: The finite topology on a complex is of Zariski type: sub simplicial com- plexes are the closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sub-graphs of a graph define closed sets of a graph but not all closed sets come from subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' On complexes coming from graphs, we could get a slightly rougher topology by declaring sets to be closed if their simplicial complexes are Whitney complexes coming from closed sub-graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finally, we have looked at locally closed sets, sets which are intersections of open and closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Compact 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Traditionally, a topological space is declared to be compact if every open cover has a finite sub-cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Compactness in this strict sense is not a very useful in finite topological spaces: using the definition, every set is compact (whether it is open, closed and even if it is neither).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It makes therefore much sense in a finite topological space to to identify compact sets with closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If topologies are considered for non-finite graphs, a set then would be compact in a complex or graph, if it is a closed and finite set defined by a finite abstract simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Again, it would make sense to consider being compact as a synonym for being finite and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In an infinite graph G, many (but not all) compact sets are represented by finite subgraphs of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Their Whitney complexes are then simplicial complexes and also closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' On a general graph (V, E) with no restrictions on V, E, one could also look at the slightly rougher topology in which finite subgraphs define closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are less closed sets then and therefore also less open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In zero dimensions, we would get the co-finite topology, which in the finite case agrees with the discrete topology and where compactness is equivalent to being finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have just seen that in general, like if one looks at topological spaces which are not so commonly used - finite topological spaces are examples - one has to be a bit more careful when using notions which involve compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The property that every open cover has a finite subcover is really not a good notion for compactness in the case of finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, a map is called proper if the inverse image of a compact set is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case of finite topologies with the standard definition of compactness, every map (even a not continuous one) would be proper, simply because the inverse image of any finite set is a finite set, which by the definition of having a finite cover, would be declared to be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, the notion of “proper” does not really say much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If one requires a compact set to be closed too, then on finite topologies, the notion of proper is the same than continuous because there, every closed set is compact and the fact that all inverse images of closed sets is closed is equivalent to continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Looking art finite topological spaces could be named “radically elementary topology” similarly as radically elementary probability theory covers a lot of traditional probability theory [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite structures are not that limiting, especially if one considers them in a non-standard frame work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In internal set theory IST for example, [48, 50] compact sets can be modeled by finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Compact simplicial complexes X therefore can be modeled by finite abstract simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Of course, the number of elements is non-standard if the set X is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If one looks at a continuous map from a compact topological space to itself, then in general there are infinitely many fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a situation like in the context of the Lefschetz fixed point theorem, one traditionally assumes that there are finitely many fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the non-standard frame work, one would just assume that the 15 FINITE TOPOLOGY number of fixed points is standard (an axiomatically defined term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This will then assure that also the sum of the Lefschetz numbers is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: In a finite topological space, every set is compact so that the classical notion is not very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every map would be proper for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It makes sense therefore to consider all closed finite sets as compact instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If one would consider infinite complexes, every finite sub-simplicial complex would be considered compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Especially, every finite subgraph graph defines a closed compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In non-standard analysis frame-works, ”compact topological spaces” can be treated like ”finite topolog- ical spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Connectivity 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph G is called path connected if for every two vertices a, b, there is a path e1 = (a, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , en = (vn−1, b) (a finite collection of edges) connecting a = v0 with b = vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A simplicial complex G is path connected if the graph G1 defined by the complex is path connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph is path connected if and only its topology O(G) is connected: here is the proof: if G = U ∪ V is the disjoint union of two open sets U, V , then there can not be simplex which contains x ∈ U and y ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that there is no path connecting x with y in the graph G1 and G is not path connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' On the other hand, if K1, K2 are not path connected components in G1, then their smallest open neighborhoods U1, U2 are disjoint and G is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In graph theory, connectivity is also called 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph is called 1-connected, if it is connected but there exists a vertex v which when removed, renders G \\ v discon- nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph K2 is 1-connected but its Barycentric refinement, the path graph P3 is not 1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A simplicial complex can be defined to be 1-connected, if is graph G1 is 1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If f : G → H is a continuous surjective map between graphs and G is 1-connected, then also H is 1-connected: to prove this, note first that H must be connected because the continuous image of a connected topological space is always connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If H was not 1-connected, all unit spheres S(x) in H would be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, also f −1(S(x)) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This does not generalize to higher connectivity in graph theory: it is possible that the continuous image of a 2-connected graph can be only 1-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As an exam- ple, take the kite graph G in which one edge has been removed from K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This graph is 2-connected, but a continuous map from G to K2 has an image that is only 1-connected but not 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: The topology of a complex is connected if and only if the complex is classically connected in the sense that the graph of the complex is path connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology of a graph on its simplicial complex is connected if and only if it is path connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex is connected if and only if its graph is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A continuous image of a connected complex is a connected complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If a graph is 1 connected, then a continuous image is still 1 connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Separation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology is called a Kolmogorov space or T0, if for any pair of distinct points x, y, there is at least one point who has a neighborhood not containing the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology O of a simplicial complex G is always a Kolmogorov space: this is clear in the case 16 OLIVER KNILL x ⊂ y or y ⊂ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case x ⊂ y, then U(y) does not contain x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If x ∩ y is not empty, then U(x) does not contain y and U(y) does not contain x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' However, the topology O is not Fr´echet = T1 if the dimension is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Fr´echet means means that for every two points x, y, there exists a neighborhood of x which does not contain y and a neighborhood of y that does not contain x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As an example, take x ̸= y with x ⊂ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now, only y can be separated from x but x can not be separated from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every neighborhood of x contains y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If the complex has positive dimension, the it is never Fr´echet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Of course, all zero-dimensional complexes are Fr´echet because the topology is the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology of a simplicial complex G is also not Hausdorff or T2 if the complex has positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In particular, the topology of a graph G is not Hausdorff if G has positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Two vertices x, y in a graph which are connected by an edge can not be separated by open sets in the complex: as x must contain U(x) and y must contain U(y) and these two open sets have an intersection U(x) ∩ U(y) which contains the edge {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology is also not normal = T4: two closed sets can in general not be separated by open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Examples are the closures x, y of two simplices x, y ∈ G that have a non- empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They are closed sets but every neighborhood of one intersects with any neighborhood of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One could think that the non-Hausdorff property is a handicap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' However, it can be a blessing in the context of connection calculus, when we consider higher order characteristics generalizing the Euler characteristic χ(G) = � x∈G ω(x) with ω(x) = (−1)dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While Euler characteristic satisfies for every subset A, B ⊂ G the valuation property χ(A ∪ B) = χ(A) + χ(B) − χ(A ∩ B), this property does no more hold in the case of Wu characteristic ω(G) = � x,y,x∩y∈G ω(x)ω(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But the valuation property will hold for open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A topology is called Alexandroff if every point x has a smallest non-empty open neighborhood U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In an Alexandroff topology, space has smallest atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Equivalently, an Alexandroff topological space has the property that arbitrary intersections of open sets are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Most topological spaces we are familiar with are not Alexandroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If a metric space is Alexandroff, it must be a discrete topological space as then, every single point needs to be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, any discrete topology and the indiscrete topology O = {∅, X} is Alexandroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any finite topology must be Alexandroff because the intersection of all open sets containing x is an open set U(x), containing the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Calling a finite topology a “finite Alexandroff topology” would be a pleonasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Still, it is good to use the name as Alexandroff was one of the firs who seriously considered finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To compare, note that the geometric realization |G| (in some Euclidean space) of a finite abstract simplicial complex G (or finite simple graph with the Whitney complex) is always a Hausdorff, because |G| is a closed subset of a Hausdorff topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We will see that looking at the topological realization of a complex loses some information like the topological nature of spheres in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Topological manifolds can in general not be described by one simplicial complex (or equivalence class of Barycentric refinements) alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is not a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Most topological spaces we look at, even compact ones like Cantor type sets can not be described by one single finite standard topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 13 13Nonstandard analysis teaches us however that we can describe it by a non-standard finite topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Standard finite topological spaces are then the spaces we look at here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The axiom system however 17 FINITE TOPOLOGY Summary: The topology of a complex or graph is Kolmogorov (T0) not Fr´echet (not T1), not Hausdorff (not T2) and not normal (not T4), but (like all finite topologi- cal spaces) is Alexandroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The non-Hausdorff property is in sharp contrast with the topology given by geometric realizations which are Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Dimension 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The maximal dimension of a simplicial complex is defined as maxx∈Gdim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The maximal dimension of a graph G is the maximal dimension of its Whitney complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have already seen that a continuous map f : G to H between simplicial complexes has the property that dim(f(x)) ≤ dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The maximal dimension of a continuous image of a complex G is therefore smaller or equal than the dimension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An open cover {Uj} of G is a set of open sets such that � j Uj = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A cover defines a ˇCech nerve graph, in which the sets Uj are the vertices and where two are connected if they simultaneously intersect in a non-empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The maximal dimension of this graph is called the dimension of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The minimum over all dimensions of covers of G is called the topological dimension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The following fact is an other reason why the topology associated to a simplicial complex or graph is the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topological dimension and the maximal dimension d are the same for every complex and every graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that we can cover the space with sets U({v}) with v ∈ V = � x∈G x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This cover can not be refined further because removing one would keep some vertex v uncovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The dimension of this cover is equal to the maximal dimension of G because if x is a simplex of dimension d, then all the open sets {U(v)}v∈G intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Therefore, the topological dimension of G is at at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every cover of G must contain all sets U({v}) with v ∈ V because otherwise {v} ∈ G would not be covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, the dimension is also at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The ˇCech graph of an open cover U of a graph or simplicial complex is defined as the graph in which the open sets U are the vertices and where two vertices are connected if they have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, the ˇCech graph of the cover {U({v}), v ∈ V } is the graph G itself and the ˇCech cover of the cover {U(x), x ∈ G} is the Barycentric refinement G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One usually looks at covers for which every of the open sets are contractible (every star U(x) is considered contractible because its closure B(x) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=') Summary: The topological dimension of a topology on a complex as defined by Lebesgue, agrees with the maximal dimension of the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The ˇCech graph of the base cover of a complex is the graph G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is the Whitney complex of a graph G, then the ˇCech graph of the cover {U(v)}v∈V (where V is the set of zero dimensional simplices) is the graph G itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Product 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every data structure, whether we deal with graphs, with simplicial complexes or with topological spaces has notions of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Shannon product G∗H [56] of two graphs G, H is the graph for which V (G∗H) = V (G)×V (H) is the Cartesian product of sets and where E(G ∗ H) = {((a, b), (c, d)), (a, c) ∈ E(G) ∪ V (G) or (b, d) ∈ E(H) ∪ V (H)}, meaning does not allow us to define the intersection of all standard open sets so that compact topological spaces do not have atoms U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18 OLIVER KNILL that two points are connected if both projections have the property that they project onto a vertex or edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Shannon product does not go over naturally to complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can of course look at the complex of the Shannon product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What is nice about the Shannon product is that it allows to see graphs as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have explored this a bit in [28, 39, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The box product topology 14 on G ∗ H is the finest topology on the simplicial complex of G ∗ H such that both the projections on G and H are both continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph topology of the product G ∗ H is in general much finer than the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can see this already for G = H = K2, where G ∗ H = K4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topologies of G and H have only 5 elements, while the topology of G ∗ H has 167 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph topology on U ∗ V is in general finer than the topology generated by the “cubes” U ∗ V where U, V are basis elements of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Some simplices in G ∗ H are of the form x∗y which is a (k+1)∗(l+1)−1-simplex if x was a k simplex and l was a l simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But not every simplex in G ∗ H is of the form x ∗ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, for G = H = K2, only the 0, 1 and 3-dimensional simplices in G ∗ Y are products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The 2-simplices (the triangles) in G ∗ H are not products because 2 = (k + 1) ∗ (l + 1) − 1 implies either k + 1 = 3 or l + 1 = 3 but there are no 2-simplices in neither G nor H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can see this also from the fact that there are 22 − 1 = 3 simplices in G and H and 24 − 1 = 15 simplices in G ∗ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Only 9 of them are of the form x ∗ y with x a simplex in G and y a simplex in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Shannon product does not preserve manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Stanley-Reisner product of two simpicial complexes G and H is defined as the Whitney complex of the graph in which the Cartesian product G × ×H are the vertices and where two different vertices (x, y), (u, v) are connected by an edge if either x ⊂ u, y ⊂ v or u ⊂ x, v ⊂ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Stanley-Reisner product of a p-manifold with a q-manifold is a (p + q)-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As the graph defining G × H is homotopic to the Shannon product, it inherits properties of the former like the K¨unneth formula or the compatibility with the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We will write more about the compatibility with higher characteristics elsewhere and especially show that they are topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can now ask whether there is not a natural ring structure on simplicial complexes which corresponds to the Shannon ring, or whether there is even a ring structure which preserves manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One problem is that if we take the Cartesian product of two simplicial complexes, we don’t have a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We would have to close it but it would not have the properties we like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Much more elegant is to expand the class from simplicial complexes to delta-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This structure is more general than simplicial sets, a popular construct which has more structure than δ-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every simplicial set of course is also a delta set by just forgetting the degeneracy maps si maps and only keep the face maps di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The disadvantage of working with δ sets is that we have to carry around not only sets of sets but also keep track of the maps di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' And the entire elegance of having a simple topology etc is gone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' δ sets also are useful when describing quivers, which generalize finite simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is a Whitney functor from quivers to δ sets generalizing the functor from graphs to simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 15 14For finitely many products, the box product topology agrees with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 15In order to talk about funtors one needs to adapt the morphisms on graphs and simplicial complexes and continuous maps are the most natural common denominator as both simplicial maps as well as graph homomorphisms are continuous maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 19 FINITE TOPOLOGY Summary: The projections from the Shannon product G∗H of two graphs to one of its factors is a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph topology of the product graph is in general much finer than the product topology in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Shannon product does not preserve topological quantities like higher characteristic or manifold properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Stanley- Reisner product however does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Stanley-Reisner product is more compatible with topology but does not provide a ring structure as associativity fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Shannon product on the other hand defines a ring and so an arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Join 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join G ⊕ H of two graphs G, H has as vertex set V (G) ∪ V (H) and as the edge set E(G) ∪ E(H) ∪ {(a, b), a ∈ V (G), b ∈ V (H)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join operation in graphs theory was first defined by Zykov [64] and does exactly what the join does for geometric realizations of the Whitney complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join with a 0-sphere is a suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, the join of two spheres is a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The proof follows from the sphere formula SG⊕H(x) = SG(x) ⊕ H and SG⊕H(y) = G ⊕ SH(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' so that by induction if each G, H, SG(x), SH(y) are spheres, then the unit sphere of any point in G ⊕ H is a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join of a graph with the 0-sphere S0 is called the suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since S0 ⊕ S0 = C4 is a cyclic graph and so a discrete sphere, the join of a graph with a cyclic graph is a double suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join of two simplicial complexes G, H can be defined also without referring to their graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The simplices in G⊕H are the union of G, H and G⊕H where x⊕y = x∪y is a k+l+1-dimensional simplex obtained by taking the disjoint union of the two sets x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If x had k+1 elements and y had l+1 elements, then x⊕y has k+1+l+1 = k+1+2 = (k+l+1)+1 elements so that k + l + 1 is the dimension of x ⊕ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, for example, if G = {a, b} is the zero sphere and H = {1, 2, 3, 4, (12), (23), (34), (41)} is a discrete circle, then G ⊕ H =} = {a, b, 1, 2, 3, 4, (12), (23), (34), (41), (a1), (a2), (a3), (a4), (a12), (a23), (a34), (a41) (b1), (b2), (b3), (b4), (b12), (b23), (b34), (b41)} is the suspension of a circle and the octahe- dron complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join operation is dual to the disjoint union G+H as addition: if G′ denotes the graph complement of G in which edges and non-edges are switched, then (G⊕H)′ = G′+H′, where G + H is the disjoint union of the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G1 is the Barycentric refinement of G, then the unit sphere S(x) of a point x ∈ G = V (G1) is of the form S+(x) ⊕ S−(x), where S+(x) = {y, x ⊂ y} is the unstable sphere and S −(x) = {y, y ⊂ x} is the stable sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a discrete manifold, where every unit sphere is a sphere, both the stable and unstable spheres are spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G, H are graphs and if x is a simplex in G and y is a simplex in H, then x ⊕ y is a simplex in G⊕H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, the Whitney complex of the Zykov join G(G⊕H) = G ⊕H is the join of the simplicial complexes which is the G ∪ H ∪ {x ⊕ y, x ∈ G, y ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology of the join by definition has as a basis the sets UG⊕H(z), where z = x ⊕ y is a simplex in G ⊕ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is UG⊕H(x ⊕ y) = UG⊕H(x) ⊕ H ∪ G ⊕ UG⊕H(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The total set of stars U(x ⊕ y) is a basis of G ⊕ H and generates the topology of G ⊕ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 20 OLIVER KNILL Summary: The embedding of G in G ⊕ H with the induced topology is a classical homeomorphism onto the image, as it is a bijection onto the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The join of the topological base in G and H defines a base for the join G ⊕ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The smallest atomic open sets U(x⊕y) in G⊕H is a basis: the base of the join contains open sets U(x)⊕H as well as open sets G ⊕ U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Subgraph 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have seen that a subcomplex H of a complex G is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology induced from G on a subcomplex H is the topology of H itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A general subset H of G can still be given a topology by taking as open sets U ∩ H with U in the topology of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This topological space agrees with the closure of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similar standard consequences hold for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any subgraph H of G has a topology which agrees with the induced topology from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A subset A of G(H) is open if and only if it is of the form O ∩ G(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The simplicial complex of a subgraph A of G is a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As in general when we take the relative topology on a closed subset K of a topological space X, the relative topology has the sets U ∩ K as open sets, where U ranges over the open sets in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' All the axioms for a topology are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The relative topology is the finest topology on K which has the property that the inclusion i : K → G is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To reformulate this, the relative topology on a subgraph H of G agrees with the graph topology of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We only have to look at a basis to see this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If x is a simplex in H, then UH(x) is the intersection UG(x) ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, the basis for the topology on H is the same than the restriction of the basis of the topology on G to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This proves the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The subgraph H can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As for any topological space on some set X we can restrict the topology on any subset Y of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, we can build a topology on any subset of the simplicial complex G of a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If a subset W ⊂ V of the vertex set of a finite simple graph G = (V, E) is given, one can look at the subgraph H generated by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means to take the largest subgraph of G which contains the vertex set W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One could look at the closure of a subgraph H as the subgraph generated by the vertex set of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is in general a much larger graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a Hamiltonian subgraph of G (a graph which passes through all vertices) for example this “closure” would be the graph itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology defined by the simplicial complex however make a subgraph naturally closed already as the simplicial complex H generated by the simplices in H is a sub-simplicial complex of the complex G of G and so closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Sub simplicial complexes correspond to closed sets in the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sub- graphs of a graph define a subclass of closed sets in the topology on the simplicial complex defined by the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The relative topology on a subgraph of a graph is the graph topology of H itself and does not use the topology of the host graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The relative topology on a subcomplex of a complex is the topology of the subcomplex itself which is the same without looking at the ambient space G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Quotient 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a graph and ∼ is an equivalence relation on vertices honoring the edges, then the set of equivalence classes H = G/ ∼ can carry a topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' First of all, the 21 FINITE TOPOLOGY equivalence relation induces an equivalence relation on complete subgraphs and x/ ∼ is the complete graph on the set of equivalence classes of V (x)/ ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Define G(H) as the set {x/ ∼, x ∈ G(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G = K2 with G = {{1, 2}, {1}, {2}} for example and ∼ identifies the two points 1, 2, we get G(H) = {{1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Assume G is a cover of H, meaning that there is a surjective graph homomorphism f : G → H, then we can see H as a quotient G/ ∼, where v ∼ w if f(v) = f(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example is the cover C8 → C4 with f(v) = v mod 4 if V (C8) = {0, 1, 2, 3, 4, 5, 6, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other example is the cover S2 → P 2 of a sufficiently large 2-sphere for which the equivalence relation defines a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Some 2-spheres are too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For an octahedron O for example, a graph with 6 elements, identifying opposite vertices produces not no projective plane but O/ ∼= K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order that an equivalence relation on the sets of simplicial complex G produces a a quotient topology, one needs to make some assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a Barycentric refinement, things are easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Having the complex G too small can make things weird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Lets look for example the cycle complex C4 = {1, 2, 3, 4, (12), (23), (34), (41)} and impose the equivalence relation where we identify the vertices 1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The quotient is no more a simple graph, but a quiver because multiple connections appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have now a graph with three vertices 1, 3, 4 and double bond connections (13) and double bond connections (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can however look at the situation in the Barycentric refinement, where the identification becomes now a figure 8 graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This situation matters if we look at Riemann-Hurwitz formulas which relate the Euler characteristic of the quotient with the Euler characteristic of the complex itself as well as using ramification points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In a case of a covering and having a group A of order |A| acting on G so that G/A is again a complex, then The Riemann-Hurwitz formula tells χ(G/A) = χ(G)/|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, if A = Z2 acts on a sphere G and H = G/A is a projective space, then χ(H) = χ(G)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But if the complex is too small like for the Octahedron complex considered above, then G/A is not a complex any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For the Barycentric refinement however it works and we get like that complexes representing a projective plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: The topology on a quotient H = G/ ∼ can be defined as usual in topology as the finest topology which makes the projection from G to the space of equivalence classes continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If H is a simplicial complex, then its topology is the quotient topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If quotients come from covers, then we have completely analogue situations like in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can for example look at the quotient of an antipodal map on a Barycentric refined sphere and get a finite topological space representing a projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Manifold 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph G is called a d-manifold if every unit sphere S(x) in G is a (d − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A d-sphere is a d-manifold such that for some vertex v, the graph G − v without v is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A graph is contractible if there exists v such that S(v) and G − v are both contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These notions can be defined also for complexes without referring to graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex G is a d-manifold if every unit sphere S(x) = B(x) \\ U(x) is a (d − 1)- sphere, where B(x) = U(x) is the unit ball the closure of U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The unit S(x) is always closed and so carries a simplicial complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A d-sphere is a d-manifold G such that G \\ U(x) is contractible for some x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex is contractible if there exists x such that S(x) and G \\ U(x) are both contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can extend contractibility to non-closed sets 22 OLIVER KNILL by defining for example an open set to be contractible if its closure is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every star U(x) is contractible with this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 16 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' All these inductive definitions are primed by the assumption that the empty graph 0 or the empty complex 0 is the (−1)-sphere and that the one point graph 1 = K1 or 1-point complex 1 is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a graph that is a d-manifold, then all its Barycentric refinements Gn are d-manifolds too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a complex which is a d-manifold, then all their Barycentric refinements are manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Examples of discrete manifolds in the sense just defined are combinatorial triangulations of a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But not all triangulations are manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The tetrahedron K4 for example is contractible and so not a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We could look at the 2-skeleton complex of K4 however and get a sphere complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Barycentric refinement of K4 is a 3-ball with 2-dimensional boundary which corresponds to the Barycentric refinement of the 2-skeleton complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us add a remark coming from the continuum: every PL manifold (a mani- fold equipped with a PL-structure) admits a combinatorial triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The question of Poincar´e from 1899, whether every smooth manifold admits a triangulation has been answered positively in the 1930ies: every smooth manifold has an essentially unique PL- Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (The converse is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are PL-manifolds which can not be smoothed or admit different smooth structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=') The question shifted then to the topological situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' See [16] and especially [55] for history or [42] for more recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also PL struc- tures do not exist in general on topological manifolds in dimensions 4 or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 4-manifolds examples were given using the Kirby-Siebenmann invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Non-PL triangulations of man- ifolds were constructed using the Edwards-Cannon double suspension theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The fact that the Barycentric refined graph G1 obtained from the Whitney complex G of a d-manifold graph G is a d-manifold can be proven by induction with respect to dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Indeed, G is a manifold if and only if G1 is a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G was not a manifold, then some unit sphere S(v) in G would not be a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But then also SG1(v) which is the Barycentric refinement of S(v) would not be a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By induction assumption (unit spheres are one dimension smaller) this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G and H are homeomorphic and G is a manifold and H is not, then every Gn is a manifold and non of the Hm are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Take a unit sphere S(v) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since its inverse image is homeomorphic to S(v) it is a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By definition, it is the boundary of a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' All unit spheres of interior points in this ball are by definition spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now, every vertex in G\\ is in the interior of the inverse of a ball B(x) in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The following statement in the summary is not true if “homeomorphic” would be replaced by “has a homeomorphic geometric realizations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of homeomorphism proposed here is probably is equivalent to PL-equivalent but we do not prove this because we don’t deal with infinity here: Summary: If G, H are homeomorphic and G is a d-manifold then H is a d-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 16We identify collapsible and contractible and use homotopic to 1 if we mean that a complex can be deformed to 1 = K1 by both expansions and contraction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While homotopic to 1 is a computationally difficult equivalence relation, contractibility is easy to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23 FINITE TOPOLOGY 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Contractible 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The concept of “contractible” entered in a crucial way in the definition of “sphere” and so in the definition of manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion 17 makes sense for general graphs and general simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is different from homotopic to 1, where one can do both homotopy extensions and contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The dunce hat graph is a concrete example of a finite simple graph which is homotopic to a point which is not contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18 The graph G is contractible if and only if G1 is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A complex G is contractible if and only if G1 is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Contractibility can be extended to non-closed sets by assuming the closure to be contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The continuous image of a contractible graph does not need to be contractible: an example is the map from the linear graph Ln+1 to the graph Cn mapping the initial and end point to the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While Ln+1 is contractible, the graph Cn is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is also not true that if H = f(G) is contractible then G is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let f map a 0-sphere G = {V = {a, b}, {}} to K1 = {{a}, {}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is continuous because both graphs have the discrete topology but the 0-sphere is not contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: As in the continuum, homotopy transformations are not continuous in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But contractions = homotopy reductions of graphs f : G → G−v or complexes G → G \\ U(x) are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the Unlike the unit spheres S(x), the balls B(x) are always contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Boundary 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A d-manifold with boundary is a graph or complex with the property that every unit sphere is either a (d − 1)-sphere or is a (d − 1) ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The same definition applies for simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If S(x) is a (d − 1) sphere, we have an interior point x, if S(x) is a (d − 1)-ball, x is a boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a manifold with boundary, the boundary is a (d − 1) manifold without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example of a manifold with boundary is a d-ball which by definition is a d-sphere with a point removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other example is a complete graph Kd+1, where all points are boundary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can include Kd+1 into the class of manifolds with boundary just in order to have Barycentric invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We want a graph to be homeomorphic to its Barycentric refinement and in general to have a graph G homeomorphic to H if there exist continuous maps Gn → Hm → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can say: Summary: If G is a manifold with boundary and G is homeomorphic to H, then H is a manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The same holds for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Duality 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Continuity is not compatible with some duality notions in graph theory or the theory of simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The operation of mapping a graph to its graph complement is in general not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Already the dimensions do not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph complement of a cyclic graph Cn is always homotopic to a sphere or then to a wedge sum of two spheres 17Again: we avoid the term collapsible used often in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18The Dunce hat can be realized as a graph G with 17 vertices, 52 edges and 36 triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its unit spheres are all either 1-spheres or homeomorphic to figure eight graphs (wedge sums of two 1-spheres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are homotopy expansions which make it contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24 OLIVER KNILL [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Only for special cases like C5, where the graph complement is the same graph, the complement operation can be made to be a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can ask however whether the graph complement operation maps homeomorphic graphs to homeomorphic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But also here the answer is no: let f : G → H be a continuous map between finite graphs like for example f(x) = x mod 4 from G = C8 to H = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Does there exist a continuous map from Gc to Hc?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph complement of C8 is homotopic to a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph complement of C4 is a disconnected union of two graphs K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since Hc is disconnected and Gc is connected and a continuous map can not map a connected space to a disconnected space (as continuity preserves the property of being connected), we can also not see the graph complement operation as a map preserving homeomorphic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us reformulate this a bit differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have identified simplicial complexes or graphs which are Barycentric refinements of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of graph complement is not at all compatible with the Barycentric refinement notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Barycentric refinement of C4 is C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But there is no topological similarity between Cc 4 which is homotopic to a 0-sphere and Cc 8 which is homotopic to a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For simplicial complexes, there is the Alexander duality operation: if G is a complex and V � x∈G x is the set of 0-dimensional simplices, then the Alexander dual of G is the complex {y ⊂ V, (V \\ y) /∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' one can ask whether there is a duality notion which corresponds to the graph com- plement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a graph, we can look at the simplicial complex G of Gc but that does have little to do with the simplicial complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can experiment with other notions like if G is an arbitrary simplicial complex and V = � x x is the vertex set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can look at the complement Gc of G in the complete complex on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is a duality notion, but of course, Gc is almost never a simplicial complex and the closure Gc of G is a complex but G → Gc is not a duality notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We see that graphs have their purpose especially with respect to arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: While interesting for other reasons, duality notions like graph comple- ment or Alexander duality are not compatible with homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topic of duality also shows that having different data structures for finite geometries is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The graph complement duality for graphs for example works well with arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It provides an isomorphism between the Sabidussy ring with join and large product as operations, and the Shannon ring with disjoint union and Shannon product as opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Edge refinement 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Edge refinement are topological transformations of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They induce topological modifications of simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is best described on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Given a graph G and an edge e = (a, b), we can refine the graph by adding a new vertex v, remove e and connect v to the intersection of S(a) and S(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When applied to the cyclic graph Cn, it produces Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When applied more generally to a discrete d-manifold, we get a new d-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that the new unit sphere S(v) is the join of the 0-sphere {a, b} and the (d − 2)-sphere S(a) ∩ S(b) and so again a (d − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The unit spheres S(a) and S(b) are not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The unit spheres of vertices w with the edge e = (a, b) in the unit sphere are themselves 25 FINITE TOPOLOGY edge refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Using induction with respect to dimension, one has now verified that edge refinements preserve basic invariants like Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In full generality, an edge refined graph Ge is homeomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' First we can define a surjective continuous map from Ge to G induced from the rule that every vertex goes itself and that the new vertex goes to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order to see that this map is continuous, check the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The inverse of every star in G is homeomorphic to a star in Ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star U(x) of x = {a} has as as an inverse image the union of the star of a and star of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star of the edge (a, v) has the empty set as an inverse image, The star of the edge (v, b) has as an inverse image the star of (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, the star of any simplex containing (a, v) has an empty inverse image while the star of any simplex containing (v, b) has as the inverse image the star of the corresponding simplex containing (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To check the other direction, we have to construct a continuous map from the Barycen- tric refinement G1 of G to Ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can take the canonical homeomorphism projection map from G1 to G and modify it so that it becomes a map from G1 to Ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We should mention that also the Dehn-Sommerville property is preserved by edge refinement and Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Dehn Sommerville spaces generalize spheres and like spheres produce a monoid under the join operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They therefore can be used to generate spaces more general than manifolds but still have many properties of manifolds like that odd-dimensional manifolds have zero Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Dehn-Sommerville d- spaces Xd [27] are inductively defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They must have the property that χ(G) = 1+(−1)d and that all their unit spheres satisfy S(x) ∈ Xd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The induction starts with X−1 = {}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Having the class Xd−1 of (d−1)-dimensional Dehn-Sommerville spaces topologically invariant immediately bootstrap to see that also d-dimensional Dehn-Sommerville spaces have the property that they are invariant under homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: The edge refinement operation produces a homeomorphic graph and so of its Whitney complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If a graph is a d-manifold, then the edge refined graph is a d-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can look at edge refinements as local Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also mentioned that Dehn-Sommerville spaces, a class of graphs generalizing spheres and like spheres forming a submonoid of all complexes, are topological in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A homeomorphic sibling of a Dehn-Sommerville space is Dehn-Sommerville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Fundamental group 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A closed curve in a graph G can be defined as a continuous map from a circular graph Cn to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that the vertices x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xn = x0 are mapped into vertices y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , yn = y0 such that either yi = yi+1 or (yi, yi+1) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The fundamental group of a graph equipped with a reference point v is defined as the equivalence classes of closed curves in G starting at v modulo curve homotopy deformations, where two curves are called curve homotopic, if they can be morphed into each other by homotopy steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A homotopy step is an operation, where an edge of the path attached to a triangle t (an embedded complete graph K3) is replaced with the two other sides or then reverses such a homotopy deformation and replaces two edges of the path in a triangle with the other edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If f : G → H is a continuous map, then a closed path maps either into a closed path or 26 OLIVER KNILL a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The following result mirrors corresponding results in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is however a statement in finite topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Replacing 1-spheres by a d-sphere S equipped with a base point, one can look at sphere embeddings continuous images of S attached to a base point in G and so look at homotopy groups πn(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sum of two such embeddings S1, S2 can be obtained by embedding a wedge sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We were once interested in graph complements of circular graphs [38] because there, all higher dimensional wedge sums of spheres appear (at least homotopically equivalent) as graphs complements of cyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It would be nice if one could use this to compute higher homotopy groups better but this has not worked yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: If f : G → H is a continuous map on graphs, it induces a group homo- morphism f∗ : π1(G) → π1(H) on the fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Euler characteristic 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Euler characteristic of a finite abstract simplicial complex G is defined as χ(G) = � x∈G ω(x), where ω(x) = (−1)dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The quantity can be seen in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is first of all a valuation, meaning that it satisfies χ(G ∪ H) = χ(G) + χ(H) − G ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If fk(G) counts the number of k-dimensional simplices then also χ(G) = �∞ k=0(−1)kfk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G = (V, E) is a graph, its Euler characteristic χ(G) is defined as the Euler characteristic of its Whitney complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If f : V → R is a locally injective function on vertices, then if(v) = 1 − χ(S−(v)) is the PoincarHopf index of f at the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By induction, one can check the Euler-Poincar´e formula χ(G) = � v if(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Applying this to the graph G1 of the Whitney simplicial complex G of a graph G and using the function f(x) = dim(x) which is locally injective, one immediately can see that the Euler characteristic of G and G1 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that if(v) = 1 − χ(S−(v)) = 1 + (−1)k = ω(v) because in the case of the dimension functional, S−(x) is the boundary sphere complex of x which has by the Euler-Gem formula the Euler characteristic 1 + (−1)k if x has dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Euler characteristic therefore preserves Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can see this also by explicitly writing down the linear map transforming the f vector (f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , fd) of G to the f vector of its Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is only one eigenvalue 1 of this linear operator T and the corresponding eigenvector of T ∗ defines the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can see from the Poincar´e-Hopf formula immediately that homotopy extensions and homotopy reductions preserve the Euler characteristic: choosing a function f which has the property that it is maximal on the added vertex, we get if(v) = 1 − χ(S−(v)) = 0 because S−(v) is contractible and because recursively one sees that contractible graphs have Euler characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can also see that edge refinements in general preserve the Euler characteristic: if e = (a, b) is an edge, then the edge refinement replaces the join of K2 with of S(a) ∩ S(b) with the join of the path graph P3 with S(a) ∩ S(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The operation just replaces a contractible part with an other contractible part meaning that the Euler characteristic of that part does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, one can show other operations like flip diagonal operations on embedded kite graphs do not change the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But flip diagonal operations does not preserve d-manifolds in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 27 FINITE TOPOLOGY 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If simplices in G are equipped with an orientation 19 one can interpret an arbitrary function f : G → R as a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For y ⊂ x, define sign(y, x) = 1 if the orientation of y matches the orientation of x restricted to y, and −1 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The exterior derivative df(x) = � y⊂x,|x|−|y|=1 sign(y, x)f(y) is a n × n matrix if G has n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Because if z ⊂ y ⊂ x with |z| = |y|−1 satisfies � y sign(z, y)sign(y, x) = 0, the matrix d satisfies d2 = 0 so that the Hodge Laplacian L = dd∗ + d∗d is block diagonal with fk × fk block matrices Lk leaving invariant the class of k-forms, functions on k-dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The kernel of Lk is called the k’th Betti number of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By using the McKean-Singer symmetry that the non-zero eigenvalues of L on even forms agrees with the non-zero eigenvalues of L on odd forms, one can see that the super trace str(A) = � k(−1)kAkk has the property that χ(G) = str(e−tL) for any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For t = 0, one has str(1) = � k(−1)kfk(G) and in the limit t → ∞, where only the kernels of Lk survives, one gets χ(G) = � k(−1)kbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The identity � k(−1)kfk(G) = � k(−1)kbk(G) is called the Euler-Poincar´e formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Betti numbers of the Barycentric refinement G1 are the same than the Betti numbers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This can be seen as a consequence of the K¨unneth formula which relates the Betti numbers of H · G with the Betti numbers of H and the Betti numbers of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Stanley-Reisner product H · G is homotop to the Shannon product H ∗ G for which one can show the K¨unneth formula by taking the product of Harmonic functions d∗fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Homotopy deformations preserve the Betti numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If H = f(G) is a continuous image of G then bk(H) ≤ bk(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' From these statements one can get immediately that homeomorphic geometries have the same Betti numbers and so the same Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Betti numbers, cohomology group, Euler characteristic are topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also the sphere spectrum � x∈G χ(S(x)) is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The valuation property χ(U∪V ) = χ(U)+χ(V )−χ(U∩V ) holds for Euler characteristic and all subsets U, V of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Euler-Poincar´e identity � k(−1)kfk(G) = � k(−1)kbk(G) can be seen by heat deformation and using McKean-Singer symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Characteristics 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Euler characteristic is the first of many higher characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The next after Euler characteristic is Wu characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is defined as ω(G) = � x∩y∈G ω(x)ω(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also all higher characteristics are invariant under Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For manifolds with boundary, it is χ(G) − χ(δG) (see [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is a multi-linear valuation but not a valua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It had been puzzling to us how the Wu characteristic behaves, even when looking at wedge sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While the Euler characteristic is invariant under homotopy and so also under homeomorphisms the Wu characteristic is only invariant under homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To see why topology is involved, we have to restate that the energy theorem tells χ(G) = � x,y∈G g(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is a quadratic identity to that ω(G) = � x,y∈G ω(x)ω(y)g(x, y)2 (see [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Because g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) is expressed in terms of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What happens is that f −1(U(x) ∩ U(y)) is an open set with the same Euler character- istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What happens is that if U, V are arbitrary open sets in the topology of G, then ω(U ∪ V ) = ω(U) + ω(V ) − ω(U ∩ V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 19There does not need to be compatibility with intersecting simplices 28 OLIVER KNILL 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Euler characteristic χ(G) as a linear combination of basic valuations fk(G) count- ing simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It satisfies the valuation formula χ(G ∪ H) = χ(G) + χ(H) − χ(G ∩ H) if G and H are arbitrary simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This formula does not hold for the Wu charac- teristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, if G is the Octahedron complex and H is the circle complex C4, then ω(G) = χ(G) = 2, ω(H) = χ(H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we look at the wedge sum G ∧ H which is G ∪ H with a common 1 point complex K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now ω(K1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We compute ω(G ∪ H) = 3 so that obviously, the valuation formula does work as in the case of Euler characteristic, where χ(G ∪ H) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now we know the solution to the puzzle: while G, H, G ∩ H are all open sets by themselves, in the topology of G ∪ H, the complexes G, H are only closed in G ∪ H and no more open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have however the identity ω(U ∩ V ) = ω(U) + ω(V ) − ω(U ∩ V ) for open sets within the topological space O in X = G ∪ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we look at the open balls A = G \\ {x}, B = H \\ {x} (they are open as an open set intersected with the complement of a closed set), and the open set C = U(x) in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now A, B, C are open sets in X and we have ω(A ∪ B ∪ C) = ω(A) + ω(B) + ω(C) − ω(A ∩ B) − ω(B ∩ C) + ω(A ∩ B ∩ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Consider the figure 8 graph X which is the wedge sum of two circular graphs X = G sup H = C4 ∧ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can explain ω(X) = 7 by putting things together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The valuation formula does not work for closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, the following formula does not work: 7 = ω(X) = ω(G) + ω(H) − ω(G ∩ H) = 0 + 0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' However, we can write G as a union of three open sets U, V, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Both U, W are linear graph without boundary which have Wu characteristic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star graph without boundary has Wu characteristic 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The intersection between U and V has Wu characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So, we have ω(X) = ω(U) + ω(V ) + ω(W) − ω(U ∩ V ) − ω(V ∩ W) = 1 + 9 + 1 − 2 − 2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We first used the old definition ω(U) = � x,y,x∩y̸=∅ ω(x)ω(y) for Wu characteristic and not the correct definition ω(U) = � x,y,x∩y∈U ω(x)ω(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is no difference between the two definitions if we deal with simplicial complexes which are closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It matters however if we deal with open sets for example, take the two open sets U = {(1, 2, 3), (1, 2)} and V = {(1, 2, 3), (2, 3)}, which are stars in the complete complex K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now W = U ∩ V = {(1, 2, 3)} is the star of the facet (1, 2, 3) which has ω(W) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have X = U ∪ V = {(1, 2), (2, 3), (1, 2, 3)} with ω(x) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have ω(U) = ω(V ) = 0 and ω(W) = 1, ωX = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The identity ω(U)+ω(V )−ω(U ∩V ) = ω(U ∪V ) is valid but only because the simplices x = (1, 2) and y = (2, 3) were not allowed to “interact”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Their intersection was not in U ∪V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A convenient way to compute Wu characteristic therefore is to write the complex as a union � j Uj of open sets, then use the inclusion exclusion property ω(G) = � j ω(Uj) − � i∩j ω(Ui ∩ Uj) + � i∩j∩k ω(Ui ∩ Uj ∩ Uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This explains again the known fact that for d-manifolds M we have ω(M) = χ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Lets assume now that G is a manifold graph with vertex set V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can cover G with the open sets U(v), v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now use that ω(U(x)) = 1 for any simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' So we have ω(G) = � x=(v)∈G ω(U(x))−� x=(v,w)∈G ω(U(x))+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' which is � x(−1)dim(x)ω(x) = χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For manifolds ω(G) = χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general we have the star formula: ω(G) = � x∈G ω(x)ω(U(x)) using the stars U(x) of the simplex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We had previously proven the formula ω(G) = � x,y∈G ω(x)ω(y)χ(U(x) ∩ U(y))2 in [37] we have here a sum over G and not a more costly 29 FINITE TOPOLOGY sum over pairs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also see the ball formula for the unit balls B(x) = U(x) which is remarkable because there is no direct relation between ω(B(x)) and ω(U(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The relation ω(G) = � x∈G ω(x)ω(B(x)) follows from � x∈G ω(x)ω(S(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (See [36] Corollary 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Wu characteristic is no homotopy invariant but a topological invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The valuation property χ(U∪V ) = χ(U)+χ(V )−χ(U∩V ) holds for all open sets, where ω(U) = � x,y,x∩y∈U ω(x)ω(y) for a set of sets U and not ω(U) = � x,y,x∩y̸=∅ ω(x)ω(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is the Gauss-Bonnet type formula ω(G) = � x∈G ω(x)ω(U(x)) complement- ing χ(G) = � x∈G ω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This allows to compute the Wu characteristic for larger spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is also an energy theorem ω(G) = � x,y g(x, y), where g(x, y) = ω(x)ω(y)ω(U(x) ∩ U(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Dynamics 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If T is a continuous map from a finite topological space O into itself then every point is eventually periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, a simplicial map T on a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The attractor of T is a finite set and on every connected component of the attractor, one just cyclically permutes points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One calls the forward attractor also the ω-limit set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case of a homeomorphism, there is also the α limit set which is the ω-limit set of the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Lefschetz fixed point theorem for graphs [22] tells that if T is a graph endomorphism T : G → G then the sum of the indices of the fixed points agrees with the Lefschetz number χT(mathcalG), the super trace � k(−1)ktr(L|Hk) of T induced each space of harmonic forms Hk = ker(Lk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This result formulated for graphs [22] obviously works for arbitrary simplicial complexes and a continuous map T : G → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If F is the set of fixed points of T and index iT(x) = ω(x)sign(T|x) with sign(T|x) being signature of the permutation of T induced on x, then the Lefschetz formula tells � x∈F iT(x) = χT(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The formula is easy to prove using the heat flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Koopman operator U : f → f(T) has as the super trace � x∈F iT(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Applying the heat flow does not change the super trace of e−tLU as non-zero eigenvalues in the odd forms and even forms agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the limit t → ∞, only the map induced on the kernel survives and this is the Lefschetz number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A special case of the Lefschetz fixed point theorem is the Brouwer fixed point theorem which applies in the case when the complex has trivial cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A special case is if G is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An even more special case is if G is a d-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can also start with an arbitrary finite topological space O and fix a pre-basis B which generates the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This defines a nerve simplicial complex G on B, where the complex consists of all subset of B which have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A continuous map f on O now defines a continuous map on the simplicial complex G so that the Lefschetz fixed point theorem applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can now define the cohomology of the topological space (equipped with the base) as the cohomology of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Lefschetz number of the super trace of the from T induced map on the cohomology is then equal to the sum of the indices of fixed points of f on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that there is an open set in O which is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30 OLIVER KNILL 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A homeomorphism T : G → G of a finite geometry can be enhanced to a sequence of homeomorphisms Tn : Gn → Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As more iterations are needed, as more Barycentric refinements are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a homeomorphism T this means specifying a sequence of permutations Tn : On → On of the topologies of Gn and then require some compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' How well the map Tn on On approximates the dynamics of Tm on Om determines the amount of regularity of smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The upgrade of a homeomorphism T : G → G to a stratified sequence of home- omorphisms Tn : Gn → Gn is motivated by various similar constructions in mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' like computing with sequences of rational numbers with a larger and larger number of digits in order to approximate real numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' to do numerical computations of partial differential equations on sequences of meshes or the concept of inverse limit in constructions like p-adic integers or then martingales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' where a stochastic process is observed on a sequence of adapted σ-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since every stochastic process given in the form of a sequence of IID random variables Xn can be assigned a compact topological space Ω a continuous function f and a homeomorphism T such that Xn = f(T n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The sigma-algebra An generated by the random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , Xn is the Borel σ algebra of a topological space On which is the product space Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If X has a finite set as range, then On is a finite topological space and An is the Borel σ algebra generated by On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: The Lefschetz fixed point theorem and so the Brouwer fixed point the- orem naturally work for continuous maps on simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It even works for a finite topological space when applied to the nerve of a pre-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order to study the dynamics of a homeomorphism, one has to pick a choice of concrete homeomor- phisms on refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The length of the orbit which one wants to compute accurately determines how many Barycentric refinement lifts are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Categorical 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Here are some general contemplations about the various categories: complexes, graphs and topologies involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite simplicial complexes form a category Sim with simplicial maps as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite graphs form a category Gra with graph homomorphisms as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite topological spaces Top form a category too where continuous maps are the morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have the Whitney map from graphs to complexes, the Alexandroff map from complexes to topological spaces and the ˇCech map from topological spaces to the nerve graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These maps are not functors because the morphisms do not correspond directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can however enlarge the class of morphisms both on Sim as well as on Gra to have morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, in order to see the Whitney map Gra → Sim as a functor between categories one has to expand the possible morphisms on graphs allowing not only graph homomorphisms but maps from one graph to an other in which edges can collapse to vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order to have a functor between Sim and Top, we enlarge the class of simplicial maps and allow also continuous maps, still order preserving but not mapping simplicial subcomplexes to simplicial subcomplexes necessarily (these are the open maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also the map assigning to a simplicial complex a graph is a functor again if one uses the larger class of morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The composition of the functors from Graphs to Complexes back to Graphs is the Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If Sim/ be the equivalence classes of complexes under Barycentric refinement and Gra/ the equivalence classes of graphs under Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Whitney map now identifies these two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A continuous map from 31 FINITE TOPOLOGY some Gn → H could serve as the morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We chose however to make more assumptions and use the notion of having H a continuous image of G to define homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Some of the graph theory literature assumes graphs are one-dimensional simplicial skeleton complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While useful for some set-up’s it is rather limiting as graphs are so much more than one dimensional objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Whitney complex reflects rather general topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are other simplicial complexes associated to graphs of course, like the graphical matroid other skeleton complexes or the neighborhood complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Topological graph theory looks at graphs embedded in two-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case, a graph naturally naturally defines a cell complex in which the two-dimensional faces are the connected components of the complement of the embedded graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This uses infinity but it allows to deal with two-dimensional complexes which have the discrete topol- ogy of the underlying surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' On a surface of degree g for example, the number of vertices v, the number of edges e and the number of faces f satisfies v − e + f = 2 − 2g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also in topological graph theory one can sometimes avoid infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of being planar for example is settled with Kuratowski’s theorem completely within finite mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' That theorem uses homeomorphism in the narrow sense as homeomorphic as one-dimensional simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Discrete CW complexes extend simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A discrete CW structure can be introduced within combinatorics, once one has defined what a k-sphere is: Start building up the geometry G0 = {} and successively attach k-balls (called cells or handles) to already existing (k − 1)-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can for example attach a 0-ball (called a vertex) to a −1 sphere (the empty graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Once the 0-dimensional part is built, we can attach 1-balls (called edges) to 0-spheres (2 disjoint points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Then one can start adding 2-dimensional balls (faces) to 1-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, one can add triangles 2-simplices to the a triangular circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Obviously, every finite abstract simplicial complex is also an abstract CW complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While simplicial complexes are natural and given just as they are, a CW complex comes with a “timeline” of how the structure has been built up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can so build also multi-graphs or quivers or more general complexes called δ-sets, which are simplicial complexes in which simplices can occur with multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Adding a bit more structure produces a subclass of δ-sets called simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In category theory this is known as a pre-sheaf on the simplex category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If on a δ-set boundary maps are defined, one has a cohomology like on simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have more recently also looked at quivers, graphs where self-loops and multiple connections can happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case, one can naturally attach δ-sets to a quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' δ-sets generalize simplicial complexes in that different sets can appear multiple times and where boundary maps are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The category of δ sets generalize the category of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The later are δ sets with more structure attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' δ sets (and so also simplicial sets) have a cohomology attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can now ask, what natural topologies can be associated to a quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have not yet investigated this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One possibility would be to do this on the sets x in the δ-complex and take the basis U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But now, the space is not even T0 any more as different points can have the same minimal open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can no more distinguish points by their minimal open neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 32 OLIVER KNILL Summary: Simplicial complexes, finite simple graphs and topological spaces can not be directly linked with their traditional morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But topology glues them together if we look at continuous maps as morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Whether we talk about a graph with continuous maps on them, simplicial complexes with continuous maps on them or finite topological spaces with continuous maps on them, we always can switch to the other two pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It requires however to change already what we mean by morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While traditionally, these three categories use different notation and jargon, topology unifies them nicely and allow us to work on finite geometries using a trinity of interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Non-standard analysis links this radically finite geometry with rather arbitrary compact topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Ringed spaces 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As in commutative algebra approaches to geometry, one can use the notion of ringed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Attach a ring F(U) to ever open set and call it a section of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Given restriction maps produces a pre-sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The usual way to rephrase this is that this is a contra-variant functor from the category of open sets with inclusion morphisms to the category of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To get a sheaf, we need existence (gluing) and uniqueness (locality) properties: Gluing is related to existence h(x)|U(x) ∩ U(y) = h(y)|U(x) ∩ U(y) then there exists a h with h|U(x) = h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Locality relates to uniqueness because h ∈ R(x) = k ∈ R(x) for all x, then we have the same h = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A simple case is to the sheaf of ring-valued continuous functions on open sets such that if V ⊂ U, the restriction of F(U) to F(V ) is a ring homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In our case, where the topology of a simplicial complex with the topology, we deal with a ringed simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The section F(x) = F(U(x)) is in this context the stalk of x and its elements are the germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Given a commutative local ring R and any function h : G → R defines already a locally ringed sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But things can be much more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The restriction maps from U(x) → U(y) if x ⊂ y do not have to be the obvious ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Actually, any ring valued matrix r(x, y) can serve as a transition map F(x) → F(y) if x ⊂ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The pre-sheave condition now means for x ⊂ y ⊂ z that the cocycle condition r(y, z)r(x, y) = r(x, z) holds and especially that r(y, x)r(x, y) = r(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Classically, when looking at general topological spaces, the stalk F(U(x)) at some point x is the direct limit F(U) over all the open sets U containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the finite topology case, the stalk at x is F(U(x)), which is just a ring attached to the star U(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A locally ringed space is a ringed space in which every stalk is a local ring, meaning that it has a unique maximal ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example of a ringed space are differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If an orientation is fixed on each simplex x ∈ G, then these are just the functions from G to the ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite geometries also allow to use the frame work of schemes in an elementary frame work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A ringed space by definition attaches to every open set a ring and its spectrum, the set of prime ideals in the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If U is an open finite set, then the space OX(U) of functions on U have as prime ideals the functions which vanish at some simplex x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The spectrum therefore is just the set of simplices in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The theory as developed for general ringed spaces can be taken over word for word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The frame work can be useful also in combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, we can take C-valued functions and require that the restriction maps from ring F(U) to the ring F(V ) if V ⊂ U is not the obvious ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Locally ringed topological space 33 FINITE TOPOLOGY can have global properties are not necessarily the obvious ones: going around a closed loop for example can produce a non-trivial map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The lack of linear structures prevents having constructs like tangent spaces in the discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' However, we have attached to each simplex a unit sphere S(x) and so a sphere bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What we can do in general is to have transition maps on spheres S(x) coming from positive dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These transitions tell what happens if one looks at S(x) as part of S(v) or S(w) if v, w ⊂ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Going from a pre-sheaf to a sheaf means to have transition maps on positive dimensional simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Having a topology on a complex allows to use sheaf theoretical concepts on finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In finite topological spaces, the ring R(x) attached to a star is called a stalk and its elements are the germs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Morse extensions 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A function f : G → R to a totally ordered space R like R of Z is called a Morse function if it is locally injective, meaning that f(x) ̸= f(y) if x ⊂ y or y ⊂ x and S−(x) = {y ∈ S(x), f(y) < f(x)} is a (k − 1)-sphere for some k ≥ 0 or then contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the former case, we have added a critical point, in the later case a regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If x is a critical point, the integer k ≥ 0 is called the Morse index of the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If xk is a fixed enumeration of points such that f(xk) ≥ f(xl) if k ≥ l, then Gn = {y, f(y) < f(xn)} is a Morse build-up an χ(Gn+1) = χ(Gn) + χ(B+(xn)) − χ(S−(xn)) = χ(Gn) + if(xn) so that the Poincar´e-Hopf formula χ(G) = � x if(x) [19] holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This formula holds for any locally injective function f but for Morse functions the Poincar´e-Hopf index if(x) ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We see that the existence of a Morse function implies that G can be seen as a CW complex in which successively k-balls 20 are attached to (k − 1)-spheres in the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Morse build-up of a graph are not homeomorphism even if we look at the step when a regular point is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can see this already from the fact that the dimension can increase without adding a critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A continuous map can not increase dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The contraction process is however a continuous process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The stable unit sphere S− f (x) is then a subgraph of S(x) and so a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can state this all in other words and say that a real valued function on a simplicial complex needs not to be continuous but that every function on the vertex set of a graph is continuous in the topology of the Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It would not be useful to enforce continuity because {f(x) < c} can be a single point which is neither open nor closed in the topology of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When we look however at the situation on the graph level with a function f : V (G) → R, then the sets Gn = {v, f(v) ≤ vn} generate graphs which are closed sets in G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In this sense any function f : V → R on a graph G is continuous in the topology of G while a function on a-priori given simplicial complex G is hardly ever continuous in that topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A function G → R becomes only continuous if we look at it as a function of the graph G1 and so using the topology of G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 20also called handles 34 OLIVER KNILL Summary: Morse theory is a concrete way to build up an abstract finite CW- complex using a Morse function as a guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While the sets Gn in a Morse build- up of a complex are neither open or closed, the topology of their graphs make them topological spaces in a refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Functions on simplicial complexes become naturally continuous when considered in the topology of the Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Calculus 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A function f : G → R can be interpreted as a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Similarly, if G is a graph, we look at functions f : G → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When restricted to k-dimensional simplices, one gets k-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Calculus can be studied on arbitrary Barycentric refinement levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Provided that orientations are fixed on G any scalar function is just a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In order to define level surfaces, we need only the very mild assumption that functions are locally injective, meaning that adjacent vertices take different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Lest look in this section at graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In case of a simplicial complex G, look at the graph G1 in which the vertex set is G and where two are connected if one is included in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A locally injective function f : G → R can be lift to a function on G1 by assigning to a simplex x the average of f over the vertices in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can also take a function f : V (G) → R and distribute its values f(v) equally to all points x in U(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This produces a new function G1 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What might happen under such a refinement of a locally injective function, that it is no more locally injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can fix this by looking at the lexicographic order of the pair (f, dim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Because in a Barycentric refinement, dim(x) ̸= dim(y) if x, y are connected in G1, this is a well defined ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can now use f on G1 to build again a level surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Given a function on the vertex set of G1 we can move the content from vertices in G1 which are sets of vertices in G to vertices of G by equally distributing the value f(x) to to all vertices v ∈ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' With f(x) = ω(x), this produces f(v) = κ(v), where κ(x) is the curvature [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The consequence � x ω(x) = � v κ(v) is the Gauss-Bonnet theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' See [18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If f is a function on a discrete d-manifold G which is locally injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Then the level surface U = {f = c} generated by the set of simplices on which f changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' See [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the topology of the complex, this is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' However, its graph defines a discrete (d − 1) manifold if it is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This manifold now carries a topology again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Note that as a set M in G, the set U is always open because if x is in M then every set y containing x is in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can still make f locally injective by replacing f with f(x) + ϵdim(x) for ϵ small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This allows us to extend f to G1 in a determined way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can now look at a variety {f1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , fm = 0} for m locally injective functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , fm in the n’th Barycentric refinement Gn as the set of simplices, where all the lifted functions of fj change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have seen that using the “dimension trick” providing a lexicographic order of the function on a higher level, we can lift any function on a graph uniquely to Barycentric refinement where we can again define level sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This level set is a graph where two simplices are connected if one is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Remarkably, by the discrete Sard theorem, we never run into singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We expect if the functions fk are lifted nicely to a Barycentric refinement, then the corresponding manifold is homeomorphic to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We expect that there could be surprises if we take a situation from the continuum, where {f1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , fm = 0} is a classical variety which is not a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case, there could be surprises near singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology depends on the choice of the Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 35 FINITE TOPOLOGY Summary: If G is a n-manifold and m ≤ n locally injective functions are given on G, then the “variety” {f1 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , fm = 0} is a well defined graph again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is either empty or a (n − k)-manifold S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Interaction energy 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Given a simplicial complex G with n elements x and any n × n matrix taking values in some ring R, we can define the internal energy of a subset A ⊂ G as ω(A) = � x,y,x∩y∈A h(x, y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The matrix h does not have to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can think of h(x, y) also as a current from x to y and ω(A) as the total current or traffic flowing overall through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now look at the matrix g(x, y) = � x,y ω(x)ω(y)ω(U(x) ∩ U(y)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This matrix gives a potential energy between the simplices x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Unlike h, the matrix g is always symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The energy theorem tells ω(G) = � x,y g(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This result assures that the total potential energy of G is the total internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This theorem generalizes an energy theorem proven before, see [36, 37, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, if h(x, y) is diagonal with h(x, x) = ω(x), then ω(A) = χ(A) is the Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case the matrix g is the inverse of the operator L(x, y) = χ(x∩y), where x is the closure of {x}, a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The frame work also captures energized simplicial complexes where h(x, x) = h(x) and h(x, y) = 0 for x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To prove the more general energy theorem, note that the map h → g is linear and that both the energy and the total sum are both linear expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We only need to verify the statement therefore for the matrix h satisfying h(x0, y0) = 1 and h(x, y) = 0 else for some fixed simplices x0, y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These n2 basis elements are fixed-points of the linear map T(h) = g and the energy relation � x,y,x=x∩y∈G h(x, y) = � x,y g(x, y) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' By linearity, the relation holds then for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To verify the statement for a basis element, note that the left hand side is 1 if x0 and y0 intersect and 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The right hand side is ω(x)ω(y)ω(U(x) ∩ U(y)) which is non-zero only if x0 and y0 intersect and both x0, y0 are contained in U(x) and U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that the union x0 ∪ y0 is contained in U(x) ∩ U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that both x and y have to contain x0 ∪y0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This means that the simplex x∩y has to contain the simplex x0 ∪y0 and so x0 ∩ y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But � x0⊂x,y0⊂y,x0∩y0⊂x∩y ω(x)ω(y) = � x0⊂x ω(x) � y0 ⊂ yω(y) = 1 ∗ 1 = 1 because the Euler characteristic of a simplex is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also have as before a curvature relation κ(x) = � y g(x, y) = ω(x)g(x, x) = ω(x)χ(U(x)) and so � x κ(x) = χ(G) which is a Gauss-Bonnet relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It can also be thought of as a Poincar´e-Hopf theorem for the locally injective function f(x) = −dim(x) because then, the atom U(x) is the stable sphere S−(x) = {y, f(y) < f(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Still, since the dimension function f is not so well visible, it is good to think of κ not as an index but as a curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The internal energy of the “atom” U(x) is up to a sign a curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summing up the curvature gives the total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can also think of the relation � x ω(x)g(x, x) as the super trace str(g) of the matrix g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the context of simplicial complexes, the notion of super trace is natural since str(1) = χ(G) is the Euler characteristic and because of the McKean-Singer formula str(e−tL) = χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' [20, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 36 OLIVER KNILL 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Actually, we would like to announce here already that arbitrary tensor energy theorems hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xm) be arbitrary ring-valued functions of m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We already had for ω2(A) = � x,y,x∩y∈A h(x, y) that g2(x, y) = ω(x)ω(y)ω2(U(x) ∩ U(y)) satisfies � x,y g2(x, y) = ω2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If G is a finite abstract simplicial complex and h(x, y, z) arbitrary R valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For a subset A ⊂ G, define the internal cubic energy ω3(A) = � x,y,z,x∩y∩z∈A h(x, y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Now define g3(x, y, z) = ω(x)ω(y)ω(z)ω3(U(x) ∩ U(y) ∩ U(z)) and think about it as the potential energy of the triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Then the total potential energy agrees with the total internal energy � x,y,z g3(x, y, z) = ω3(G) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This works also with more interaction like quartic ω4(A) = � x,y,z,w,x∩y∩z∩w∈A h(x, y, z, w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For g4(x, y, z, w) = ω(x)ω(y)ω(z)ω(w)ω(U(x) ∩ U(y) ∩ U(z) ∩ U(w)), the total potential energy agrees with the total internal energy � x,y,z,z g4(x, y, z, w) = ω4(G) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Summary: Given an interaction transfer rule between m intersecting simplices in a simplicial complex G, we can assign internal energies k tuples of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The total energy can be expressed also as the sum of all potential energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The internal m-energy of a set ω(A) = � � j xj∈A h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xm) is associated to closed sets, the potential energy uses open sets gm(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xk) = � j ω(xj)ω(�k j=1 U(xj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The energy theorem assures that the total internal energy is the total potential energy ωm(G) = � x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=',xk g(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case m = 1 and h(x) = ω(x), where the total energy is the Euler characteristic, this is a Gauss-Bonnet theorem χ(G) = � x ω(x)ω(U(x)), where U(x) is the smallest open set containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the case h(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' , xm) = �m k=1 ω(xk), the energy ω(G) is the m-th characteristic, a topological invariant for the complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Unlike for m = 1, which gave the Euler characteristic, we have no homotopy invariant however for m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Remarks 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Inspired by Alexandroff and Zariski, we have revisited here at the finite topology on a simplicial complex G defined by stars and especially for simplicial complexes coming from a finite simple graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 21 The analogy to algebra is that the vertices of the graph play the role of the maximal ideals and that the simplices play the role of prime ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A basis for the topology is the set of stars of a simplex, the set of simplices which 21Since open sets are still quite local, the drawbacks of Zariski topology appear not really relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 37 FINITE TOPOLOGY contain x, as well as an added empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Unlike any topology on the vertex set V like the one given by distance which would render the graph completely disconnected, our topology honors connectivity and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It shares the non-Hausdorff property with the Zariski topology on prime ideals of a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closed sets in our graph topology are exactly the simplicial complexes of subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' They play the role of algebraic subsets of a variety in algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The closure of a set A of simplices is the smallest abstract simplicial complex which contains the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One of the motivations to look at finite topologies is that there are finite simple spaces G, H for which the geometric realization |G|, |H| are homeomorphic but which are not home- omorphic in the finite topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Open sets U(x), U(y) be entangled in a complicated way in the finite topology if the dimension is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But these entanglements are not always visible when looking at the topology induced from Euclidean distance in geometric realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology of the Euclidean realization is not sophisticated enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In other words, there are triangulations of topological manifolds which have the manifold as a geometric realizations but which are not discrete manifolds as defined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The definition of homeomorphism is motivated by the notion of piecewise linear map in topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A map f : |G| → |H| be- tween geometric realizations of simplicial complexes is PL, if there is a piecewise linear map between Barycentric refinements |Gn| → |Hm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A PL homeomorphism is then a simplicial map such that there is a homeomorphism |Gn| → |Hm| for some refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This looks equivalent to what we do here in the finite but leaves finite mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As we are not interested in infinity here, we do not bother showing the equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other motivation has been to answer the question why higher characteristics like Wu characteristic ω(G) = � x,y,x∩y∈G ω(x)ω(y) is a topological notion while Euler characteristic χ(G) = � x ω(x) is more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We will write about this more in the future but one of the key facts is that ω(U ∪V ) = ω(U)+ω(V )−ω(U ∩V ) holds for open sets but not for closed sets or sets which are neither closed nor open in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For Euler characteristic this valuation formula holds for all subsets U, V of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Euler characteristic does super count simplices, while Wu characteristic does super count intersecting simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This interacting points should not be separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In indeed, if an intersecting pair x, y is in the intersection U ∩ V of two open sets it must be in both sets U and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other mystery which still needs more investigation is the notion of analytic torsion A(G) = � k Det(Lk)k(−1)k+1, where Lk are the blocks of the Hodge Laplacian L = D2 = (d+d∗)2 of the Whitney complex and Det is the pseudo determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We wrote this as a super determinant SDet(D) = � k Det(Dk)(−1)k of the Dirac operator D with Dirac blocks Dk = d∗ kdk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Analytic torsion can be defined for any simplicial complex but so far it has been accessible only in 2 cases: the first is when G is homotopic to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In that case A(G) = |V | where V = � x∈G x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The second case was when G = |V ||V ′| for even dimensional spheres and G = |V |/|V ′| for odd dimensional spheres, where V ′ is the number of maximal simplices in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But analytic torsion is not a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Even for manifolds like a torus, the analytic torsion changes if we make a Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As already in the etymology of the name, a non-trivial fundamental group makes the functional A(G) more complicated as we can get torsion terms along non-contractible closed loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Still it is not only that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we make homotopy extensions of a sphere which are not homeomorphisms, the torsion formula for the sphere in terms of |V | and |V ′| disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 38 OLIVER KNILL 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite and so Alexandroff topologies can be interesting from a physics point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' First of all, there is local interaction of simplices which are contained in each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can not separate such points using open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The fact that we have smallest non-empty open Planck units U(x) is some sort a space quantization or atoms of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we look at a manifold with a very fine triangulation, both the lack of the Hausdorff topology and the Alexandroff feature are hardly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The situation is also present in floating point arithmetic, when a computer deals with small numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Every point has a smallest neighborhood which can no more be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also can not separate two points which are too close from each other even so they are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Points which are identified by the equal- tolerance parameter define the machine graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' With a machine precision log10(252) ∼ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='65 the distance below which two numbers are identified is about 2−46 = 2−52−1+7 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='42109 · 10−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The definition of homeomorphism is motivated by the fact that if we have two finite topological spaces coming from a finite abstract simplicial complex and a continuous sur- jective map f : X → Y and a continuous surjective map g : Y → X and the unit spheres S(x) have homeomorphic pre-images and unit balls of locally maximal simplices have balls as images, then X, Y are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 22 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The given definition of homeomorphism within finite topology has shifted a bit while writing down this text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We first tried to avoid unit spheres S(x) and balls but failed to prove some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Using unit sphere S(y) in the definition allows the use of induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For the unit balls of maximal simplices, there is no interesting topology as they are balls and we just require that the inverse image of such a unit ball is a ball (which is precisely defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is postulated that all d-dimensional balls are homeomorphic but it could also be proven from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It also seems to be necessary to ask such a requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We need it for example that if G is a manifold and H is homeomorphic, then H is a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can imagine continuous surjective maps going both ways which collapse substantial parts of the topology somewhere in the interior of the manifold so that the inverse image of S(y) could become a complicated object in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Looking at stars U(x) = W +(x) and cores V (x) = {x} = W −(x) of simplices is also motivated by connection calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have seen for example that the Green function matrix g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) is always the inverse to the matrix L(x, y) = χ(V (x)∩V (y)), where χ(A) = � x∈A ω(y) is the Euler characteristic of an arbitrary subset of G and ω(x) = (−1)dim(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While V (x)∩V (y) which is always a simplicial complex, are closed, the sets U = U(x) ∩ U(y) are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Euler characteristic of the closure B(x) = U is in general different from the Euler characteristic of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Actually χ(B(x)) = χ(U(x)) + χ(S(x)) where S(x) is the boundary of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23 22We wonder whether it is true in general: does already the existence of two continuous surjective maps f : X → Y, g : Y → X force X, Y to be homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Using the axiom of choice, one can invert the surjections and have injections, showing with Cantor-Schroeder-Bernstein that the cardinalities are the same so that there is a bijection between the topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the non-Alexandroff case, where points can be written as intersections of open sets, this should then give a homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 23This phenomenon prevented us for some time to find the Green star formula for the Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 39 FINITE TOPOLOGY 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Both the star U(x) and the core V (x) = {x} = W −(x) can be seen as measurable sets in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we close a topology O under complements and countable intersections and unions, we get the Borel σ algebra A as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This set A still does not cover all subsets of G if G has dimension 2 or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that if {e = (a, b)} is a single non-maximal simplex, then it is neither open nor closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The smallest open set containing it is U(e), the smallest closed set containing it is the simplicial complex {e, {a}, {b}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can look for probability theory on the graph and look for example at the measure fk(A)/fk(G) counting the fraction of k-dimensional simplices in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For probability theory on finite set, see [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the literature, one often a “graph” as a topological space that is obtained as a geometric realization as a one-dimensional simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A more topological ap- proach is to look at the geometric realization of its Whitney complex in which all the complete subgraphs Kn+1 are realized as simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can then also look at other simplicial com- plexes attached to a graph, similarly as one can attach other topologies to Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The notion of homeomorphism could be extended to such cases too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are simplicial complexes and so graphs that are not homeomorphic but which have homeomorphic realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An example is a double suspension of a rational homology sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is not topologically equivalent to a sphere in our sense but in a geometric realization, it is by the double suspension theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' While also in the discrete, any manifold that is a suspension of a manifold must be a sphere, in the discrete, a suspension of a non-manifold is by definition not a discrete manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have searched for notions of homeomorphism within finite combinatorics for a while like [23], where we looked at ˇCech type notions like the nerve of an open cover and asked that two homeomoprhic graphs have isomorphic nerves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In 2016 we experimented (motivated by the Zarisiki topology) with the concept of having the closed subgraphs play the role of closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have not taken it too seriously: do we want to work with topological spaces that are non-Hausdorff?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We decided now it is better to work with the topology generated by the star basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' When reviewing the Lusternik-Schnirelmann category, where open covers play a role, the concept fits better with what one does in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Since 2016, we have also realized more the importance of stars U(x) = U(x) and cores x = W −(x) = {x} of simplices as they form a hyperbolic structure and because g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) and L(x, y) = χ(W −(x) ∩ W −(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the current notation, we would write this Green-Star formula as g(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The fact that U(x) ∩ U(y) can be topologically quite complicated even so both sets U(x), U(y) are smallest open sets and have contractible closures, the intersection U(x)∩U(y) can be rather complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The local smallest Planck units U(x) in a complex can be entangeled in a complicated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also, the simplicial complex belonging to the closure of U(x) ∩ U(y) can be topologically very different from U(x) ∩ U(y)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 24 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are many interesting open questions and many opportunities for experimen- tation or further explorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can ask for example about the fraction |O(G)|/2|G(G)| telling us in a graph what fraction of subsets of the simplicial complex are open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As the number of open sets and closed sets agree, this is equivalent to count the number of sub- graphs of a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Numerical computations become quickly too hard to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have 24This was a reason to drove us almost insane in 2016 while looking for the Green star formula as the formula with the closure U(x) ∩ U(y) worked in most cases and especially small complexes, but that it had rare failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 40 OLIVER KNILL φ(C4) = 48/256 and φ(C5) = 124/1024 and φ(C6) = 323/4096 and φ(K1) = 1, φ(K2) = 5/8, φ(K3) = 19/128 and φ(K4) = 167/32768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us add a comment on the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The paper [1], was dedicated to Emmy Noether, assumes that the space is locally finite spaces which under a global compactness assumption means finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff already identifies discrete T0 spaces with partially ordered sets and identifies closes sets as simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' He notes that if in an Alexan- droff space, two smallest open sets U(x) = U(y) agree, then x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The reason is that then x ∈ U(y) and so y ⊂ x and y ∈ U(x) and so x ⊂ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' He calls simplicial complexes vollst¨andige Mengensystems = complete set systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The completion of an arbitrary set of sets is the closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Interestingly, the void {} = empty set is not considered of this type, even so today we consider this to be a simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is technically a finite set of sets which is closed under the operation of taking finite non-empty subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also in topology, the empty set is a closed set as in any topological space we consider ∅, X to be clopen = closed and open and define connectedness as the property that the only clopen sets are ∅ and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The modern point of view is to see the void 0 as the (−1) dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff also notes that pre-base of stars centered at 0-dimensional simplices define what we would call today a ˇCech graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In our terminology we would say that every locally finite simplicial complex coming from a graph G has a ˇCech cover (the pre-base) whose graph is G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It has also a ˇCech cover (coming from the base) which is the Barycentric refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff also notes that a continuous map can be lifted to Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' He however refers to the geometric realization as a polyhedron in order to define something analog to homeo- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff also reformulates the construction of the cohomology ring following Alexander-ˇCech and Whitney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Finite topologies spaces were picked up again as such in the 1960ies like [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Strong for example shows that for any finite topological space, there is a unique minimal base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Connectedness and path connectedness are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' That a continuous self-map f on a finite topological space that is either injective or surjective must be a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Strong shows already that on finite topological spaces, continuity is equivalent with simplicial map: x ⊂ y if and only if f(x) ⊂ f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Strong equips the space HG of continuous maps G → H with the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It is ordered with g ≤ f if for all points g(x) ≤ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If f ≤ g then f, g are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The connectivity components of HG are the homotopy classes of maps from G to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' [44] starts with Finite topological spaces have more interesting topological properties than one might suspect at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Indeed, McCord points out that for every finite topological space, there is a finite simplicial complex which is a weak homotopy equivalent in the sense that the induced maps on all homotopy groups are isomorphisms (meaning for π0 which is not equipped with a group structure, that the number of connected components are the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' McCord is known also for a version of the nerve theorem stating that the homotopy type of a nice topological space is encoded in the ˇCech nerve of a nice open cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This certainly applies for finite topological spaces and the cover coming from minimal open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What is needed for example is that the intersection of two such sets is either contractible or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The ˇCech nerve of a cover has been introduced by Alexandroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 41 FINITE TOPOLOGY 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Code 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The following few Mathematica lines allow to compute the topology of a complex or the topology of the Whitney complex of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We see that the number of topologies on the cyclic graph Cn is the Lucas number L(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We then display the code for Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � Closure [ A ]:= If [A=={},{},Delete [ Union [ Sort [ Flatten [Map[ Subsets ,A ] , 1 ] ] ] , 1 ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Whitney [ s ]:= If [ Length [ EdgeList [ s ]]==0 ,Map[{#}&, VertexList [ s ] ] , Map[ Sort , Sort [ Closure [ FindClique [ s , Infinity , All ] ] ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' UU[ G , x ]:=Module[{U={}},Do[ If [ SubsetQ [G[ [ k ] ] , x ] , U=Append[U,G[ [ k ] ] ] ] , { k , Length [G] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='U] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Basis [ G ]:= Table [UU[G,G[ [ k ] ] ] , { k , Length [G] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' SubBasis [ G ]:=Module[{V=Union [ Flatten [G] ] } , Table [UU[G,{V[ [ k ] ] } ] , { k , Length [V ] } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' UnitSpheres [ G ]:=Module[{B=Basis [G] } , Table [Complement[ Closure [B [ [ k ] ] ] , B [ [ k ] ] ] , { k , Length [B ] } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' UnitBalls [ G ]:=Map[ Closure , Basis [G ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Cl [ U , A ]:=Module[{V=U} ,Do[V=Union [Append[V, Union [V [ [ k ] ] ,A[ [ l ] ] ] ] ] , { k , Length [V]} ,{ l , Length [A] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='V] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Topology [ G ]:=Module[{V=B=Basis [G] } , Do[V=Cl [V,B] , { Length [ Union [ Flatten [G ] ] ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Append[V, { } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' GraphBasis [ s ]:= Basis [ Whitney [ s ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' GraphTopology [ s ]:= Topology [ Whitney [ s ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' N u l l i t y [ Q ]:=Length [ NullSpace [Q ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Fvector [ G ]:= Delete [ BinCounts [Map[ Length ,G] ] , 1 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' ToGraph [ G ] :=Module[{ n=Length [G] , v , e , s } , v=Range[ n ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' e ={};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Do[ If [ ( SubsetQ [G[ [ k ] ] ,G[ [ l ] ] ] | | SubsetQ [G[ [ l ] ] ,G[ [ k ] ] ] ) && Not[G[ [ k]]==G[ [ l ] ] ] , e=Append[ e , v [ [ k]]−>v [ [ l ] ] ] ] , {k , n} ,{ l , k+1,n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' s=UndirectedGraph [ Graph [ v , e ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' BarycentricGraph [ s ]:=ToGraph [ Whitney [ s ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' BarycentricComplex [ G ]:= Whitney [ ToGraph [ s ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' w[ x ]:=−(−1)ˆLength [ x ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Wu1[ A ]:= Total [Map[w,A ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Chi=Wu1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Wu2[ A ]:=Module[{ a=Length [A] } ,Sum[ x=A[ [ k ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sum[ y=A[ [ l ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [MemberQ[A, Intersection [ x , y ] ] , 1 , 0 ] ∗w[ x ]∗w[ y ] , { l , a }] ,{ k , a } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Wu=Wu2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Wu3[ A ]:=Module[{ a=Length [A] } ,Sum[ x=A[ [ k ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sum[ y=A[ [ l ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sum[ z=A [ [ o ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [MemberQ[A, Intersection [ x , y , z ] ] , 1 , 0 ] ∗w[ x ]∗w[ y ]∗w[ z ] , { o , a }] ,{ l , a }] ,{ k , a } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' FastChi [ A ]:=Module[{UU=Basis [A] } ,Sum[w[A[ [ k ] ] ] ∗ Chi [UU[ [ k ] ] ] , { k , Length [A ] } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' FastWu [ A ]:=Module[{UU=Basis [A] } , Sum[w[A[ [ k ] ] ] ∗Wu[UU[ [ k ] ] ] , { k , Length [A ] } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' FastWu3 [ A ]:=Module[{UU=Basis [A] } ,Sum[w[A[ [ k ] ] ] ∗ Wu3[UU[ [ k ] ] ] , { k , Length [A ] } ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Suspension [ G ]:=Module[{ q=Max[ Flatten [G]]+1 , n=Length [G] } , Closure [ Union [ Table [Append[G[ [ k ] ] , q ] , { k , n } ] , Table [Append[G[ [ k ] ] , q+1] ,{k , n } ] ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' JoinAddition [ A , B ]:=Module[{ q=Max[ Flatten [A] ] ,Q,G=A} ,Q=Table [B [ [ k ]]+ q , { k , Length [B ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Do[G=Append[G, Union [A [ [ a ] ] ,Q[ [ b ] ] ] ] , { a , Length [A]} ,{ b , Length [Q] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='G=Union [G,Q] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [A=={},G=B ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [B=={},G=A] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' DoubleSuspension [ G ]:= Suspension [ Suspension [G ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' WuBetti [ G ]:=Module[{ Cohomology2 , n , n2 ,G2, l l , ln , dd1 , dd2 , LL2 , L2 , dd , br , D2,DD} , n=Length [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' length [ x ]:=Length [ x [ [ 1 ] ] ] + Length [ x [ [ 2 ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G2={};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' IS=Intersection ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Do[ If [ Length [ IS [G[ [ k ] ] ,G[ [ l ] ] ] ] > 0 ,G2=Append[G2,{G[ [ k ] ] ,G[ [ l ] ] } ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' n2=Length [G2 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G2=Sort [G2, length [#1]< length [#2] & ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' l l = Map[ length ,G2 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' ln=Union [ l l ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' br=Prepend [ Table [Max[ Flatten [ Position [ l l , ln [ [ k ] ] ] ] ] , { k , Length [ ln ] } ] , 0 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' d e r i v a t i v e 1 [ { x , y }]:=Table [{ Sort [ Delete [ x , k ] ] , y} ,{k , Length [ x ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' dd1=Table [0 ,{ n2 } ,{ n2 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Do[ u=d e r i v a t i v e 1 [G2 [ [m ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [ Length [ u] >0 , Do[ r=Position [G2, u [ [ k ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [ r !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='={} , dd1 [ [m, r [ [ 1 , 1 ] ] ] ] = ( − 1 ) ˆ k ] , { k , Length [ u ] } ] ] , {m, n2 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' d e r i v a t i v e 2 [ { x , y }] :=Table [{ x , Sort [ Delete [ y , k ] ] } , { k , Length [ y ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' dd2 = Table [0 ,{ n2 } ,{ n2 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Do[ u = d e r i v a t i v e 2 [G2 [ [m ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [ Length [ u] >0 , Do[ r=Position [G2, u [ [ k ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If [ r !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='={} , dd2 [ [m, r [ [ 1 , 1 ] ] ] ] = ( − 1 ) ˆ ( Length [G2 [ [m, 1 ] ] ] + k ) ] , {k , Length [ u ] } ] ] , {m, n2 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' dd = dd1 + dd2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' D2=dd+Transpose [ dd ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' L2 =D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='D2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' LL2=Table [ Table [ L2 [ [ br [ [ k ]]+ i , br [ [ k ] ] + j ] ] , { i , br [ [ k+1]]−br [ [ k ] ] } , { j , br [ [ k + 1]] − br [ [ k ] ] } ] , {k , Length [ br ] −1}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Cohomology2=Map[ NullSpace , LL2 ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Map[ Length , Cohomology2 ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � � 42 OLIVER KNILL 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As example computations, we compute the number of elements in the topology of a circle Cn where |G| = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can show by induction in n that the number of open sets in the topology is L(2n), where L(n) is the Lucas number defined by L(0) = 2, L(1) = 1, L(2) = 3 and the recursion L(n + 1) = L(n) + L(n − 1) is the Fibonacci recursion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (The only difference is that for the Lucas numbers, the entry L(0) = 2, while for the Fibonacci numbers, the entry F(0) = 1 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=') � Table [ Length [ GraphTopology [ CycleGraph [ k ] ] ] , { k , 4 , 7 } ] Table [ LucasL [2 n ] , { n , 4 , 7 } ] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Here we compute the Euler characteristic and Wu characteristic of star graphs: � Table [ s=StarGraph [ k ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' { Chi [ Whitney [ s ] ] ,Wu[ Whitney [ s ] ] } , { k , 3 , 1 0 } ] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Here we compute the Wu characteristic of the basis of a random graph � s=RandomGraph [ { 1 5 , 5 4 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Map[Wu, GraphBasis [ s ] ] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is the code for Figure 1 � e={1−>2,2−>3,3−>1,3−>4,3−>6,3−>8,8−>9,8−>10};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' V=ViewVertical ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' s=UndirectedGraph [ Graph [ e ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' BG=BarycentricGraph ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A=GraphPlot3D [ s , ViewPoint−>{1,−3,−1},V− >{1 , −1 , −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' B=GraphPlot3D [BG[BG[ s ] ] , ViewPoint−>{0,−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5,−2},V− >{1 ,0 ,0}];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' S=GraphicsRow [ {A,B} ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Export [ ” f i g u r e 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' pdf ” ,S , ”PDF” ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Show[ S ] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We illustrate Gauss-Bonnet ω(G) = � x∈G ω(x)ω(U(x)) = � x∈G w(x)ω(B(x)) and � x∈G ω(x)ω(S(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These formulas hold for any simplicial complex G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have seen an analog formula χ(G) = � x∈G ω(x)χ(U(x)) for Euler characteristic ω1(G) = χ(G) before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But it holds for any higher characteristic ωm(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � s=RandomGraph [{44 , 2 2 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' { Timing [ FastWu [G] ] , Timing [Wu[G] ] } U=Basis [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' S=UnitSpheres [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' B=UnitBalls [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' {Wu[G] ,Sum[w[G[ [ k ] ] ] ∗Wu[U [ [ k ] ] ] , { k , Length [G] } ] , Sum[w[G[ [ k ] ] ] ∗Wu[ S [ [ k ] ] ] , { k , Length [G] } ] , Sum[w[G[ [ k ] ] ] ∗Wu[B [ [ k ] ] ] , { k , Length [G] } ] } � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For small complexes, the direct Wu computation is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But already if G has several hundred entries, the fast Wu computation is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the first example, where the complex had 59 elements, the direct computation was 4 times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the third of the following cases the fast Wu computation took 23 seconds while the Wu computation took 90 sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The complex had 1355 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � s=RandomGraph [{14 , 3 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' { Timing [ FastWu [G] ] , Timing [Wu[G] ] } s=RandomGraph [{44 , 2 2 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' { Timing [ FastWu [G] ] , Timing [Wu[G] ] } s=RandomGraph [{54 , 4 2 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' { Timing [ FastWu [G] ] , Timing [Wu[G] ] } � � 43 FINITE TOPOLOGY 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, we measure ω(U(x)) + χ(B(x)) − χ(S(x)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is equivalent to ω(U(x)) ≥ χ(U(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is something, we have not been able to explain yet: � s=RandomGraph [{14 , 3 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U=Basis [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' S=UnitSpheres [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' B=UnitBalls [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Map[ Chi , U] − Map[ Chi , B] + Map[ Chi , S ] Map[Wu, U] − Map[ Chi , U] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also have an energy theorem for Wu characteristic ω(G) = � x,y g2(x, y) , where g2(x, y) = ω(x)ω(y)ω(U(x) ∩ U(y)) is a Green function matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Unlike in the case of Euler characteristic, where g1(x, y) = ω(x)ω(y)χ(U(x) ∩ U(y)) was unimodular, the matrix g2 is no more unimodular in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The determinant is in general not 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For manifolds, we compute it here for a 3-sphere, a double suspension of a cyclic graph C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � s=RandomGraph [{20 , 4 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' n=Length [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' U=Basis [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' g2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗Wu[ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{Wu[G] , Total [ Flatten [ g2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g2 [ [ k , k ] ] , { k , n } ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [ Det [ g2 ] ] � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There would be a lot more to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We can look for example at the sphere Green matrix s2(x, y) = ω(x)ω(y)ω(S(x) ∩ S(y)) and compare it with the ball Green matrix b2(x, y) = ω(x)ω(y)ω(B(x) ∩ B(y)) and the star Green matrix g2(x, y) = ω(x)ω(y)ω(U(x) ∩ U(y)) for which we see that the trace and the total sum of all entries and the determinant are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Is there some significance to the nullities we see in the ball or sphere Green function entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If Wu is replaced with Euler we get Green function matrices s1, b,g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � s=RandomGraph [ { 2 0 , 4 0 } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' G=Whitney [ s ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' n=Length [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='U=Basis [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' S=UnitSpheres [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' B=UnitBalls [G] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' s1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [ S [ [ k ] ] , S [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' s2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [ S [ [ k ] ] , S [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{ Chi [G] , Total [ Flatten [ s1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ s1 [ [ k , k ] ] , { k , n } ] ,Det [ s1 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{Wu[G] , Total [ Flatten [ s2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ s2 [ [ k , k ] ] , { k , n } ] ,Det [ s2 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' b1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [B [ [ k ] ] ,B [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' b2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [B [ [ k ] ] ,B [ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{ Chi [G] , Total [ Flatten [ b1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ b1 [ [ k , k ] ] , { k , n } ] ,Det [ b1 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{Wu[G] , Total [ Flatten [ b2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ b2 [ [ k , k ] ] , { k , n } ] ,Det [ b2 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' g1=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Chi [ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' g2=Table [w[G[ [ k ] ] ] ∗ w[G[ [ l ] ] ] ∗ Wu[ Intersection [U[ [ k ] ] ,U[ [ l ] ] ] ] , { k , n} ,{ l , n } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{ Chi [G] , Total [ Flatten [ g1 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g1 [ [ k , k ] ] , { k , n } ] ,Det [ g1 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [{Wu[G] , Total [ Flatten [ g2 ] ] ,Sum[w[G[ [ k ] ] ] ∗ g2 [ [ k , k ] ] , { k , n } ] ,Det [ g2 ] } ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' a Print [{ N u l l i t y [ g1 ] , N u l l i t y [ b1 ] , N u l l i t y [ s1 ] , N u l l i t y [ g2 ] , N u l l i t y [ b2 ] , N u l l i t y [ s2 ] } ] � � 44 OLIVER KNILL 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Gauss-Bonnet formulas for higher characteristic allows to compute higher invariants more quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It still needs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The homology 3 sphere is implemented with a simplicial complex of 392 simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It has Euler and Wu characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Its suspension has Euler and Wu characteristic 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The double suspension again has Euler and Wu characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We know by the double suspension theorem that the double suspension of the homology sphere has a geometric realization that is homeomorphic to a 5 sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We unfortunately can not compute the Wu cohomology yet as the complex is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We suspect that Wu cohomology can distinguish the double suspension of the homology sphere from the 5 sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The following computation still needs a few minutes to compute, even so we use the Gauss-Bonnet version for Wu characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Already in the case of the Suspension of the homology sphere, the fast Wu computation is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For small complexes, the direct computation is faster because the fast Wu procedure requires to pre-compute a basis of the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � onesphere=Whitney [ CycleGraph [ 4 ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' moebius = Whitney [ GraphComplement [ CycleGraph [ 7 ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' c y l i n d e r=Whitney [ UndirectedGraph [ Graph [ {1−>2,2−>3,3−>4,4−>1,5−>6,6−>7,7−>8,8−>5,1−>5,5−>2,2−>6,6−>3,3−>7,7−>4,4−>8,8−>1}]]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' twosphere=Suspension [ onesphere ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' threesphere=DoubleSuspension [ onesphere ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' homologyB ={{1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='9} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='{1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2 ,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' {13 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='14 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='15 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='16}};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' homologysphere=Closure [ homologyB ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [ FastWu [ homologysphere ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [ FastWu [ Suspension [ homologysphere ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Print [ FastWu [ DoubleSuspension [ homologysphere ] ] ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Here are some computations of Wu Betti numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We could not yet complete the computation of the Wu cohomology for the double suspension of the homnology sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' � WuBetti [ onesphere ] (∗ {0 ,1 ,1} ∗) WuBetti [ twosphere ] (∗ {0 ,0 ,1 ,0 ,1} ∗) WuBetti [ threesphere ] (∗ {0 ,0 ,0 ,1 ,0 ,0 ,1} ∗) WuBetti [ moebius ] (∗ {0 ,0 ,0 ,0 ,0} ∗) WuBetti [ c y l i n d e r ] (∗ {0 ,0 ,1 ,1 ,0} ∗) WuBetti [ homologysphere ] (∗ not yet able to compute ∗) WuBetti [ DoubleSuspension [ homologysphere ] ] (∗ d i t o ∗) � � 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Example 1: For the smallest positive dimensional example G = K2, we have the simplicial complex G = {(1), (2), (1, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The basis has three elements and consists of {{(1), (1, 2)}, {(2), (1, 2)}, {(1, 2)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The corresponding unit spheres are {{(2)}, {(1)}, {(1), (2)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology has 5 elements {{}, {(1), (1, 2)}, {(2), (1, 2)}, {(1, 2)}, G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any path graph is homeomorphic to this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A homotopy reduction f : K2 → K1 given by 1− > 1, 2− > 1 is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The topology in K1 is {{}, (1)} and the inverse of every of the open sets is an open set in K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There 45 FINITE TOPOLOGY is no continuous surjective map from K1 to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Any Barycentric refinement of K1 is K1 and a map on finite spaces can not increase cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also, any continuous map can only decrease or preserve dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Example 2: All cyclic graphs Cn with n ≥ 4 are homeomorphic but Cn is not homeomorphic to a path graph Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There is no homeomorphism as there would have to be a continuous surjective map f : Cn → Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' This is not possible because there are two unit spheres S(x) in Pm for which the inverse image has 2 elements (a 0-sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' As a 0-sphere S0 is not homeomorphic to a 1-point graph (there is not even a surjective map from K1 to S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can also see that Cn is not homeomorphic to Pm because the Euler characteristic does not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One can also see it from the fact that the fundamental groups do not match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Example 3: the definition of homeomorphism in finite spaces can be used to pro- duce constructive verifications that two spaces are homeomorphic or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' To do so, cover both spaces with balls which intersect in balls then try to match the balls up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Obviously, if the maximal dimension of the two spaces is different they can not be homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Let us assume that we have two complexes G and H which are homeomorphic and both have maximal dimension d, then the number of connected components of d-dimensional maximal balls must be the same in both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Two star graphs S(n) and S(m) with different number of rays can not be homeomorphic for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The star graph S(n) has n different open one-dimensional balls, while S(m) has m different connectivity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Questions 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The definition of “homeomorphism” proposed here seems have all the properties we want: it has invariants like Euler characteristic, Wu characteristic, Lusternik-Schnirelmann category (the minimal number of contractible sets which cover the space), Betti numbers, Wu Betti numbers, cup length, Lebesgue dimension, connectivity type, separation properties or being a manifold are the same for homeomorphic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' There are other notions which are not topological invariants like the number of k- dimensional simplices fk, the eigenvalues of some Hodge Laplacian, curvature, inductive dimension, average simplex cardinality, Dehn-Sommerville invariants for non-manifolds, the Fermi characteristic φ(G) = � x ω(x) which agrees with the determinant of the connection Laplacian det(L) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For Dehn-Sommerville, especially related to Gauss-Bonnet curvatures [18, 25, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have shown for example that the Dehn-Sommerville property is invariant under edge refinement, the join operation and Barycentric refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also Poincar´e-Hopf [19, 32, 33, 31] can be reformulated more conveniently in a topological frame work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' A) One thing we could not explore yet whether the relative Wu characteristic [25, 26] ω(G, H) for a subcomplex H defined as � x∈G,y∈H,x∩y̸=∅ ω(x)ω(y) is depends on the topology of H and on the embedding in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Wu characteristic ω(G) = � x∩y̸=∅ ω(x)ω(y) itself is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We would also like to know to compute the Wu characteristic in a classical manner without triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Obviously, just looking at the structure of open covers does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' What matters also are the local dimensions, the dimensions of the covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' For example, if we glue two manifolds along a k-dimensionial part, then the dimension of this connection matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If two graphs C4 are glued along a point (one calls this a wedge sum), we get the figure 8 graph G with ω(G) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' If we glue it along an edge, we get a digital figure 8 graph H and ω(H) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' These two graphs are not homeomorphic because 46 OLIVER KNILL there are unit sphere which are not homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The vertex degrees of G are 2 or 4 while the vertex degree of H are 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In the context of calculus, there are questions about the minimal number of critical points of locally injective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In general, a critical point is a point x for which S− f (x) = {y ∈ S(x), f(y) < f(x)} is non-contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Points for which the Poincar´e-Hopf index if(x) = 1 − χ(S− f (x)) ̸= 0 are critical points but there can also be critical points of index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Lusternik-Schnirelman inequality assures cup(G) + 1 ≤ cat(G) ≤ cri(G) where cup is the cup length (a homotopy and so topological invariant) and where cri counts the minimal number of critical points, which a locally injective function can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Morse inequalities count the minimal number of critical points of a Morse function can have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Also here, one can ask whether this number is a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' More generally one can ask whether the numbers ck counting the minimal number of Morse critical points of index k is topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' The Morse inequalities produce the general bound � k(−1)kbk ≤ � k(−1)kvk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' B) As the main focus of this note was a definition of homeomorphism, it would be good to know more about redundancies in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We have played with various versions of the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We tried first not to make any requirement about the unit ball and only have the condition for unit spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' An other modification would be to avoid talking about unit spheres and balls and ask that every unit ball B(x) is homeomorphic to f −1B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' (We have asked this only for locally maximal simplices where the unit ball is a ball) This implies that the boundary S(x) is homeomorphic to the boundary of the inverse f −1B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It would also imply the for a maximal simplex where B(x) is a ball, the inverse f −1B(x) is a ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We did not want to use this as a definition however because B(x) is the same so that we have have no induction to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' It still make the definition local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' One could then postulate that all balls of the same dimension are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' But that is less elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also tried to play with the requirement that f −1(B(x)) is contractible (which is weaker than requiring it to be a ball) but we had difficulty from this to establish for example that if two spaces are homeomorphic and one is a manifold, the other must be a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' C) The property that H = f(Gn) is a continuous image was defined as the property that any unit sphere S(x) is homeomorphic to f −1(S(x)) for all x ∈ H and that the inverse of the unit ball B(x) is a ball in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Is this already is enough to establish that also G is a continuous image of some g(Hm)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' In dimensions 0 and 1 it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' D) The Green function matrix gm(x, y) is still a bit of an enigma in the case when it is defined by a general function hm(x, y) defining the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We see for example that g2 is identically zero if h2 is anti-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' We also see that if h(x, y) = 1 everywhere, then g is invertible and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Diskrete R¨aume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' 2, 2, 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Alexandrov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdE1T4oBgHgl3EQfaAQ0/content/2301.03156v1.pdf'} +page_content=' Combinatorial topology.' 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Sato3, 4 +1Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, 50018 Zaragoza (Spain) +2ARAID Foundation, 50018 Zaragoza (Spain) +3Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8577, Japan +4Max Planck Institute for the Structure and Dynamics of Matter, +Luruper Chaussee 149, 22761 Hamburg, Germany +(Dated: January 6, 2023) +Non-equilibrium steady states are created when a periodically driven quantum system is also +incoherently interacting with an environment – as it is the case in most realistic situations. The +notion of Floquet engineering refers to the manipulation of the properties of systems under periodic +perturbations. Although it more frequently refers to the coherent states of isolated systems (or to the +transient phase for states that are weakly coupled to the environment), it may sometimes be of more +interest to consider the final steady states that are reached after decoherence and dissipation take +place. In this work, we propose a computational method to find the multicolor periodic perturbations +that lead to the final steady states that are optimal with respect to a given predefined metric, such +as for example the maximization of the temporal average value of some observable. We exemplify +the concept using a simple model for the nitrogen-vacancy center in diamond: the goal in this case +is to find the driving periodic magnetic field that maximizes a time-averaged spin component. We +show that, for example, this technique permits to prepare states whose spin values are forbidden in +thermal equilibrium at any temperature. +Exploring novel materials in search of desired prop- +erties and functionalities is one of the most important +tasks of material sciences and engineering, as it can sig- +nificantly impact fundamental sciences and practical ap- +plications. For example, the conversion efficiency of solar +cells has been significantly enhanced over the past several +decades through the discovery of various types of mate- +rials [1–3]. Likewise, thanks to the exploration in a vast +materials space, various superconducting materials have +been found [4–7]. In addition to these examples, various +materials explorations have been conducted toward the +realization of desired material properties and functional- +ities in the equilibrium phase [8–10]. +Recently, the exploration and design of material func- +tionalities has been extended to the nonequilibrium phase +of matter under the presence of optical or magnetic driv- +ings. In the seminal work by Oka and Aoki [11], for exam- +ple, the light-induced anomalous Hall effect in graphene +has been theoretically studied in terms of the Floquet +picture, suggesting the emergence of topological states of +matter. Inspired by this work, various groups have inves- +tigated the emergence of new material properties under +electromagnetic drivings. The design of material func- +tionalities in the nonequilibrium phase has thus become +a full new field of research, that is often called Floquet +engineering [12–17]. +In most theoretical works about Floquet engineering, +the states of the target system have been investigated by +considering the time-periodic solutions of the Schr¨odinger +equation. +However, real materials are surrounded by +their environment, and those Floquet states, which are +the time-periodic solutions of the Schr¨odinger equation, +may decay quickly and not be relevant. In fact, recent +theoretical and experimental studies suggest that the re- +alization of the Floquet states can be significantly dis- +turbed by their interaction with the environment [18–22]. +For a practical description of such driven systems, a the- +ory of open-quantum systems under periodic driving has +to be considered. However, understanding such driven +nonequilibrium phases is significantly more difficult. +Recently, we have demonstrated [17] an approach to +Floquet engineering based on the use of quantum optimal +control theory (QOCT) [23–27]: the idea was to allow for +multicolor periodic driving, rather than the monochro- +matic ones that are normally assumed, and to use the +tools of QOCT to find the amplitudes of the various fre- +quency components that optimize a given target property +of the system – in that work, the goal was to modify at +will the (pseudo) band structure of graphene. +However, that work also ignored the effect of the envi- +ronment, and therefore, the found optimal states would +only live in a transient prethermalized phase. To realize +the Floquet control of material properties and functional- +ities in systems more tightly coupled to an environment, +going beyond the conventional Floquet analysis for iso- +lated systems, we extend here that previous concept of +Floquet engineering to open-quantum systems. For this +purpose, we first discuss how to apply optimal control +theory for nonequilibrium steady states of open-quantum +systems under periodic driving, based on a quantum mas- +ter equation. +We then apply the introduced optimal- +control procedure to a model of the NV center of di- +amond under periodic driving, demonstrating that, for +example, driven open quantum systems under optimized +fields may display exotic properties that are forbidden in +the equilibrium phase. Although to our knowledge, no +previous work has attempted the optimization of NESSs +with respect to the external drivings, a related work [28] +arXiv:2301.02004v1 [cond-mat.mtrl-sci] 5 Jan 2023 + +2 +has recently demonstrated the use of automatic differen- +tiation to optimize steady states with respect to internal +system or bath parameters. +Method. +In order to manipulate the nonequilibrium +steady states, we solve the following optimization prob- +lem. Our first assumption is to consider, as master equa- +tion, a Lindblad-type equation [29, 30] with time-periodic +external fields: +˙ρ(t) = − i [H(t), ρ(t)] ++ +� +ij +γij +� +Vijρ(t)V † +ij − 1 +2{V † +ijVij, ρ(t)} +� +. +(1) +Here, the Hamiltonian H(t + T) = H(t) is periodic +with time period T. +We consider it to be composed +of a field-free and and a periodic perturbation part: +H(t) = H0 + g(u, t)V , where g(u, t) = g(u, t + T) is +some T-periodic real function parametrized by the set +u = u1, . . . , uP – the control parameters. The incoherent +part of the evolution is determined by the set of Lind- +blad operators Vij, which we will assume in the following, +without loss of generality, to be the transition operators +Vij = |Ei⟩⟨Ej|, where |Ei⟩ are the field free Hamiltonian +eigenvectors. +We should warn that the previous equation is not uni- +versally valid. In fact, the problem of deriving valid mas- +ter equations for systems with time-dependent Hamil- +tonians is still an open research area. The equation of +Lindblad can only be rigorously derived if the Hamilto- +nian is time independent – and even then, it rests on +several additional conditions, most notably Markov’s ap- +proximation. Various authors have tackled the problem +of deriving master equations for driven systems [31]. In +some circumstances, Lindblad-type equations with time- +dependent Hamiltonians such as Eq. (1) are appropri- +ate [32], and have been used for various purposes [33–35]. +The previous equation is a simplified version of the so- +called Floquet-Lindblad equation [36]. We will work with +it as working hypothesis; furthermore, the optimization +procedured described below can be easily generalized to +more complex master equations. +A Lindblad equation such as the one above can always +be written as a linear equation in Liouville space: +˙ρ(t) = L(u, t)ρ(t) , +(2) +where we now consider ρ(t) to be in vectorized form, i.e +it is a N 2-dimensional complex vector vector, where N is +the dimension of the underlying Hilbert space [37]. The +Lindbladian L(u, t) is the N 2 × N 2 dimensional operator +that results of transforming Eq. (1) into this space. We +split it as: +L(u, t) = L0 + g(u, t)V . +(3) +Let us call ρu(t) to the periodic solution (i.e. ρu(0) = +ρu(T)) of Eq. (2) for a set of parameters u. This solution +corresponds to a non-equilibrium steady-state (NESS). +Note that, in principle, there could be more than one +steady state, but we will consider here that it is unique. +We then consider the time-average function +F(ρ) = 1 +T +� T +0 +dt ˜A(ρ(t)) , +(4) +for some function of density matrices ˜A – in practice, this +will typically be the expectation value of some operator +A: ˜A(ρ) = Tr[Aρ]. The problem that we attempt to solve +is the optimization of function: +G(u) = F(ρu) , +(5) +subject perhaps to some constraint on the parameters u. +Such +class +of +optimization +problems +for +time- +dependent processes that can be controlled by the ma- +nipulation of external handles is the object of (quantum, +in this case) optimal control theory (QOCT). Any func- +tion optimization algorithm requires a method for the +computation of the function; in addition, many efficient +algorithms will also require a method for the computa- +tion of its gradient. Computing the function G essentially +amounts to obtaining the NESS. In the following, we will +show one possible way to do this, and also derive one +expression for the gradient. Note that since +G(u) = 1 +T +� T +0 +dt Tr[Aρu(t)], +(6) +the gradient components may then be computed as: +∂G +∂uk += 1 +T +� T +0 +dt Tr[A∂ρu +∂uk +(t)], +(7) +and therefore the problem in fact amounts to finding +some procedure to compute the derivatives ∂ρu +∂uk . +Let us first rewrite Eq. (2) elementwise: +˙ρα(t) = +� +β +Lαβ(u, t)ρβ(t) +(8) +and consider the Fourier transform of these objects: +ρα(t) = +� +n +ρα,neiωnt, +(9) +ρα,n = 1 +T +� T +0 +dt e−iωntρα(t), +(10) +Lαβ(u, t) = +� +n +Lαβ,n(u)eiωnt, +(11) +Lαβ,n(u) = 1 +T +� T +0 +dt e−iωntLαβ(u, t), +(12) +where ωn = +2π +T n , +n = 0, 1, . . . , N − 1. +In the fre- +quency domain, the Lindblad equation, Eq. (2), can then +be rewritten as [38]: +� +β +N−1 +� +m=0 +[Lαβ,n−m(u) − iδnmδαβωm] ρβ,m = 0. +(13) + +3 +And, by further defining the following operator +Lαn,βm(u) = Lαβ,n−m(u) − iδnmδαβωm, +(14) +we finally rewrite Eq. (2) as: +� +β +N−1 +� +m=0 +Lαn,βm(u)ρβ,m = 0. +(15) +This is a linear homogeneous equation; the solution +(the nullspace or kernel, assuming that it has dimension +one), will be the periodic solution that we are after, the +NESS [39]. We now need some procedure to find +∂ρ +∂uk . +Taking variations of Eq. (15) with respect to the param- +eters u, we get: +L(u) ∂ρ +∂uk +(u) = − ∂L +∂um +(u)ρu. +(16) +This is a linear equation that would provide ∂ρu +∂uk . How- +ever, note that since L(u) has a non-empty kernel (given +precisely by ρu), it cannot be solved straightforwardly. In +fact, it does not have a unique solution: If x is a solution +of +L(u)x = − ∂L +∂um +(u)ρ(u), +(17) +x + µρu is also a solution for any µ. +To remove this +arbitrariness, we impose the normalization condition, +Trρu = 1 for any u, and therefore: +Tr∂ρu +∂uk += 0. +(18) +To find +∂ρu +∂uk in practice, we may then take the follow- +ing two steps: First, we compute a solution of the linear +equation, Eq. (17), with the least-squares method, by im- +posing that the solution x0 is perpendicular to the kernel, +i.e.: x† +0 · ρu = 0. Then, we update the solution with the +condition, Eq. (18). The required solution is obtained as: +∂ρu +∂uk += x0 − (Trx0)ρu. +(19) +Once we have +∂ρu +∂uk , we can evaluate the gradient in +Eq. (7). Armed with this procedure to compute this gra- +dient, one can perform the optimization of function G(u) +with many efficient algorithms. +Results. In the following, we will use the previous equa- +tions with the following model of the NV center of dia- +mond [33, 40]: +H(u, t) = H0 + V (u, t), +(20) +H0 = −BsSz + NzS2 +z + Nxy(S2 +x − S2 +y), +(21) +V (u, t) = −gx(t)BdSx − gy(t)BdSy. +(22) +The model definition must be completed with the +definition of the dissipative part: +we take γij += +0 +2 +4 +6 +8 +10 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +< Sz > += 3 +< Sz > += 3 + 0.090 +FIG. 1. +Thermal average of Sz, as a function of the inverse +temperature β = +1 +kBT . The value at β = 3, used in the text +for the rest of the calculations, is singled out. Inset: structure +of the Nitrogen vacancy defect in diamond. +γe−βEi/(e−βEi +e−βEj) and γii = 0, where β = 1/(kBT) +is the inverse of the temperature, and γ is a rate con- +stant [41]. The reason for choosing this model is the work +of Ikeda et al. [33], who studied the NESSs of this system +under circularly polarized light (gx(t) = cos(ωt); gy(t) = +sin(ωt)). In that work, the high-frequency approximation +was used in order to derive simplified expressions for the +NESS. Here, the goal would be to parametrize functions +gx = gx(u, t) and gy = gy(u, t), and find the parameters u +that result in a NESS that maximizes the time-averaged +value of some observable (for example, Sz). +Following Ikeda et al. [33], we set the units of the model +by fixing Nz = 1; the rest of the parameters of the model +are then given by: Nxy = 0.05, Bs = 0.3, Bd = 0.1, γ = +0.2 (see [40] for a review on the NV diamond centers, +this and other models, and the typical values that these +constants may take). +First, let us consider the field-free value of Sz; the ther- +mal average of Sz, ⟨Sz⟩β, is shown in Fig. 1 as a function +of the inverse temperature β. One can see how at zero +temperature (β → ∞), ⟨Sz⟩β → 0, reflecting the fact that +the ground-state value of Sz is also zero: ⟨ψ0|Sz|ψ0⟩ = 0. +As the temperature increases, the population of the first +excited state grows, and therefore the thermal average +of Sz also grows, since ⟨ψ1|Sz|ψ1⟩ ≈ 1. However, if the +temperature is increased further, the population of the +second excited state also starts to grow, and the ther- +mal average starts to decrease, as ⟨ψ2|Sz|ψ2⟩ ≈ −1. In +the limit of infinite temperature (β → 0), the thermal +average approaches zero again, as that limit involves an +equally populated ensemble of all three states. Note then +that a thermal control of Sz, i.e. the manipulation of the +value of Sz via a variation of the temperature, is limited +to the range 0 < ⟨Sz⟩β < 0.14. + +c +V +N4 +However, as we will show, if a periodic perturbation +is added, this range can be enlarged, and one may reach +NESSs with larger or smaller values of the (time aver- +aged) Sz. In the following, let us fix β = 3, and seek for +the drivings that are capable of producing those NESSs. +The first step is to set a parametrized form for the time- +dependent functions gx and gy used in Eq. (22); the sim- +plest choice is to use Fourier expansions: +gx(u, t) = u0 + +M +� +n=1 +[u2n cos(ωnt) + u2n−1 sin(ωnt)] ,(23) +gy(u, t) = u2M+1 + +M +� +n=1 +[u2M+1+2n cos(ωnt)+ +u2M+2n sin(ωnt)] . +(24) +The control parameters are therefore the Fourier co- +efficients of the temporal shape of the two magnetic +fields, u0, . . . u4M+1. The index M determines the cut- +off frequency ωM, whereas all the Fourier frequencies are +ωn = nω0 for n = 1, . . . , M. A choice must then be made +on the fundamental frequency ω0, which is of course re- +lated to the period that we choose for the external field +ω0 = 2π +T . In this work, we have chosen ω0 = 0.5 Nz, and +M = 4, such that the cutoff frequency is ωM = 2.0 Nz. +By defining the control functions in this parametrized +manner, we effectively constrain the final solution to a +given domain of validity – in this case setting a maxi- +mum frequency. This would be consistent with any ex- +perimental realization of this concept, as in practice the +time-dependent magnetic fields would also be constrained +in frequencies due to technological limitations. +The optimization of function (6) may then be started +using any gradient-based algorithm – the one that we +have used for these calculations is the Sequential Least- +Squares Quadratic Programming (SLSQP) algorithm [42] +as implemented in the NLOPT library [43]. Note that we +have not performed an unconstrained maximization for +all possible values of parameters uj, but we have added +a constraint on the amplitudes of each frequency compo- +nent: +|uj| ≤ κ +for any j. +(25) +Such a constraint would also be present in an experiment. +The chosen algorithm permits to include this constraint. +Fig. 2 shows the results of one optimization; in this +case the amplitudes were constrained using κ = 4.0. The +optimization is started with random fields (shown in the +top panel, with dashed lines), and then proceeds itera- +tively until the fields that optimize the temporal aver- +age of Sz are found (shown in the top panel, with solid +lines). In the bottom panel, the evolutions in time of Sz +are shown, once again for the initial guess and for the +optimized case. It can be seen how the optimized fields +lead to significantly higher values of Sz – both with re- +spect to the initial random fields, and with respect to +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +g (t) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t/T +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +< Sz > (t) +FIG. 2. +Top: +Optimized (solid lines) and initial guess +(dashed lines) temporal shapes of the time-dependent mag- +netic fields gx (red) and gy (blue). Bottom: Evolution of ⟨Sz⟩ +when using the initial guess (dashed line) and the optimal +fields (solid line). The green line represent the thermal aver- +age at β = 3. +0 +1 +2 +3 +4 +5 +6 +0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +Sz += 0.025 += 0.8 += 0.025 += 0.8 +FIG. 3. +Maximized (blue) and minimized (red) values of +the time-averaged Sz expectation value, ⟨⟨Sz⟩⟩, as a function +of the amplitude bound κ. +The various curves correspond +to different values of the rate constant γ, which are doubled +from γ = 0.025 to γ = 0.8. The shaded region marks the only +allowed values of Sz in thermal equilibrium (thus, for example +⟨Sz⟩β > 0). +the thermal value (shown as a straight green line in the +plot). +In fact, the time-averaged value of Sz achieved +in this way (≈ 0.38) is higher than the maximum that +can be achieved in equilibrium phase by modifyng the +temperature (≈ 0.14, as discussed above). +The final optimized value of function G (i.e. +of the +time averaged value of Sz) obviously depends on how we +constrain the periodic functions. +For example, on the +bound κ that we set on the amplitudes. Fig. 3 shows the + +5 +optimal value obtained as a function of that bound (red +curves), for various values of the dissipation constant γ. +Obviously, if the bound is set to a very small value, the +presence of the periodic field barely modifies the ther- +mal average (of around 0.09, for the chosen temperature +value, β = 3). However, if the bound is relaxed to higher +values, the average can be significantly increased, up to +a saturation value that depends on γ: the higher the γ, +the lower the value of the optimized ⟨⟨Sz⟩⟩. +This can +be understood physically, as a faster dissipation drives +with more strength the system towards its thermal equi- +librium state. Finally, we have attempted to minimize +the time average of Sz, wondering whether one can engi- +neer states with the in principle forbidden negative spin +values. In Fig. 3 we display the obtained optimal values, +also as a function of the amplitude bound (red curves). +It may be seen how, if sufficiently big amplitudes are al- +lowed, one may actually obtain negative values – which +are forbidden in thermal equilibrium, as it can be seen in +Fig. 1. +Summary and Outlook. We have developed an opti- +mal control scheme for the nonequilibrium steady states +of open quantum systems under time-periodic drivings, +aiming to control the properties of matter in nonequi- +librium phases. We derived an expression for the gra- +dient vectors of physical observables in NESSs with re- +spect to the parameters of the external periodic fields, +and we employed these derived gradient vectors for the +optimization of observables of the diamond NV center un- +der external periodic magnetic fields. We confirmed that +the time-averaged value of the spin component, Sz, can +be controled with the proposed optimal control sheme. +Furthermore, we demonstrated that this technique can +be used to find “exotic” NESSs, such as states that dis- +play properties that are forbidden in equilibrium phases: +As shown in Fig. 3, the z-spin component of the opti- +mized NESS can be outside the range of values allowed +in equilibrium – for example, it may be negative, which +is impossible at any temperature. +Having established an optimal control scheme for +NESSs under periodic driving, the field parameters can +be added as novel degrees of freedom for material explo- +rations aimed to endow the materials with desired prop- +erties and functionalities. This extends the concept of +material exploration, from equilibrium to nonequilibrium +situations. Because the present optimization scheme is +based on the steady state solutions of a master equation, +such as Lindblad’s equation [Eq. (1)], the relaxation and +dissipation effects are naturally included in the optimiza- +tion procedure. Hence, the engineering of material prop- +erties based on the proposed scheme can be seen as an +extension of the more common Floquet engineering usu- +ally based on the steady solutions of the time-dependent +Schr¨odinger equation without taking into account the re- +laxation and dissipation effects. The optimal control of +NESSs proposed in this work shows how the difficulties +of Floquet engineering due to the relaxation and dissipa- +tion effects can be overcome, and the natural inclusion +of these effects opens a path to the control of material +properties with experimentally realizable fields. +AC +acknowledges +support +from +Grant +PID2021-123251NB-I00 +funded +by +MCIN/AEI/10.13039/501100011033. SAS acknowledges +the support from JSPS KAKENHI Grant Numbers +JP20K14382. +∗ acastro@bifi.es +[1] A. Polman, M. Knight, E. C. Garnett, B. Ehrler, and +W. C. 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Home, Quantum harmonic oscillator state synthesis +by reservoir engineering, Science 347, 53 (2015). +[35] M. Hartmann, D. Poletti, M. Ivanchenko, S. Denisov, and +P. H¨anggi, Asymptotic floquet states of open quantum +systems: the role of interaction, New Journal of Physics +19, 083011 (2017). +[36] T. N. Ikeda, K. Chinzei, and M. Sato, Nonequilib- +rium steady states in the Floquet-Lindblad systems: +van Vleck’s high-frequency expansion approach, SciPost +Phys. Core 4, 033 (2021). +[37] T. F. Havel, Robust procedures for converting among +lindblad, kraus and matrix representations of quantum +dynamical semigroups, Journal of Mathematical Physics +44, 534 (2003). +[38] These equations are easily reached using the following +two formulas: +1 +T +� T +0 +dt ˙ρα(t)e−iωnt = iωnρα,n, +and +1 +T +� T +0 +dt Lαβ(u, t)ρβ(t)e−iωnt = +N−1 +� +n=0 +Lαβ,n−m(u)ρβ,m. +. +[39] Other procedures could be used to compute the NESS, +sometimes also called “asymptotic Floquet states”, such +as for example simply propagating the equation for a long +time, as the system should decay to the steady state. +[40] L. Rondin, J.-P. Tetienne, T. Hingant, J.-F. Roch, +P. Maletinsky, and V. Jacques, Magnetometry with +nitrogen-vacancy +defects +in +diamond, +Reports +on +Progress in Physics 77, 056503 (2014). +[41] Notice that this dissipation model ensures the detailed +balance condition, γije−βEj = γjie−βEi. +[42] D. Kraft, Algorithm 733: Tomp–fortran modules for op- +timal control calculations, ACM Trans. Math. Softw. 20, +262–281 (1994). +[43] S. G. Johnson, The nlopt nonlinear-optimization pack- +age, http://github.com/stevengj/nlopt. + diff --git a/QtA0T4oBgHgl3EQfDf99/content/tmp_files/load_file.txt b/QtA0T4oBgHgl3EQfDf99/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..da1af3e5ac345662509dfd12900b824db9919b52 --- /dev/null +++ b/QtA0T4oBgHgl3EQfDf99/content/tmp_files/load_file.txt @@ -0,0 +1,596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf,len=595 +page_content='Floquet engineering non-equilibrium steady states Alberto Castro1, 2, ∗ and Shunsuke A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Sato3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 4 1Institute for Biocomputation and Physics of Complex Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' University of Zaragoza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 50018 Zaragoza (Spain) 2ARAID Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 50018 Zaragoza (Spain) 3Center for Computational Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' University of Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Tsukuba 305-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Japan 4Max Planck Institute for the Structure and Dynamics of Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Luruper Chaussee 149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 22761 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Germany (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 2023) Non-equilibrium steady states are created when a periodically driven quantum system is also incoherently interacting with an environment – as it is the case in most realistic situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The notion of Floquet engineering refers to the manipulation of the properties of systems under periodic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Although it more frequently refers to the coherent states of isolated systems (or to the transient phase for states that are weakly coupled to the environment), it may sometimes be of more interest to consider the final steady states that are reached after decoherence and dissipation take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In this work, we propose a computational method to find the multicolor periodic perturbations that lead to the final steady states that are optimal with respect to a given predefined metric, such as for example the maximization of the temporal average value of some observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We exemplify the concept using a simple model for the nitrogen-vacancy center in diamond: the goal in this case is to find the driving periodic magnetic field that maximizes a time-averaged spin component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We show that, for example, this technique permits to prepare states whose spin values are forbidden in thermal equilibrium at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Exploring novel materials in search of desired prop- erties and functionalities is one of the most important tasks of material sciences and engineering, as it can sig- nificantly impact fundamental sciences and practical ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' For example, the conversion efficiency of solar cells has been significantly enhanced over the past several decades through the discovery of various types of mate- rials [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Likewise, thanks to the exploration in a vast materials space, various superconducting materials have been found [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In addition to these examples, various materials explorations have been conducted toward the realization of desired material properties and functional- ities in the equilibrium phase [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Recently, the exploration and design of material func- tionalities has been extended to the nonequilibrium phase of matter under the presence of optical or magnetic driv- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the seminal work by Oka and Aoki [11], for exam- ple, the light-induced anomalous Hall effect in graphene has been theoretically studied in terms of the Floquet picture, suggesting the emergence of topological states of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Inspired by this work, various groups have inves- tigated the emergence of new material properties under electromagnetic drivings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The design of material func- tionalities in the nonequilibrium phase has thus become a full new field of research, that is often called Floquet engineering [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In most theoretical works about Floquet engineering, the states of the target system have been investigated by considering the time-periodic solutions of the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' However, real materials are surrounded by their environment, and those Floquet states, which are the time-periodic solutions of the Schr¨odinger equation, may decay quickly and not be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In fact, recent theoretical and experimental studies suggest that the re- alization of the Floquet states can be significantly dis- turbed by their interaction with the environment [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' For a practical description of such driven systems, a the- ory of open-quantum systems under periodic driving has to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' However, understanding such driven nonequilibrium phases is significantly more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Recently, we have demonstrated [17] an approach to Floquet engineering based on the use of quantum optimal control theory (QOCT) [23–27]: the idea was to allow for multicolor periodic driving, rather than the monochro- matic ones that are normally assumed, and to use the tools of QOCT to find the amplitudes of the various fre- quency components that optimize a given target property of the system – in that work, the goal was to modify at will the (pseudo) band structure of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' However, that work also ignored the effect of the envi- ronment, and therefore, the found optimal states would only live in a transient prethermalized phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' To realize the Floquet control of material properties and functional- ities in systems more tightly coupled to an environment, going beyond the conventional Floquet analysis for iso- lated systems, we extend here that previous concept of Floquet engineering to open-quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' For this purpose, we first discuss how to apply optimal control theory for nonequilibrium steady states of open-quantum systems under periodic driving, based on a quantum mas- ter equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We then apply the introduced optimal- control procedure to a model of the NV center of di- amond under periodic driving, demonstrating that, for example, driven open quantum systems under optimized fields may display exotic properties that are forbidden in the equilibrium phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Although to our knowledge, no previous work has attempted the optimization of NESSs with respect to the external drivings, a related work [28] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='02004v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='mtrl-sci] 5 Jan 2023 2 has recently demonstrated the use of automatic differen- tiation to optimize steady states with respect to internal system or bath parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In order to manipulate the nonequilibrium steady states, we solve the following optimization prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Our first assumption is to consider, as master equa- tion, a Lindblad-type equation [29, 30] with time-periodic external fields: ˙ρ(t) = − i [H(t), ρ(t)] + � ij γij � Vijρ(t)V † ij − 1 2{V † ijVij, ρ(t)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (1) Here, the Hamiltonian H(t + T) = H(t) is periodic with time period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We consider it to be composed of a field-free and and a periodic perturbation part: H(t) = H0 + g(u, t)V , where g(u, t) = g(u, t + T) is some T-periodic real function parametrized by the set u = u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' , uP – the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The incoherent part of the evolution is determined by the set of Lind- blad operators Vij, which we will assume in the following, without loss of generality, to be the transition operators Vij = |Ei⟩⟨Ej|, where |Ei⟩ are the field free Hamiltonian eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We should warn that the previous equation is not uni- versally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In fact, the problem of deriving valid mas- ter equations for systems with time-dependent Hamil- tonians is still an open research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The equation of Lindblad can only be rigorously derived if the Hamilto- nian is time independent – and even then, it rests on several additional conditions, most notably Markov’s ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Various authors have tackled the problem of deriving master equations for driven systems [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In some circumstances, Lindblad-type equations with time- dependent Hamiltonians such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (1) are appropri- ate [32], and have been used for various purposes [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The previous equation is a simplified version of the so- called Floquet-Lindblad equation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We will work with it as working hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' furthermore, the optimization procedured described below can be easily generalized to more complex master equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' A Lindblad equation such as the one above can always be written as a linear equation in Liouville space: ˙ρ(t) = L(u, t)ρ(t) , (2) where we now consider ρ(t) to be in vectorized form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='e it is a N 2-dimensional complex vector vector, where N is the dimension of the underlying Hilbert space [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The Lindbladian L(u, t) is the N 2 × N 2 dimensional operator that results of transforming Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (1) into this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We split it as: L(u, t) = L0 + g(u, t)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (3) Let us call ρu(t) to the periodic solution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' ρu(0) = ρu(T)) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (2) for a set of parameters u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' This solution corresponds to a non-equilibrium steady-state (NESS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Note that, in principle, there could be more than one steady state, but we will consider here that it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We then consider the time-average function F(ρ) = 1 T � T 0 dt ˜A(ρ(t)) , (4) for some function of density matrices ˜A – in practice, this will typically be the expectation value of some operator A: ˜A(ρ) = Tr[Aρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The problem that we attempt to solve is the optimization of function: G(u) = F(ρu) , (5) subject perhaps to some constraint on the parameters u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Such class of optimization problems for time- dependent processes that can be controlled by the ma- nipulation of external handles is the object of (quantum, in this case) optimal control theory (QOCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Any func- tion optimization algorithm requires a method for the computation of the function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' in addition, many efficient algorithms will also require a method for the computa- tion of its gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Computing the function G essentially amounts to obtaining the NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the following, we will show one possible way to do this, and also derive one expression for the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Note that since G(u) = 1 T � T 0 dt Tr[Aρu(t)], (6) the gradient components may then be computed as: ∂G ∂uk = 1 T � T 0 dt Tr[A∂ρu ∂uk (t)], (7) and therefore the problem in fact amounts to finding some procedure to compute the derivatives ∂ρu ∂uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Let us first rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (2) elementwise: ˙ρα(t) = � β Lαβ(u, t)ρβ(t) (8) and consider the Fourier transform of these objects: ρα(t) = � n ρα,neiωnt, (9) ρα,n = 1 T � T 0 dt e−iωntρα(t), (10) Lαβ(u, t) = � n Lαβ,n(u)eiωnt, (11) Lαβ,n(u) = 1 T � T 0 dt e−iωntLαβ(u, t), (12) where ωn = 2π T n , n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the fre- quency domain, the Lindblad equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (2), can then be rewritten as [38]: � β N−1 � m=0 [Lαβ,n−m(u) − iδnmδαβωm] ρβ,m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (13) 3 And, by further defining the following operator Lαn,βm(u) = Lαβ,n−m(u) − iδnmδαβωm, (14) we finally rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (2) as: � β N−1 � m=0 Lαn,βm(u)ρβ,m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (15) This is a linear homogeneous equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the solution (the nullspace or kernel, assuming that it has dimension one), will be the periodic solution that we are after, the NESS [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We now need some procedure to find ∂ρ ∂uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Taking variations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (15) with respect to the param- eters u, we get: L(u) ∂ρ ∂uk (u) = − ∂L ∂um (u)ρu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (16) This is a linear equation that would provide ∂ρu ∂uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' How- ever, note that since L(u) has a non-empty kernel (given precisely by ρu), it cannot be solved straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In fact, it does not have a unique solution: If x is a solution of L(u)x = − ∂L ∂um (u)ρ(u), (17) x + µρu is also a solution for any µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' To remove this arbitrariness, we impose the normalization condition, Trρu = 1 for any u, and therefore: Tr∂ρu ∂uk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (18) To find ∂ρu ∂uk in practice, we may then take the follow- ing two steps: First, we compute a solution of the linear equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (17), with the least-squares method, by im- posing that the solution x0 is perpendicular to the kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' : x† 0 · ρu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Then, we update the solution with the condition, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The required solution is obtained as: ∂ρu ∂uk = x0 − (Trx0)ρu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (19) Once we have ∂ρu ∂uk , we can evaluate the gradient in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Armed with this procedure to compute this gra- dient, one can perform the optimization of function G(u) with many efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the following, we will use the previous equa- tions with the following model of the NV center of dia- mond [33, 40]: H(u, t) = H0 + V (u, t), (20) H0 = −BsSz + NzS2 z + Nxy(S2 x − S2 y), (21) V (u, t) = −gx(t)BdSx − gy(t)BdSy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (22) The model definition must be completed with the definition of the dissipative part: we take γij = 0 2 4 6 8 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='14 < Sz > = 3 < Sz > = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='090 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Thermal average of Sz, as a function of the inverse temperature β = 1 kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The value at β = 3, used in the text for the rest of the calculations, is singled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Inset: structure of the Nitrogen vacancy defect in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' γe−βEi/(e−βEi +e−βEj) and γii = 0, where β = 1/(kBT) is the inverse of the temperature, and γ is a rate con- stant [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The reason for choosing this model is the work of Ikeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' [33], who studied the NESSs of this system under circularly polarized light (gx(t) = cos(ωt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' gy(t) = sin(ωt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In that work, the high-frequency approximation was used in order to derive simplified expressions for the NESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Here, the goal would be to parametrize functions gx = gx(u, t) and gy = gy(u, t), and find the parameters u that result in a NESS that maximizes the time-averaged value of some observable (for example, Sz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Following Ikeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' [33], we set the units of the model by fixing Nz = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the rest of the parameters of the model are then given by: Nxy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='05, Bs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='3, Bd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='2 (see [40] for a review on the NV diamond centers, this and other models, and the typical values that these constants may take).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' First, let us consider the field-free value of Sz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the ther- mal average of Sz, ⟨Sz⟩β, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 1 as a function of the inverse temperature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' One can see how at zero temperature (β → ∞), ⟨Sz⟩β → 0, reflecting the fact that the ground-state value of Sz is also zero: ⟨ψ0|Sz|ψ0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' As the temperature increases, the population of the first excited state grows, and therefore the thermal average of Sz also grows, since ⟨ψ1|Sz|ψ1⟩ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' However, if the temperature is increased further, the population of the second excited state also starts to grow, and the ther- mal average starts to decrease, as ⟨ψ2|Sz|ψ2⟩ ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the limit of infinite temperature (β → 0), the thermal average approaches zero again, as that limit involves an equally populated ensemble of all three states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Note then that a thermal control of Sz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the manipulation of the value of Sz via a variation of the temperature, is limited to the range 0 < ⟨Sz⟩β < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' c V N4 However, as we will show, if a periodic perturbation is added, this range can be enlarged, and one may reach NESSs with larger or smaller values of the (time aver- aged) Sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the following, let us fix β = 3, and seek for the drivings that are capable of producing those NESSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The first step is to set a parametrized form for the time- dependent functions gx and gy used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the sim- plest choice is to use Fourier expansions: gx(u, t) = u0 + M � n=1 [u2n cos(ωnt) + u2n−1 sin(ωnt)] ,(23) gy(u, t) = u2M+1 + M � n=1 [u2M+1+2n cos(ωnt)+ u2M+2n sin(ωnt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (24) The control parameters are therefore the Fourier co- efficients of the temporal shape of the two magnetic fields, u0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' u4M+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The index M determines the cut- off frequency ωM, whereas all the Fourier frequencies are ωn = nω0 for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' A choice must then be made on the fundamental frequency ω0, which is of course re- lated to the period that we choose for the external field ω0 = 2π T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In this work, we have chosen ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='5 Nz, and M = 4, such that the cutoff frequency is ωM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 Nz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' By defining the control functions in this parametrized manner, we effectively constrain the final solution to a given domain of validity – in this case setting a maxi- mum frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' This would be consistent with any ex- perimental realization of this concept, as in practice the time-dependent magnetic fields would also be constrained in frequencies due to technological limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The optimization of function (6) may then be started using any gradient-based algorithm – the one that we have used for these calculations is the Sequential Least- Squares Quadratic Programming (SLSQP) algorithm [42] as implemented in the NLOPT library [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Note that we have not performed an unconstrained maximization for all possible values of parameters uj, but we have added a constraint on the amplitudes of each frequency compo- nent: |uj| ≤ κ for any j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (25) Such a constraint would also be present in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The chosen algorithm permits to include this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 2 shows the results of one optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' in this case the amplitudes were constrained using κ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The optimization is started with random fields (shown in the top panel, with dashed lines), and then proceeds itera- tively until the fields that optimize the temporal aver- age of Sz are found (shown in the top panel, with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In the bottom panel, the evolutions in time of Sz are shown, once again for the initial guess and for the optimized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' It can be seen how the optimized fields lead to significantly higher values of Sz – both with re- spect to the initial random fields, and with respect to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 g (t) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 t/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='5 < Sz > (t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Top: Optimized (solid lines) and initial guess (dashed lines) temporal shapes of the time-dependent mag- netic fields gx (red) and gy (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Bottom: Evolution of ⟨Sz⟩ when using the initial guess (dashed line) and the optimal fields (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The green line represent the thermal aver- age at β = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 0 1 2 3 4 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='4 Sz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='025 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='025 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Maximized (blue) and minimized (red) values of the time-averaged Sz expectation value, ⟨⟨Sz⟩⟩, as a function of the amplitude bound κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The various curves correspond to different values of the rate constant γ, which are doubled from γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='025 to γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The shaded region marks the only allowed values of Sz in thermal equilibrium (thus, for example ⟨Sz⟩β > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' the thermal value (shown as a straight green line in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In fact, the time-averaged value of Sz achieved in this way (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='38) is higher than the maximum that can be achieved in equilibrium phase by modifyng the temperature (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='14, as discussed above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The final optimized value of function G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' of the time averaged value of Sz) obviously depends on how we constrain the periodic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' For example, on the bound κ that we set on the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 3 shows the 5 optimal value obtained as a function of that bound (red curves), for various values of the dissipation constant γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Obviously, if the bound is set to a very small value, the presence of the periodic field barely modifies the ther- mal average (of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content='09, for the chosen temperature value, β = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' However, if the bound is relaxed to higher values, the average can be significantly increased, up to a saturation value that depends on γ: the higher the γ, the lower the value of the optimized ⟨⟨Sz⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' This can be understood physically, as a faster dissipation drives with more strength the system towards its thermal equi- librium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Finally, we have attempted to minimize the time average of Sz, wondering whether one can engi- neer states with the in principle forbidden negative spin values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 3 we display the obtained optimal values, also as a function of the amplitude bound (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' It may be seen how, if sufficiently big amplitudes are al- lowed, one may actually obtain negative values – which are forbidden in thermal equilibrium, as it can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Summary and Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We have developed an opti- mal control scheme for the nonequilibrium steady states of open quantum systems under time-periodic drivings, aiming to control the properties of matter in nonequi- librium phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We derived an expression for the gra- dient vectors of physical observables in NESSs with re- spect to the parameters of the external periodic fields, and we employed these derived gradient vectors for the optimization of observables of the diamond NV center un- der external periodic magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' We confirmed that the time-averaged value of the spin component, Sz, can be controled with the proposed optimal control sheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Furthermore, we demonstrated that this technique can be used to find “exotic” NESSs, such as states that dis- play properties that are forbidden in equilibrium phases: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' 3, the z-spin component of the opti- mized NESS can be outside the range of values allowed in equilibrium – for example, it may be negative, which is impossible at any temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Having established an optimal control scheme for NESSs under periodic driving, the field parameters can be added as novel degrees of freedom for material explo- rations aimed to endow the materials with desired prop- erties and functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' This extends the concept of material exploration, from equilibrium to nonequilibrium situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Because the present optimization scheme is based on the steady state solutions of a master equation, such as Lindblad’s equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' (1)], the relaxation and dissipation effects are naturally included in the optimiza- tion procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' Hence, the engineering of material prop- erties based on the proposed scheme can be seen as an extension of the more common Floquet engineering usu- ally based on the steady solutions of the time-dependent Schr¨odinger equation without taking into account the re- laxation and dissipation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' The optimal control of NESSs proposed in this work shows how the difficulties of Floquet engineering due to the relaxation and dissipa- tion effects can be overcome, and the natural inclusion of these effects opens a path to the control of material properties with experimentally realizable fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} +page_content=' AC acknowledges support from Grant PID2021-123251NB-I00 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtA0T4oBgHgl3EQfDf99/content/2301.02004v1.pdf'} 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Game +Aviv Shamsian * 1 Aviv Navon * 1 2 Neta Glazer 1 2 Kenji Kawaguchi 3 +Gal Chechik 1 4 Ethan Fetaya 1 +Abstract +Auxiliary learning is an effective method for en- +hancing the generalization capabilities of trained +models, particularly when dealing with small +datasets. However, this approach may present +several difficulties: (i) optimizing multiple objec- +tives can be more challenging, and (ii) how to +balance the auxiliary tasks to best assist the main +task is unclear. In this work, we propose a novel +approach, named AuxiNash, for balancing tasks +in auxiliary learning by formalizing the problem +as generalized bargaining game with asymmetric +task bargaining power. Furthermore, we describe +an efficient procedure for learning the bargaining +power of tasks based on their contribution to the +performance of the main task and derive theoret- +ical guarantees for its convergence. Finally, we +evaluate AuxiNash on multiple multi-task bench- +marks and find that it consistently outperforms +competing methods. +1. Introduction +When training deep neural networks with limited labeled +data, generalization can be improved by adding auxiliary +tasks. In this approach, called Auxiliary learning (AL), +the auxiliary tasks are trained jointly with the main task, +and their labels can provide a signal that is useful for the +main task. AL is beneficial because auxiliary annotations +are often easier to obtain than annotations for the main task. +This is the case when the auxiliary tasks use self-supervision +(Oliver et al., 2018; Hwang et al., 2020; Achituve et al., +2021), or their annotation process is faster. For example, +learning semantic segmentation of an image may require +careful and costly annotation, but can be improved if learned +jointly with a depth prediction task, whose annotations can +be obtained at scale (Standley et al., 2020). +*Equal contribution +1Bar-Ilan University, Ramat Gan, Is- +rael +2Aiola, Herzliya, Israel +3National University of Sin- +gapore +4Nvidia, +Tel-Aviv, +Israel. +Correspondence +to: +Aviv Shamsian , Aviv Navon +. +Auxiliary learning has a large potential to improve learn- +ing in the low data regime, but it gives rise to two main +challenges: Defining the joint optimization problem and +performing the optimization efficiently. (1) First, given a +main task at hand, it is not clear which auxiliary tasks would +benefit the main task and how tasks should be combined +into a joint optimization objective. For example, Standley +et al. (2020) showed that depth estimation is a useful aux- +iliary task for semantic segmentation but not the opposite. +In fact, adding semantic segmentation as an auxiliary task +harmed depth estimation performance. This suggests that +even close-related tasks may interfere with each other. (2) +Second, training with auxiliary tasks involves optimizing +multiple objectives simultaneously; While training with mul- +tiple tasks can potentially improve performance via better +generalization, it often underperforms compared to single- +task models. +Previous auxiliary learning research focused mainly on the +first challenge: namely, weighting and combining auxiliary +tasks (Lin et al., 2019). The second challenge, optimizing +the main task in the presence of auxiliary tasks, has been +less explored. Luckily, this problem can be viewed as a case +of optimization in Multi-Task Learning (MTL). In MTL, +there is extensive research on controlling optimization such +that every task would benefit from the others. Specifically, +several studies proposed algorithms to aggregate gradients +from multiple tasks into a coherent update direction (Yu +et al., 2020; Liu et al., 2021a; Navon et al., 2022). We +see large potential in bringing MTL optimization ideas into +auxiliary learning to address the optimization challenge. +Here we propose a novel approach named AuxiNash that +takes inspiration from recent advances in MTL optimization +as a cooperative bargaining game (Nash-MTL, Navon et al., +2022). The idea is to view a gradient update as a shared +resource, view each task as a player in a game, and have +players compete over making the joint gradient similar to +their own task gradient. In Nash-MTL, tasks play a symmet- +ric role, since no task is particularly favorable. This leads +to a bargaining solution that is proportionally fair across +tasks. In contrast, task symmetry no longer holds in aux- +iliary learning, where there is a clear distinction between +the primary task and the auxiliary ones. As such, we pro- +pose to view auxiliary learning as an asymmetric bargaining +arXiv:2301.13501v1 [cs.LG] 31 Jan 2023 + +Auxiliary Learning as an Asymmetric Bargaining Game +Figure 1. Illustrative example: A regression problem in R2 with +two auxiliary tasks, one helpful and one harmful. AuxiNash suc- +ceeds in using the helpful auxiliary task and disregards the harmful +one, as demonstrated by how it learns to weigh different tasks: The +left panel shows the preference vector p during optimization. As a +result, AuxiNash converges to a solution with large proximity to +the optimal solution, far superior to that obtained from optimizing +with the main tasks alone (right panel). See Section 6.1 for further +details. +game. Specifically, we consider gradient aggregation as a +cooperative bargaining game where each player represents a +task with varying bargaining power. We formulate gradient +update using asymmetric Nash bargaining solution which +takes into account varying task preferences. By generalizing +Nash-MTL to asymmetric games with AuxiNash, we can +efficiently direct optimization solution towards various areas +of the Pareto front. +Determining the task preferences that result in optimal per- +formance is a challenging problem for several reasons. First, +the relationship between tasks can change during the op- +timization process, making it difficult to know in advance +what preferences to use. This means that the process of +preference tuning needs to be automated during training. +Second, using a grid search to find the optimal preferences +can be computationally expensive, and the complexity of +the search increases exponentially with the number of tasks. +To overcome these limitations, we propose a method for +efficiently differentiating through the optimization process +and using this information to automatically optimize task +preferences during training. This can improve the perfor- +mance of the primary task, and make it more efficient to +find the optimal preferences. +We theoretically analyze the convergence of AuxiNash and +show that even if the preference changes during the opti- +mization process we are still guaranteed to converge to a +Pareto stationary point. Finally, we show empirically on +several benchmarks that AuxiNash achieves superior re- +sults to previous auxiliary learning and multi-task learning +approaches. +Contributions: +This paper makes the following contri- +butions: (1) We introduce AuxiNash - a novel approach +for auxiliary learning based on principles from asymmetric +bargaining games. (2) We describe an efficient method to +dynamically learn task preferences during training. (3) We +theoretically show that AuxiNash is guaranteed to converge +to a Pareto stationary point. (4) We conduct extensive exper- +iments to demonstrate the superiority of AuxiNash against +multiple baselines from MTL and auxiliary learning. +2. Related Work +Auxiliary Learning. +Learning with limited amount of +training data is challenging since deep learning models tend +to overfit to the training data and as a result can generalize +poorly (Ying, 2019). One approach to overcome this limi- +tation is using auxiliary learning (Chen et al., 2022; Kung +et al., 2021). Auxiliary learning aims to improve the model +performance on primary task by utilizing the information of +related auxiliary tasks (Dery et al., 2022; Chen et al.). Most +auxiliary learning approaches use a linear combination of +the main and auxiliary losses to form a unified loss (Zhai +et al., 2019; Wen et al., 2020). Fine-tuning the weight of +each task loss may be challenging as the search space of the +grid search grows exponentially with the number of tasks. +To find the beneficial auxiliary tasks, recent studies utilized +the auxiliary task gradients and measure their similarity with +the main task gradients (Lin et al., 2019; Du et al., 2018; +Shi et al., 2020). Navon et al. (2021) proposed to learn a +non-linear network that combines all losses into a single +coherent objective function. +Multi-task Learning. +In multi-task learning (MTL) we +aim to solve multiple tasks by sharing information between +them (Caruana, 1997; Ruder, 2017), usually through joint +hidden representation (Zhang et al., 2014; Dai et al., 2016; +Pinto & Gupta, 2017; Liu et al., 2019b). Previous studies +showed that optimizing a model using MTL helps boost +performances while being computationally efficient (Sener +& Koltun, 2018; Chen et al., 2018). However, MTL presents +a number of optimization challenges such as: conflicting +gradients (Wang et al., 2020; Yu et al., 2020), and flatten +areas in the loss landscape (Schaul et al., 2019). These chal- +lenges may result with performance degradation compare +with single task learning. Recent studies proposed novel +architectures (Misra et al., 2016; Hashimoto et al., 2017; +Liu et al., 2019b; Chen et al., 2020) to improve MTL while +others focused on aggregating the gradients of the tasks +such that it is agnostic to the optimized model (Liu et al., +2021b; Javaloy & Valera, 2021). Yu et al. (2020) proposed +to overcome the conflicting gradients problem by subtract- +ing normal projection of conflicted task before forming an +update direction. Most gradient based methods aim to mini- +mize the average loss function. Liu et al. (2021a) suggested + +Main +Aux. helpful +Aux. harmful +Optimal +Train optimal +AuxiNash +0.6 +1.5 +Preference +0.4 +1.0 +0.2 +0.0 +0 +500 +1000 +0.5 +1.0 +1.5 +Step +W1Auxiliary Learning as an Asymmetric Bargaining Game +Figure 2. Task preferences: By varying the preference vector p, +we show that AuxiNash can control the trafe-off between tasks. +Compared with Nash-MTL, AuxiNash achieves a wider range of +diverse solutions, an important property for auxiliary learning. See +Section 6.2 for further details. +an approach that will decrease every task loss in addition to +the average loss function. The closest work to our approach +is Nash-MTL (Navon et al., 2022). The authors proposed +a principled approach to dynamically weight the losses of +different tasks by incorporating concepts from game theory. +Bi-level Optimization. +Bi-Level Optimization (BLO) +consists of two nested optimization problems (Liao et al., +2018; Liu et al., 2021c; Vicol et al., 2022). The outer opti- +mization problem is commonly referred to as the upper-level +problem, while the inner optimization problem is referred +to as the lower-level problem (Sinha et al., 2017). BLO +is widely used in a variety of deep learning applications, +spanning hyper-parameter optimization (Foo et al., 2007; +MacKay et al., 2019), meta learning (Franceschi et al., +2018), reinforcement learning (Zhang et al., 2020; Yang +et al., 2019), and multi-task learning (Liu et al., 2022; Navon +et al., 2021). A common practice to derive the gradients of +the upper-level task is using the implicit function theorem +(IFT). However, applying IFT involves the calculation of an +inverse-Hessian vector product which is infeasible for large +deep learning models. Therefore, recent studies proposed di- +verse approaches to approximate the inverse-Hessian vector +product. Luketina et al. (2016) proposed approximating the +Hessian with the identity matrix, where other line of works +used conjugate gradient (CG) to approximate the product +(Foo et al., 2007; Pedregosa, 2016; Rajeswaran et al., 2019). +We use a truncated Neumann series and efficient vector- +Jacobian products, as it was empirically shown to be more +stable than CG (Liao et al., 2018; Lorraine et al., 2020; +Raghu et al., 2020). +3. Background +3.1. Nash Bargaining Solution +We will first give a quick introduction to cooperative +bargaining games. In a bargaining game, a set of K players +jointly decide on an agreed-upon point in the set A of +all agreement points. +If failing to reach an agreement, +the game default to the disagreement point D. +Each +player i ∈ [K] := {1, ..., K} is equipped with a utility +function ui : A ∪ {D} → R, which they wish to maximize. +Intuitively, each player has a different objective, and +each tries to only maximize their own personal utility. +However, we generally assume that there are points in the +agreement set that are mutually beneficial to all players, +compared to the disagreement point, and as such the +players are incentivized to cooperate. The main question +is on which point in the agreement set will they decide upon. +Denote by U = {(u1(x), ..., uK(x)) : x ∈ A} ⊂ RK the +set of the utilities of all possible agreement points and d = +(u1(D), ..., uK(D)). The set U is assumed to be convex and +compact. Furthermore, we assume that there exists a point +u ∈ U for which ∀i : ui > di. Nash (1953) showed that +under these assumptions, there exists a unique solution to the +bargaining game, which satisfies the following properties: +Pareto optimality, symmetry, independence of irrelevant +alternatives, and invariant to affine transformations (see +Sz´ep & Forg´o, 1985, and the supplementary material for +more details). This unique solution, referred to as the Nash +Bargaining solution (NBS), is given by +u∗ = arg max +u∈U +� +i +log(ui − di) +(1) +s.t. ∀i : ui > di +As shown in Navon et al. (2022), NBS properties are suitable +for the multi-task learning setup, even if the invariance +to affine transformations implies that gradient norms are +ignored. The symmetry assumption, however, implies that +each player is interchangeable which is not the case for +auxiliary learning. Naturally, our main concern is the main +task and the auxiliaries are there to support it, not compete +with it. Thus, we wish to discard the symmetry assumption +for the auxiliary learning setup. +3.2. Generalized Bargaining Game +Kalai (1977) generalized the NBS to the asymmetric case. +First, define a preference vector to control the relative trade- +off between tasks p ∈ RK with pi > 0 and � +i pi = 1 (see +Figure 2). Similar to the symmetric case, the Generalized +Nash Bargaining Solution (GNBS) maximizes a weighted + +Muiti-MNiST +0.91 +0.8 +preference +0.89 +C + AC( +0.6 +AsymNash +Task 2 +NashMTL +1 +0.4 +0.87 +ask +0.2 +0.85 +0.87 +0.89 +0.91 +0.93 +Task 1 ACCAuxiliary Learning as an Asymmetric Bargaining Game +product of utilities, +u∗ = arg max +u∈U +� +i +pi log(ui − di) +(2) +s.t. ∀i : ui > di +. +The symmetric case is a special case of GNBS with uniform +preferences pi = 1/K, ∀i ∈ [K]. +3.3. Bargaining Game for Multi-task learning +Recently, Navon et al. (2022) formalized multi-task learning +as a bargaining game as follows. Let θ ∈ Rd denote the +parameters of a network f(·; θ). At each MTL optimiza- +tion step, we search for an update direction ∆θ. Define +the agreement set U = {∆θ | ∥∆θ∥ ≤ 1} as the set of +all possible update directions. The disagreement point d +is defined to be equal to zero, i.e., to stay with the current +parameters and terminate the optimization process. Let +gi denote the gradient of θ w.r.t. the loss of task i (for +each i ∈ [K]). The utility function for task i is defined as +ui(∆θ) = g⊤ +i ∆θ, i.e., the directional derivative in direc- +tion ∆θ. Navon et al. (2022) assumed that the gradients +g1, ..., gK are linearly independent if θ is not Pareto station- +ary, and we adopt that assumption in our analysis. Under +this assumption, Navon et al. (2022) show that the solution +for the bargaining game at any non-Pareto stationary point +θ is given by ∆θ = � +i αigi, where the weight vector α +satisfies +G⊤Gα = 1/α. +(3) +Here, G is the d × K matrix whose i-th column is the i-th +task gradient gi, and 1/α is the element-wise reciprocal. +4. Generalized Bargaining Game for +Auxiliary Learning +In this section, we first extend the result from Navon et al. +(2022) to the asymmetric case. Next, we describe a method +to learn the preference vector p. +4.1. Generalized Bargaining Solution +We prove the following claim, which generalizes the claim +of Navon et al. (2022) to asymmetric games. +Claim 4.1. Let p ∈ RK ++ with � +i pi = 1. +The so- +lution to the generalized bargaining problem ∆θ∗ += +arg max∆θ∈U +� +i pi log(∆θ⊤gi) is given by (up to scal- +ing) � +i αigi at any non-Pareto stationary point θ, where +α ∈ RK ++ is the solution to G⊤Gα = p/α where p/α is the +element-wise reciprocal. +Proof. We define F(∆θ) = � +i pi log(∆θ⊤gi) and have +∇F = �K +i=1 +pi +∆θT gi gi. Note that for all ∆θ with ui(∆θ) > +0 for all i ∈ [K] the utilities are monotonically increasing +in ∥∆θ∥, hence the optimal solution lies on the boundary +of U, and ∇F|∆θ∗ is parallel to ∆θ∗. This implies that +�K +i=1 +pi +∆θ⊤gi gi = λ∆θ for some λ > 0. From the linear +independence assumption, we have for the optimal solution +∆θ = � +i αigi, thus ∀i, ∆θ⊤gi = +pi +λαi . Setting λ = 1 +(as we ignore scale), the solution to the bargaining game +is reduced to finding α ∈ RK ++ for which ∀i, ∆θ⊤gi = +� +j αjg⊤ +j gj = pi/αi. Equivalently, the optimal solution is +given by α ∈ RK ++ such that G⊤Gα = p/α where p/α is +the element-wise reciprocal. +Given a preference vector p, we solve G⊤Gα = p/α by +expressing it as the solution to an optimization problem. We +first solve a convex relaxation which we follow by a concave- +convex procedure (CCP) (Yuille & Rangarajan, 2003; Lipp +& Boyd, 2016), similar to Navon et al. (2022) for solving +G⊤Gα = p/α w.r.t. α. See Appendix C for full details. +4.2. Optimizing the Preference Vector +The derivation in the previous section allows us to learn +using a known preference vector p. Unfortunately, in most +cases, the preference vector is not known in advance. One +simple solution is to treat the preferences pi as hyperparam- +eters and set them via grid search. However, this approach +has two significant limitations. First, as the number of hyper- +parameters increases, grid search becomes computationally +expensive as it scales exponentially. Second, it is possi- +ble that the optimal preference vector would vary during +optimization, hence using a fixed p would be sub-optimal. +To address these issues, we develop an approach for dynam- +ically learning the task preference vector during training. +This reduces the number of hyperparamters the user needs +to tune to one (the preference update rate) and dynamically +adjusts the preference to improve generalization. We do this +by formulating the problem as bi-level optimization, which +we discuss next. +Let LT denote the training loss and LV denote the general- +ization loss, given by the loss of the main task on unseen +data, i.e., LV = ℓval +main. In the auxiliary learning setup, we +wish to optimize p such that a network f(·; θ) optimized +with LT (·; p) would minimize LV . Formally, +p∗ = arg min +p LV (θ∗(p)), s.t. θ∗(p) = arg min +θ +LT (θ, p) +Using the chain rule to get the derivative of the outer prob- +lem, we get +∂LV (p, θ∗(p)) +∂p += ∂LV +∂p +� �� � +=0 ++∂LV +∂θ +∂θ∗ +∂p = ∂LV +∂θ +∂θ∗ +∂α(p) +∂α(p) +∂p +As we can compute ∂LV +∂θ +by simple backpropagation, the +main challenge is to compute +∂θ∗ +∂α(p) and ∂α(p) +∂p . + +Auxiliary Learning as an Asymmetric Bargaining Game +Figure 3. Visualization of the update direction: We show the update direction (blue) obtained by AuxiNash on three gradients in R3. We +rescaled the update directions for better visibility, showing only the direction. We further show the size of the projection (red) of the +update to each gradient direction (black). By varying the preference vector, we observe the change in the obtained update direction. +Importantly, we note that the effect on the update direction is non-trivial, as p only affects the update implicitly through the bargaining +solution α. +To compute +∂θ∗ +∂α(p) we can (indirectly) differentiate through +the optimization process using the implicit function theorem +(IFT) (Liao et al., 2018; Lorraine et al., 2020; Navon et al., +2021): +∂θ∗ +∂α(p) = − +� ∂2LT +∂θ∂θ⊤ +�−1 +∂2LT +∂θ∂α(p)⊤ +(4) +Since computing the inverse Hessian directly is intractable, +we use the algorithm proposed by Lorraine et al. (2020) +to efficiently estimate the inverse-Hessian vector product. +This approach uses the Neumann approximation with an +efficient vector-Jacobian product. Thus, we can efficiently +approximate the first term, ∂LV +∂θ +∂θ∗ +∂α(p). We note that in prac- +tice, as in customary, we do not optimize till convergence +but perform a few gradient updates from the previous value. +For further details, see Vicol et al. (2022) that recently +examined how this affects the bi-level optimization process. +For the second term, ∂α(p) +∂p , we derive a simple analytical +expression using the IFT in the following proposition: +Proposition 4.2. For any (p, α) satisfying G⊤Gα = p/α, +there exists an open set U ⊆ RK containing p such that +there exists a continuously differentiable function ˆα : U → +RK satisfying all of the following properties: (1) ˆα(p) = α, +(2) G⊤Gˆα(¯p) = ¯p/ˆα(¯p) for all ¯p ∈ U, and (3) +∂ ˆα(p) +∂p += +� +G⊤G + Λ0 +�−1 Λ1. +(5) +Here Λ0, Λ1 ∈ RK×K are the diagonal matrices defined by +(Λ0)ii = pi/α2 +i ∈ R and (Λ1)ii = 1/αi ∈ R for i ∈ [K]. +We refer the readers to Appendix B for the proof. +Putting everything together, we obtain the following effi- +cient approximation, +∂LV (p, θ∗(p)) +∂p += −∂LV +∂θ +� ∂2LT +∂θ∂θ⊤ +�−1 +∂2LT +∂θ∂α(p)⊤ × +� +G⊤G + Λ0 +�−1 +Λ1 +(6) +Algorithm 1 AuxiNash +Input: θ – initial parameter vector, p – initial preference +vector, {ℓi}K +i=1 – differentiable loss functions, η, ηp – learning +rates +for T = 1, ..., N do +for t = 1, ..., Np do +Compute task gradients gi = ∇θℓi +Set G the matrix with columns gi +Solve for α: G⊤Gα = p/α +Update the parameters θ ← θ − ηGα +end for +Evaluate ∇pLV using Eq. 6 +Update p ← p − ηp∇pLV +end for +Return: θ. +We note that this approximation can be computed in a rela- +tively efficient manner, with the cost of only several back- +propagation operations to estimate the vector-Jacobian prod- +uct (we use 3 in our experiments). We also note that the +matrix +� +G⊤G + Λ0 +� +that we invert is of size K ×K, where +K is the number of tasks that is generally relatively small. +In practice, we use a separate batch from the training set +to estimate the generalization loss LV . We further discuss +this design choice and provide an empirical evaluation in +Section 6.3. During the optimization process, we alternate +between optimizing θ and optimizing p. Specifically, we +update p once every Np optimization steps over θ. We set +Np = 25 in our experiments. The AuxiNash algorithm is +summarized in Alg. 1. +5. Analysis +We analyze the convergence properties of our proposed +method in nonconvex optimization. We adopt the following +three assumptions from Navon et al. (2022): +Assumption 5.1. We assume that for a sequence {θ(t)}∞ +t=1 +generated by our algorithm, the set of the gradient vectors +g(t) +1 , ..., g(t) +K at any point on the sequence and at any partial +limit are linearly independent unless that point is a Pareto +stationary point. + +NashMTL +AuxiNash (.9,.05,.05) +AuxiNash (.05,.9,.05) +AuxiNash (.05,.05,.9) +92 +92 +92 +92 +g1 +g1 +g1 +g1 +93 +93 +93 +g3Auxiliary Learning as an Asymmetric Bargaining Game +Table 1. NYUv2. Test performance for three tasks: semantic segmentation, depth estimation, and surface normal. Values are averages over +3 random seeds. +Segmentation +Depth +Surface Normal +mIoU ↑ Pix Acc ↑ +Abs Err ↓ Rel Err ↓ +Angle Distance ↓ +Within t◦ ↑ +∆% ↓ +Mean +Median +11.25 +22.5 +30 +STL +38.30 +63.76 +0.6754 +0.2780 +25.01 +19.21 +30.14 +57.20 +69.15 +LS +38.43 +64.36 +0.5472 +0.2184 +29.57 +25.42 +20.50 +44.85 +58.20 +8.69 +PCGrad +39.25 +64.95 +0.5389 +0.2141 +28.66 +24.26 +21.99 +47.00 +60.31 +5.66 +CAGrad +39.25 +65.15 +0.5385 +0.2155 +26.11 +20.95 +26.96 +53.66 +66.37 +−1.46 +Nash-MTL +39.83 +66.00 +0.5235 +0.2075 +25.32 +19.87 +28.86 +55.87 +68.27 +−4.76 +GCS +38.96 +64.35 +0.5769 +0.2293 +29.57 +25.53 +20.64 +44.68 +57.99 +9.54 +OL-AUX +40.51 +65.49 +0.6652 +0.2614 +24.65 +18.72 +30.92 +58.37 +70.12 −2.88 +AuxiLearn +38.63 +64.20 +0.5415 +0.2173 +29.98 +25.29 +20.03 +43.94 +57.17 +9.15 +AuxiNash (ours) +40.79 +66.79 +0.5092 +0.2042 +24.90 +19.31 +29.83 +57.07 +69.27 +−6.80 +Assumption 5.2. We assume that all loss functions are +differentiable, bounded below and that all sub-level sets are +bounded. The input domain is open and convex. +Assumption 5.3. We assume that all the loss functions are +L-smooth, +∥∇ℓi(x) − ∇ℓi(y)∥ ≤ L∥x − y∥. +(7) +Since even single-task non-convex optimization might only +admits convergence to a stationary point, the following the- +orem proves convergence to a Pareto stationary point when +both θ and p are optimized concurrently: +Theorem 5.4. Suppose that Assumptions 5.1, 5.2, and 5.3 +hold. +Let {θ(t)}∞ +t=1 be the sequence generated by the +update rule θ(t+1) = θ(t) − µ(t)∆θ(t) where ∆θ(t) = +�K +i=1 α(t) +i g(t) +i +is the weighted Nash bargaining solution +(G(t))⊤G(t)α(t) = p(t)/α(t) where p(t) can be any discrete +distribution. Set µ(t) = +1 +K +�K +i=1 p(t) +i (Lα(t) +i )−1. The se- +quence {θ(t)}∞ +t=1 has a subsequence that converges to a +Pareto stationary point θ∗. Moreover all the loss functions +(ℓ1(θ(t)), ..., ℓK(θ(t))) converge to (ℓ1(θ∗), ..., ℓK(θ∗)). +See full proof in Appendix B. +6. Experiments +In this section, we compare AuxiNash with different ap- +proaches from multi-task and auxiliary learning. We use +variety of datasets and learning setups to demonstrate the +superiority of AuxiNash. To encourage future research +and reproducibility, we will make our source code publicly +available. Additional experimental results and details are +provided in Appendix D. +Baselines. +We compare AuxiNash with natural baselines +from recent auxiliary and multi-task learning works. The +compared methods includes (1) Single-task learning (STL), +which trains a model using the main task only; (2) Linear +scalarization (LS) that minimizes the sum of losses � +k ℓk; +(3) GCS (Du et al., 2018), an auxiliary learning approach +that uses gradient similarity between primary and auxiliary +tasks; (4) OL-AUX (Lin et al., 2019), an auxiliary learning +approach that adaptively changes the loss weight based on +the gradient inner product w.r.t the main task; (5) AuxiLearn +(Navon et al., 2021), an auxiliary learning approach that dy- +namically learns non-linear combinations of different tasks; +(6) PCGrad (Yu et al., 2020), an MTL method that removes +gradient components that conflict with other tasks; (7) CA- +Grad (Liu et al., 2021a), an MTL method that optimizes for +the average loss while explicitly controlling the minimum +decrease rate across tasks; (8) Nash-MTL (Navon et al., +2022), an MTL approach that is equivalent to AuxiNash but +with a fixed pi = 1/K weighting. +Evaluation +We report the common evaluation metrics for +each task. Since MTL methods treat each task equally, +and these may vary in scale, we also report the overall +relative multi-task performance ∆%. ∆% is defined as +the performance drop compared to the STL performance. +Formally, ∆% = +1 +K +�K +k=1(−1)δk(Mm,k − Mb,k)/Mb,k. +We denote Mb,k and Mm,k as the performance of STL and +the compared method on task k, respectively. δk = 0 if +a lower value is better for the metric Mk and 1 otherwise +(Maninis et al., 2019). In all experiments, we report the +mean value based on 3 random seeds. +It is important to note that for MTL models, we present +the results of a single model trained on all tasks. For auxil- +iary learning methods, we trained a unique model per task, +treating it as the main task and using the remaining tasks as +auxiliaries. + +Auxiliary Learning as an Asymmetric Bargaining Game +Table 2. Cityscapes. Test performance for three tasks: 19-class +semantic segmentation, 10-class part segmentation, and disparity. +Semantic Seg. +Part Seg. +Disparity +mIoU ↑ Pix Acc ↑ +mIoU ↑ Pix Acc ↑ +Abs Err ↓ ∆% ↓ +STL +48.64 +91.01 +53.60 +97.62 +1.108 +LS +37.66 +88.63 +40.92 +96.98 +1.105 +9.84 +PCGrad +39.10 +89.31 +41.71 +97.14 +1.133 +9.28 +CAGrad +39.45 +89.04 +51.95 +97.54 +1.098 +4.66 +Nash-MTL +51.14 +91.59 +56.99 +97.87 +1.066 +−3.23 +GCS +37.45 +88.62 +41.14 +96.97 +1.124 +10.19 +OL-AUX +27.63 +89.34 +51.12 +97.52 +1.397 +15.16 +AuxiLearn +36.18 +88.24 +40.51 +96.95 +1.141 +11.3 +AuxiNash +52.52 +91.91 +58.53 +97.93 +1.027 +−5.15 +6.1. Illustrative Example +We start with an illustrative example, showing that Auxi- +Nash can utilize helpful auxiliaries while ignoring harmful +ones. +We adopt a similar problem setup as in Navon et al. (2021) +and consider a regression problem with parameters W T = +(w1, w2) ∈ R2, fully shared among tasks. The optimal pa- +rameters for the main and helpful auxiliary tasks are W ⋆, +while the optimal parameters for the harmful auxiliary are +˜W ̸= W ⋆. The main task is sampled from a Normal distri- +bution N(W ⋆T x, σmain), with σmain > σh where σh denotes +the standard deviation for the noise of the helpful auxiliary. +The change in the task preference throughout the optimiza- +tion process is depicted in the left panel of Figure 1. Auxi- +Nash identify the helpful tasks and fully ignore the harmful +ones. In addition, Figure 1 right panel presents the main +task’s loss landscape, along with the optimal solution (W ⋆, +marked ▲), the optimal training set solution of the main task +alone (■) and the solution obtained by AuxiNash (marked +�). While using the training data alone with no auxiliary +information yields a solution that generalizes poorly, Aux- +iNash converges to a solution with large proximity to the +optimal solution W ⋆, +6.2. Controlling Task Preference +In this section, we wish to better understand the relationship +between the preference p and the obtained solution. We note +the preference vector only implicitly affects the optimization +solution through the bargaining solution α. +Here, we show that controlling the preference vector can be +used to steer the optimization outcome to different parts of +the Pareto front, compared to the NashMTL baseline. We +consider MTL setup with 2 tasks and use the Multi-MNIST +(Sabour et al., 2017) dataset. In Multi-MNIST two images +from the original MNIST dataset are merged into one by +placing one at the top-left corner and the other at the bottom- +right corner. The tasks are defined as image classification +of the merged images. We run AuxiNash 11 times with +varying preference vector values p and fix it throughout +the training. For both tasks we report the classification +accuracy. For Nash-MTL we run the experiments with +different seed values. For both methods we train a variant +of LeNet model for 50 epochs with Adam optimizer and +1e − 4 as the learning rate. +Figure 2 shows the results. AuxiNash reaches a diverse set +of solutions across the Pareto front while Nash-MTL solu- +tions are all relatively similar due to its symmetry property. +6.3. Analyzing the Effect of Auxiliary Set +Table 3. The effect of auxiliary set: We report the mean IoU, along +with the % change w.r.t STL performance. +Cityscapes +NYUv2 +mIoU ↑ +Change % ↑ +mIoU ↑ +Change % ↑ +STL +48.64 +38.30 +STL Partial +45.97 +−5.48 +36.54 +−4.59 +AuxiNash +52.52 +7.97 +40.79 +6.50 +AuxiNash Aux. Set +51.81 +6.51 +38.94 +1.67 +One important question is on what data to evaluate the +generalization loss LV . It would seem intuitive that one +would need a separate validation set for that since estimating +LV on the training data may be biased. In practice, some +previous works use a held-out auxiliary set (Navon et al., +2021), while others use a separate batch from the training +set (Liu et al., 2019a; 2022). While using an auxiliary set +might be more intuitive, it requires reducing the available +amount of training data which can be detrimental in the +low-data regime we are interested in. +We empirically evaluate this using the NYUv2 (Silberman +et al., 2012) and Cityscapes (Cordts et al., 2016) datasets. +See Section 6.4 for more details. We choose semantic seg- +mentation as the main task for both datasets. We compare +the following methods (i) STL: single task learning using +the main task only, (ii) STL Partial: STL using only 90% +of the training data, (iii) AuxiNash: our proposed method +where we optimize the preference vector using the entire +training set, (iv) AuxiNash Aux. Set: our proposed method, +where we optimize the preference vector using 10% of the +data, allocated from the training set. +We report the mean-IoU metric (higher is better) along with +the relative change from the performance of the STL method. +The results suggest that the drawback of sacrificing some of +the training data overweighs the benefit of using an auxiliary +set. This result aligns with the observation in Liu et al. +(2022). + +Auxiliary Learning as an Asymmetric Bargaining Game +6.4. Scene Understanding +We follow the setup from Liu et al. (2019b; 2022); Navon +et al. (2022) and evaluate AuxiNash on the NYUv2 and +Cityscapes datasets (Silberman et al., 2012; Cordts et al., +2016). The indoor scene NYUv2 dataset (Silberman et al., +2012) contains 3 tasks: 13 classes semantic segmentation, +depth estimation, and surface normal prediction. The dataset +consists of 1449 RGBD images captured from diverse in- +door scenes. +We also use the Cityscapes dataset (Cordts et al., 2016) with +3 tasks (similar to Liu et al. (2022)): 19-class semantic +segmentation, disparity (inverse depth) estimation, and 10- +class part segmentation (de Geus et al., 2021). To speed up +the training phase, all images and label maps were resized +to 128 × 256. +For all methods, we train SegNet (Badrinarayanan et al., +2017), a fully convolutional model based on VGG16 archi- +tecture. We follow the training and evaluation procedure +from Liu et al. (2019b; 2022); Navon et al. (2022) and train +the network for 200 epochs with Adam optimizer (Kingma +& Ba, 2015). We set the learning rate to 1e − 4 and halved +it after 100 epochs. +The results are presented in Table 1 and Table 2. Observing +the results, we can see our approach AuxiNash outperforms +other approaches by a significant margin. It is also important +to note that several methods achieve a positive %∆ score, +meaning they performed worse than simply ignoring the +auxiliary tasks and training on the main task alone. We +believe this is due to the difficulties presented by optimizing +with multiple objectives. +6.5. Semi-supervised Learning with SSL Auxiliaries +Table 4. CIFAR10-SSL. Test performance for classification with a +varying number of labeled data. Values are averages over 3 random +seeds. +CIFAR10-SSL-5K CIFAR10-SSL-10K +STL +79.31 ± 0.31 +83.75 ± 0.18 +LS +83.17 ± 0.54 +86.16 ± 0.39 +PCGrad +82.71 ± 0.16 +86.17 ± 0.34 +CAGrad +85.89 ± 0.63 +87.82 ± 0.28 +Nash-MTL +86.69 ± 0.14 +88.68 ± 0.14 +GCS +83.09 ± 0.34 +86.47 ± 0.56 +OL-AUX +81.44 ± 1.06 +85.49 ± 0.73 +AuxiLearn +82.83 ± 0.57 +85.52 ± 0.57 +AuxiNash (ours) +87.01 ± 0.52 +88.81 ± 0.34 +In semi-supervised learning one generally trains a model +with a small amount of labeled data, while utilizing self- +supervised tasks as auxiliaries to be optimized using unla- +beled training data. +We follow the setup from Shi et al. (2020) and evaluate +AuxiNash on a Self-supervised Semi-supervised Learning +setting (Zhai et al., 2019). We use CIFAR-10 dataset to form +3 tasks. We set the supervised classification as the main task +along with two self-supervised learning (SSL) tasks used +as auxiliaries: (i) Rotation (ii) Exempler-MT. In Rotation, +we randomly rotate each image by [0◦, 90◦, 180◦, 270◦] and +optimize the network to predict the angle. In Exempler-MT +we apply a combination of three transformations: horizontal +flip, gaussian noise, and cutout. Similarly to contrastive +learning, the model is trained to extract invariant features +by encouraging the original and augmented images to be +close in their feature space. For the supervised task we +randomly allocate samples from the training set. We repeat +this experiment twice with 5K and 10K labeled training +examples. The results are presented in Table 4. AuxiNash +significantly outperforms most baselines. +6.6. Audio Classification +Table 5. Speech Commands. Test accuracy for speech classifica- +tion, for models trained with 1000 and 500 training examples. +SC-500 +SC-1000 +STL +95.8 ± 0.1 +96.4 ± 0.1 +LS +95.7 ± 0.2 +96.7 ± 0.1 +PCGrad +95.7 ± 0.1 +96.7 ± 0.1 +CAGrad +95.7 ± 0.2 +95.7 ± 0.1 +Nash-MTL +95.7 ± 0.2 +96.6 ± 0.3 +GCS +96.3 ± 0.1 +96.9 ± 0.1 +OL-AUX +96.2 ± 0.2 +96.9 ± 0.1 +AuxiLearn +96.0 ± 0.1 +97.0 ± 0.1 +AuxiNash (ours) +96.4 ± 0.1 +97.2 ± 0.1 +We evaluate AuxiNash on the speech commands (SC) +dataset (Warden, 2018), which consists of ∼ 50K speech +samples of specific keywords. The data contains 30 dif- +ferent keywords, and each speech sample is one second +long. We use a subset of the SC containing audio samples +for only the 10 numbering keywords (zero to nine). As a +pre-processing step, we use a short-time Fourier transform +(STFT) to extract a spectrogram for each example, which +we then fed to a convolutional neural network (CNN). We +evaluate AuxiNash on 10 one-vs-all binary classification +tasks. We repeat the experiment with a training set of sizes +500 and 1000. The results are presented in Table 5. + +Auxiliary Learning as an Asymmetric Bargaining Game +7. Conclusion and Future Work +In this work, we formulate auxiliary learning as an asym- +metric bargaining game and use game-theoretical tools to +derive an efficient algorithm. We adapt and generalize recent +advancements in multi-task learning to auxiliary learning +and show how they can be automatically tuned to get a +significant improvement in performance. +We evaluated AuxiNash on multiple datasets with different +learning setups and show that it outperforms previous ap- +proaches by a significant margin. Across all experiments, +it is noticeable that MTL methods perform better than aux- +iliary learning ones although the former treat equally the +primary task and the auxiliary tasks. We suspect that this is +caused by conflicting gradients and by the fact that gradient +norms may vary significantly across tasks. These results +emphasize the connection between auxiliary learning and +multi-task optimization. In many examples, the benefit +of the auxiliary task was diminished or even completely +negated by poor optimization. Thus, we suggest that auxil- +iary learning research should be closely aligned with MTL +optimization research to effectively utilize auxiliary tasks. +8. Acknowledgements +This study was funded by a grant to GC from the Israel +Science Foundation (ISF 737/2018), and by an equipment +grant to GC and Bar-Ilan University from the Israel Science +Foundation (ISF 2332/18). AN and AS are supported by a +grant from the Israeli higher-council of education, through +the Bar-Ilan data science institute (BIU DSI). +References +Achituve, I., Maron, H., and Chechik, G. Self-supervised +learning for domain adaptation on point clouds. 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Furthermore, we assume that there exists a point u ∈ U for which +∀i : ui > di. Nash (1953) showed that under these assumptions, the two-player bargaining problem has a unique solutionthat +satisfies the following properties or axioms: Pareto optimality, symmetry, independence of irrelevant alternatives, and +invariant to affine transformations. This was later extended to multiple players (Sz´ep & Forg´o, 1985). +Axiom A.1. Pareto optimality: The agreed solution must not be dominated by another option, i.e. there cannot be any +other agreement that is better for at least one player and not worse for any of the players. +Axiom A.2. Symmetry: The solution should be invariant to permuting the order of the players. +Axiom A.3. Independence of irrelevant alternatives (IIA): If we enlarge the of possible payoffs to ˜U ⊋ U, and the +solution is in the original set U, u∗ ∈ U, then the agreed point when the set of possible payoffs is U will stay u∗. +Axiom A.4. Invariance to affine transformation: If we transform each utility function ui(x) to ˜ui(x) = ci · ui(x) + bi +with ci > 0 then if the original agreement had utilities (y1, ..., yk) the agreement after the transformation has utilities +(c1y1 + b1, ..., ckyk + bk) +B. Proofs +Proof of Proposition 4.2. Define a function F(¯p, ¯α) = G⊤G¯α − ¯p/¯α ∈ RK where and ¯p ∈ RK ++ and ¯α ∈ RK ++ are +independent variables of F with R+ being the set of strictly positive real numbers. Here, ¯p/¯α represents the coordinate-wise +operation, i.e., ¯p/¯α ∈ RK with (¯p/¯α)i = ¯pi/¯αi ∈ R for i ∈ [K]. Then, we have +∂F(¯p, ¯α) +∂ ¯α +��� +(¯p,¯α)=(p,α) = G⊤G − ∂(¯p/¯α) +∂ ¯α +��� +(¯p,¯α)=(p,α) = G⊤G + Λ0, +where the last equality follows from ∂ ¯p/¯α +∂ ¯α += −Λ0 since +∂(¯p/¯α)i +∂ ¯αj += +� ∂(¯pj/¯αj) +∂ ¯αj +if i = j +0 +otherwise += +� +−¯pi/¯α2 +i +if i = j +0 +otherwise +Similarly, +∂F(¯p, ¯α) +∂¯p +��� +(¯p,¯α)=(p,α) = −∂(¯p/¯α) +∂¯p +��� +(¯p,¯α)=(p,α) = −Λ1, +since +∂(¯p/¯α)i +∂¯pj += +� ∂(¯pj/¯αj) +∂ ¯pj +if i = j +0 +otherwise += +� +1/¯αi +if i = j +0 +otherwise +Thus, F is continuously differentiable and ∂F (¯p,¯α) +∂ ¯α +|(¯p,¯α)=(p,α) = G⊤G+Λ0 is invertible since G⊤G is positive semi-definite +and Λ0 is positive definite due to the condition of pi > 0 for all i ∈ [K]. Therefore, the function F at (p, α) satisfies the +condition of the implicit function theorem, which implies the statement of this proposition as +∂ ˆα(p) +∂p += − +�∂F(¯p, ¯α) +∂ ¯α +�−1 ∂F(¯p, ¯α) +∂¯p +��� +(¯p,¯α)=(p,α) = +� +G⊤G + Λ0 +�−1 Λ1. +(8) +B.1. Proof of Theorem 5.4 +We adopt the following implicit assumption (or the design of algorithm) from (Navon et al., 2022): if we reach a Pareto +stationary solution at some step, the algorithm halts. We will also use the following Lemma. +Lemma B.1. (Lemma A.1 of Navon et al., 2022) If L is differential and L-smooth (assumption 5.3) then L(θ′) ≤ L(θ) + +∇L(θ)⊤(θ′ − θ) + L +2 ∥θ′ − θ∥2. + +Auxiliary Learning as an Asymmetric Bargaining Game +Proof. We first note that if for some step, we reach a Pareto stationary solution the algorithm halts and sequence stays fixed +at that point and therefore converges; Next, we assume that we never get to an exact Pareto stationary solution at any finite +step. Since �K +i=1 p(t) +i += 1 and ∆θ(t) = G(t)α(t), +||∆θ(t)||2 = (α(t))⊤(G(t))⊤G(t)α(t) = (α(t))⊤(p(t)/α(t)) = +K +� +i=1 +α(t) +i +· (p(t) +i /α(t) +i ) = 1. +For each loss ℓi for i ∈ [K], using Lemma B.1 and (g(t) +i )⊤∆θ(t) = (g(t) +i )⊤G(t)α(t) = p(t) +i /α(t) +i , +ℓi(θ(t+1)) ≤ ℓi(θ(t)) − µ(t)∇ℓi(θ(t))⊤∆θ(t) + L +2 ||µ(t)∆θ(t)||2 +(9) += ℓi(θ(t)) − µ(t) p(t) +i +α(t) +i ++ (µ(t))2L +2 +(10) +We average over the above inequality over all losses and get for L(θ) = 1 +K +�K +i=1 ℓi(θ): +L(θ(t+1)) ≤ L(θ(t)) − µ(t) 1 +K +K +� +i=1 +p(t) +i +α(t) +i ++ (µ(t))2L +2 += L(θ(t)) − L(µ(t))2 + (µ(t))2L +2 += L(θ(t)) − L(µ(t))2 +2 +. +(11) +By rearranging, this shows that L(θ(t+1)) ≤ L(θ(t)) and L(µ(t))2 +2 +≤ L(θ(t)) − L(θ(t+1)). From the first inequality, the +sequence (L(θ(t)))t is non-increasing. As L(θ(t)) ∈ R is bounded below and (L(θ(t)))t is non-increasing, the monotone +convergence theorem concludes that the sequence (L(θ(t)))t converges to a finite limit. Since a convergent sequence is +a Cauchy sequence, L(µ(t))2 +2 +≤ L(θt) − L(θt+1) → 0 as t → ∞. This implies that µ(t) → 0 as t → ∞. It follows +that +1 +K +�K +i=1 p(t) +i /α(t) +i +→ 0 as t → ∞. Since p(t) +i +> 0 and α(t) +i +> 0, it implies that maxi p(t) +i +≥ 1/K. If we define +jt = arg maxi p(t) +i +we will get that p(t) +jt /α(t) +jt → 0 and therefore α(t) +jt → ∞. We can conclude that ||α(t)|| → ∞. +We will now show that ||p(t)/α(t)|| is bounded for t → ∞. As the sequence L(θ(t)) is decreasing we have that the sequence +θ(t) is in the sublevel set {θ : L(θ) ≤ L(θ0)} which is closed and bounded and therefore compact. If follows that there +exists M < ∞ such that ||g(t) +i || ≤ M for all t and i ∈ [K]. We have for all i and t, |p(t) +i /α(t) +i | = |(g(t) +i )T ∆θ(t)| ≤ ||g(t) +i || ≤ +M < ∞, and so ||p(t)/α(t)|| is bounded. Combining these two results we have ||p(t)/α(t)|| ≥ σK((G(t))⊤G(t))||α(t)|| +where σK((G(t))⊤G(t)) is the smallest singular value of (G(t))⊤G(t). Since the norm of α(t) goes to infinity and the norm +p(t)/α(t) is bounded, it follows that σK((G(t))⊤G(t)) → 0. +Now, since {θ : L(θ) ≤ L(θ0)} is compact there exists a subsequence θ(tj) that converges to some point θ∗. As +σK((G(t))T G(t)) → 0 we have from continuity that σK(G⊤ +∗ G∗) = 0 where G∗ is the matrix of gradients at θ∗. This +means that the gradients at θ∗ are linearly dependent and therefore θ∗ is Pareto stationary by assumption 5.1. Since all loss +functions ℓi are the are differentiable, all loss functions ℓi are continuous. Thus, we have from the continuity of ℓi that +ℓi(θ(t)) +t→∞ +−−−→ ℓi(θ∗). +C. Solving G⊤Gα = p/α +Here, we describe how to efficiently approximate the optimal solution for G⊤Gα = p/α through a sequence of convex +optimization problems. We define a βi(α) = g⊤ +i Gα, and wish to find α such that αi = pi/βi for all i, or equivalently +log(αi) + log(βi(α)) − log(pi) = 0. Denote ϕi(α) = log(αi) + log(βi) − log(pi) and ϕ(α) = � +i ϕi(α). With that, our +goal is to find a non-negative α such that ϕi(α) = 0 for all i. We can write this as the following optimization problem +min +α +� +i +ϕi(α) +(12) +s.t.∀i, +− ϕi(α) ≤ 0 +αi > 0 +. + +Auxiliary Learning as an Asymmetric Bargaining Game +Table 6. Speech Commands. Test performance on each one of the digit keywords. The train data-set consist of 1000 samples, uniformly +distributed between the 10 classes. Values are averaged over 3 random seeds. +Speech Commands - 1000 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Mean +STL +97.0 96.7 95.8 96.3 96.4 96.3 96.7 96.6 96.0 96.6 +96.4 +LS +97.0 97.2 96.2 97.1 97.0 96.2 96.9 97.3 95.8 96.8 +96.7 +PCGrad +97.0 97.3 96.2 96.6 96.7 96.7 96.8 97.3 95.8 96.9 +96.7 +CAGrad +97.2 97.7 95.6 97.0 97.0 96.6 97.0 97.2 95.8 97.0 +96.8 +Nash-MTL 97.1 96.8 96.4 96.6 96.9 96.6 96.8 96.9 96.0 96.8 +96.6 +GCS +97.3 97.8 96.2 97.0 97.4 96.2 96.8 97.3 96.2 97.1 +96.9 +OL-AUX +97.2 97.7 96.0 97.2 96.9 96.5 97.0 97.5 96.2 97.1 +96.9 +AuxiLearn +97.3 97.7 96.4 97.1 97.2 96.6 97.0 97.3 96.3 97.1 +97.0 +AuxiNash +97.4 98.1 96.5 97.2 97.1 97.0 97.1 97.6 96.3 97.6 +97.2 +One can see that the constraints of this problem are convex and linear while the objective is concave. To overcome this +challenge we follow Navon et al. (2022) and use the CCP to modify the concave objective into sequence of convex +optimization problems. For further details please refer to Section 3.2 in Navon et al. (2022). +D. Experimental Details +Unless stated otherwise for AuxiNash we update the preference vector p every 25 optimization steps using SGD optimizer +with momentum of 0.9 and learning rate of 5e − 3. +D.1. Illustrative Example +We follow a similar setup as in Navon et al. (2021). We consider a regression problem with parameters W T = (w1, w2) ∈ R2, +shared among tasks, with no task-specific parameters. The optimal parameters for the main and helpful auxiliary tasks are +W ⋆T = (1, 1), while the optimal parameters for the harmful auxiliary are ˜W T = (−1, −4). The main task is sampled +from a Normal distribution N(W ⋆T x, σmain), with σmain = 20 · σh where σh = 0.25 denotes the standard deviation for the +noise of the helpful auxiliary. We use 1000 training and train the model using Adam optimizer and learning-rate 1e − 2 and +batch-size of 256 for 1000 epochs. +D.2. CIFAR10-SSL +We follow the training and evaluation procedure proposed by previous works (Shi et al., 2020). We use the CIFAR10 dataset +and divide the dataset to train/val/test splits each containing 45K/5K/10K respectively. We allocate 5K/10K samples as +our labeled dataset for the supervised main task. Following Shi et al. (2020) we use the wide resnet (WRN-28-2) architecture +which is ResNet with depth 28 and width 2. We train WRN for 50K iterations using Adam optimizer with learning rate of +5e − 3 and 256 batch size. Lastly, we use the main task performance on the validation set for early stopping. +D.3. Speech Commands +We used the Speech Command dataset, an audio dataset of spoken words designed to train and evaluate keyword spotting +systems. We repeat the experiment twice with 1000 and 500 training samples distributed uniformly across 10 classes. For +validation and test we used the original dataset split containing 3643 and 4107 samples respectively. For all methods we use +a CNN network with 3 Convolution layers with a linear layer as the classifier. We train the model for 200 epochs with Adam +optimizer, and a learning rate of 1e − 3. We present the per-task results for both experiments in Table 7 and Table 6. +D.4. Scene Understanding +We follow the training and evaluation protocol presented by previous studies (Liu et al., 2019b; 2022; Navon et al., 2022). +For all methods we train SegNet (Badrinarayanan et al., 2017) model for 200 epochs with Adam optimizer. We use learning + +Auxiliary Learning as an Asymmetric Bargaining Game +Table 7. Speech Commands. Test performance on each one of the digit keywords. The train data-set consist of 500 samples, uniformly +distributed between the 10 classes. Values are averaged over 3 random seeds. +Speech Commands - 500 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Mean +STL +96.4 97.0 95.0 95.7 96.3 95.5 95.9 95.8 95.3 95.4 +95.8 +LS +95.9 96.9 94.8 95.8 96.0 95.3 95.9 96.4 95.3 95.6 +95.7 +PCGrad +96.0 97.1 95.0 95.7 95.9 95.8 95.8 96.0 94.7 95.3 +95.7 +CAGrad +96.4 97.1 95.0 95.9 96.3 95.9 96.0 96.1 95.0 95.4 +95.9 +Nash-MTL 96.6 97.4 94.1 95.7 96.3 95.4 95.9 96.2 94.8 95.2 +95.7 +GCS +96.4 97.6 95.4 96.3 96.5 95.8 96.1 96.5 96.0 97.0 +96.3 +OL-AUX +96.2 97.2 94.6 96.0 96.4 95.7 95.9 96.4 96.1 97.5 +96.2 +AuxiLearn +96.2 97.5 94.8 96.3 96.3 95.8 96.1 96.3 95.3 95.8 +96.0 +AuxiNash +96.6 97.8 95.3 96.7 97.2 96.0 96.2 96.7 95.6 95.6 +96.4 +Table 8. NYUv2. Test performance using fixed p. The preference for the main task, pmain, is in parentheses. Values are averaged over 3 +random seeds. +Segmentation +Depth +Surface Normal +mIoU ↑ Pix Acc ↑ +Abs Err ↓ Rel Err ↓ +Angle Distance ↓ +Within t◦ ↑ +∆% ↓ +Mean +Median +11.25 +22.5 +30 +STL +38.30 +63.76 +0.6754 +0.2780 +25.01 +19.21 +30.14 +57.20 +69.15 +AuxiNash (0.7) +41.36 +66.48 +0.5214 +0.2138 +24.46 +18.78 +30.80 +58.33 +70.42 +−7.62 +AuxiNash (0.8) +41.27 +66.47 +0.5322 +0.2172 +24.34 +18.67 +31.05 +58.53 +70.64 +−7.56 +AuxiNash (0.9) +41.35 +66.32 +0.5493 +0.2197 +24.39 +18.61 +31.23 +58.64 +70.68 −7.28 +AuxiNash +40.79 +66.79 +0.5092 +0.2042 +24.90 +19.31 +29.83 +57.07 +69.27 +−6.80 +rate of 1e − 4 for the first 100 epochs, then reduced it to 5e − 5 for the remaining epochs. We use a batch size of 2 and 8 +for NYUv2 and CityScapes respectively. During training we apply data augmentations for all compared methods, more +specifically, we use random scale and random horizontal flip as augmentations. Following Liu et al. (2019b; 2022); Navon +et al. (2022) we report the test performance averaged over the last 10 epochs. +D.5. Controlling Task Preference +We use the Multi-MNIST dataset that consits of 120K training samples and 20K testing samples. We allocate 12K training +examples to form validation set. We use a variant of LeNet as the trained model. Specifically, the model contains 3 CNN +layers channels followed by 2 fully connected layers. We train all methods using Adam optimizer with learning rate 1e − 4 +for 50 epochs and 256 batch size. For AuxiNash we repeat this experiment 11 times with varying preferences. More +specifically, we select the first task preference from p1 ∈ {0.01, 0.1, 0.2, 0.25, 0.4, 0.5, 0.6, 0.75, 0.8, 0.9, 0.99} and set +p2 = 1 − p1. We run the experiment equal amount of time with randomly selected seeds for Nash-MTL. +E. Additional Results +E.1. Fixed Preference Vector +Our method, AuxiNash, dynamically adjusts the preference vector throughout the optimization process. It is possible, +however, to fix the preference vector to its initial value and train a model over a grid of such preferences. As discussed in the +main text, this procedure does not scale well as the number of grid search values grows exponentially with the number of +tasks. We also note that it is possible that the optimal preference changes during the optimization process, depending on the +optimization dynamics. + +Auxiliary Learning as an Asymmetric Bargaining Game +Figure 4. Learned task preference for Cityscapes dataset. The title of each panel indicates the main task. +Here we present an ablation study of optimizing AuxiNash with fixed preference p. We use the NYUv2 dataset and train the +model with pmain ∈ {0.9, 0.8, 0.7}. The preference for the two auxiliary tasks is equal, e.g., paux,i = 0.1 when pmain = 0.8. +The results are presented in Table 8. +E.2. Learned Preferences +To better understand the learned preference and its dynamics, we observe the change in preference vectors throughout +the optimization process. We use the scene understanding experiment of Section 6.4, specifically the Cityscapes dataset. +The learned preferences are presented in Figure 4. The title of each panel indicates the main task. Here, AuxiNash learns +preferences that are aligned with our intuition, i.e., the main task’s preference is the largest. + +Semantic Seg. +Part Seg. +Depth +0.8 +0.8 +Semantic Seg. +1.0 +Part Seg. +0.8 +0.6 +Depth +0.6 +nce +0.6 +@0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +25000 +0 +50000 +25000 +50000 +0 +25000 +50000 +0 +Step +Step +Step \ No newline at end of file diff --git a/RdFRT4oBgHgl3EQfLDee/content/tmp_files/load_file.txt b/RdFRT4oBgHgl3EQfLDee/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0f83d071890c7cfb865793079c0956495b1ee5c --- /dev/null +++ b/RdFRT4oBgHgl3EQfLDee/content/tmp_files/load_file.txt @@ -0,0 +1,1721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf,len=1720 +page_content='Auxiliary Learning as an Asymmetric Bargaining Game Aviv Shamsian * 1 Aviv Navon * 1 2 Neta Glazer 1 2 Kenji Kawaguchi 3 Gal Chechik 1 4 Ethan Fetaya 1 Abstract Auxiliary learning is an effective method for en- hancing the generalization capabilities of trained models, particularly when dealing with small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' However, this approach may present several difficulties: (i) optimizing multiple objec- tives can be more challenging, and (ii) how to balance the auxiliary tasks to best assist the main task is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' In this work, we propose a novel approach, named AuxiNash, for balancing tasks in auxiliary learning by formalizing the problem as generalized bargaining game with asymmetric task bargaining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Furthermore, we describe an efficient procedure for learning the bargaining power of tasks based on their contribution to the performance of the main task and derive theoret- ical guarantees for its convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Finally, we evaluate AuxiNash on multiple multi-task bench- marks and find that it consistently outperforms competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Introduction When training deep neural networks with limited labeled data, generalization can be improved by adding auxiliary tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' In this approach, called Auxiliary learning (AL), the auxiliary tasks are trained jointly with the main task, and their labels can provide a signal that is useful for the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' AL is beneficial because auxiliary annotations are often easier to obtain than annotations for the main task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' This is the case when the auxiliary tasks use self-supervision (Oliver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Hwang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Achituve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=', 2021), or their annotation process is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' For example, learning semantic segmentation of an image may require careful and costly annotation, but can be improved if learned jointly with a depth prediction task, whose annotations can be obtained at scale (Standley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Equal contribution 1Bar-Ilan University, Ramat Gan, Is- rael 2Aiola, Herzliya, Israel 3National University of Sin- gapore 4Nvidia, Tel-Aviv, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdFRT4oBgHgl3EQfLDee/content/2301.13501v1.pdf'} +page_content=' Correspondence to: Aviv Shamsian 0 and +a period p > 0 such that +0 ≤ U(t) ≤ K, +U(t + p) = U(t), +(4) + +3 +for t ≥ 0. +The interest in the above assumption is that, as we will show +in later sections, when U(t) verifies the above assumption, +L ends up converging in probability to a relatively simple +form, which relaxes the difficulties of the quantitative char- +acterization of the reliability of complex networked systems. +Note that the periodicity of system utility has been observed +in various practical applications due to the nature of human +behavior with respect to service demand. For example, in +cellular networks, it was shown that user traffic typically +exhibits a periodical pattern [23]. Such patterns have been +also found in data that we gathered from a large-scale cellular +network, as will be illustrated in Section V-A. It is worth +noting that such trends are not exclusive to cellular networks. +For instance, the periodic behavior has also been witnessed in +the electricity demands in power grid networks [24]. Accord- +ingly, our theoretical derivations are not constrained to cellular +networks settings, but rather can be leveraged for various other +application scenarios. +III. MATHEMATICAL ANALYSIS +We decompose the time horizon T in eq. (3) into multiple +stages. Specifically, we let Dn = �n +j=1(Xj + Yj) and we +rewrite the expected utility loss of the system as +L = +lim +n→+∞ +� Dn +0 +U(t)W(t)dt +Dn +. +(5) +Next, by multiplying by 1 +n both the numerator and denomina- +tor, we end up with +L = +lim +n→+∞ +1 +n +�n +j=1 +� Dj +Dj−1 U(t)W(t)dt +1 +n +�n +j=1(Xj + Yj) +. +(6) +Noting that W(t) is equal to 0 by definition in every interval +[Dj, Dj + Xj], we can rewrite the expected loss as +L = +lim +n→+∞ +1 +n +�n +j=1 +� Dj +Dj−Yj U(t)dt +1 +n +�n +j=1(Xj + Yj) +. +(7) +Clearly, the challenging part of the evaluation of the expected +loss is the numerator. To deal with this, we leverage the +periodicity of the function U(t). +Lemma 1. If U(t) is a periodic function of period p, then the +expected loss L can be rewritten as +L = +lim +n→+∞ +1 +n +�n +j=1 +� D[p] +j +D[p] +j −Yj U(t)dt +1 +n +�n +j=1(Xj + Yj) +, +(8) +where D[p] +j += Dj mod p is the remainder of the Euclidean +division of Dj by p (i.e., the least positive residue). +Proof. To prove this result, we first note that +Dj = kp + (Dj mod p), +(9) +where k ∈ N. Next, we apply a change of variable to the +integral in eq. (7), letting t′ = t − kp. By doing so, eq. (7) +can be rewritten as follows +L = +lim +n→+∞ +1 +n +�n +j=1 +� D[p] +j +D[p] +j −Yj U(t′ + kp)dt′ +1 +n +�n +j=1(Xj + Yj) +(a) += +lim +n→+∞ +1 +n +�n +j=1 +� D[p] +j +D[p] +j −Yj U(t′)dt′ +1 +n +�n +j=1(Xj + Yj) +, +(10) +where (a) results from the periodicity of U(t). Then, by +interchanging t′ and t, we can confirm the lemma. +The next step in the analysis consists of finding the distri- +bution of D[p] +j . To do so, we first rewrite D[p] +j +as follows +D[p] +j += (X[p] +j ++ Y [p] +j +) mod p, +(11) +where +X[p] +j += ( +j +� +k=1 +Xk) mod p, +Y [p] +j += ( +j +� +k=1 +Yk) mod p. +(12) +Next, we investigate the distribution of X[p] +j +more closely. +Theorem 1. Let fX[p] +j (x) denote the probability distribution +function of X[p] +j . We have that fX[p] +j (x) converges uniformly +to the uniform distribution on [0, p]. Precisely, +sup +x∈[0,p] +|fX[p] +j (x) − 1 +p| ≤ Cαj +p , +(13) +where +α = +λp +� +λ2p2 + 4π2 , +(14) +C = +∞ +� +n=1 +λ2p2 + 4π2 +λ2p2 + 4π2n2 . +(15) +Proof. The proof revolves around a Fourier analysis of the +distribution of X[p] +j . In essence, we first analyze the behavior +of the Fourier coefficients of the random variables making +up X[p] +j . Then, by leveraging the scalability property of the +exponential distribution, we can derive the desired results. For +conciseness, the details of the proof are reported in Appendix +A. +In this theorem, we showed that X[p] +j +converges uniformly +to a uniform distribution as j becomes large. Additionally, the +bounds we derived imply that ∇α(λ, p) ≥ 0 for λ, p ≥ 0. This +means that the speed of convergence to a uniform distribution +decreases as either p or λ increases. Figure 2 illustrates this +trend, where α varies and its contour levels are shown. To +understand this trend, we can examine how the cumulative +density function of the anomalies’ inter-arrival times changes + +4 +(a) Zero Gaussian added. +(b) One Gaussian added. +(c) Ten Gaussians added. +Fig. 1: Illustrations of the smoothing phenomenon. +with increasing λ. For any interval I = [0, a] where a is any +positive real number, Pr(Xk ∈ I ) = 1−exp(−λa) increases +as λ grows for k ∈ N∗. Based on this and the expression for +X[p] +j +given by eq. (12), we can conclude that more realizations +of Xk are needed to span all the possible values from [0, p]. +Furthermore, as p increases, the range of values where X[p] +j +can fall also increases. Hence, using the expression for X[p] +j , +we can see that more realizations of Xk are also needed in +this case to span all the possible values of X[p] +j +from [0, p]. In +both cases, larger values of j are needed in order to converge. +Fig. 2: Illustration of α in function of λ and p. +Next, we recall that our goal is to characterize the distribution +of D[p] +j . To do so, we need to take into account the distribution +of Y [p] +j +, which can be quite general as we impose no restriction +on FY (t). To alleviate this difficulty, we provide below an +essential lemma to our analysis. +Lemma 2. Let A be a random variable defined on [0, p] +satisfying +sup +x∈[0,p] +|fA(x) − 1 +p| ≤ C, +(16) +where fA(x) is the probability density function of A. Let B be +a random variable of arbitrary distribution defined on [0, p] +independent of A. Then, +sup +x∈[0,p] +|fZ(z) − 1 +p| ≤ sup +x∈[0,p] +|fA(x) − 1 +p| ≤ C, +(17) +where Z = A + B mod p. +Proof. Given the independence between A and B, the proof +revolves around the notion of probability distributions’ convo- +lution. Then, by leveraging the particularity of the distribution +of A along with the definition of the modulo function, we can +derive the desired results. For conciseness, the details of the +proof are reported in Appendix B. +Remark 1. The above lemma has an important interpretation: +when a random variable (RV) B independent of A is added to +the random variable A, it results in a smoothing effect when +the modulo p function is applied. Thus, the distribution of Z = +A + B mod p becomes even closer to a uniform distribution +than A mod p. To illustrate this, we consider an exponentially +distributed RV A of rate λ = 10, and we add to this RV 0, 1, +and 10 independent standard Gaussian RVs. The results are +reported in Fig. 1, where the distribution of the sum of the RVs +modulo p is plotted in each case, and the smoothing effect is +shown. +With the above lemma in mind, we can now tackle the char- +acterization of the distribution of D[p] +j +in the next proposition. +Proposition 1. Let fD[p] +j (z) denote the probability distribution +function of D[p] +j . We have that fD[p] +j (z) converges uniformly +to the uniform distribution on [0, p]. Precisely, +sup +z∈[0,p] +|fD[p] +j (z) − 1 +p| ≤ Cαj +p . +(18) +Proof. To prove the proposition, we recall from eq. (11) that +D[p] +j +can be written as the sum of two RVs defined on [0, p]. +Then, due to the independence between X[p] +j +and Y [p] +j +, we can +leverage Lemma 2 to conclude the proposition. +Remark 2. The trend seen in Fig. 1 is confirmed by Propo- +sition 1, which shows that the convergence of the distribution +D[p] +j +to a uniform distribution happens exponentially fast. This +suggests that a relatively small number of anomalies/repair + +0.16 +0.14 +Probability density function +0.12 +0.10 +0.08 +0.06 +0.04 +0.02 +0.00 +0 +2 +4 +6 +8 +10 +X0.12 +Probability density function +0.10 +0.08 +0.06 +0.04 +0.02 +0.00 +0 +2 +4 +6 +8 +10 +X0.10 +Probability density function +0.08 +0.06 +0.04 +0.02 +0.00 +0 +2 +4 +9 +8 +10 +X8 - +0.8 +-9 +0.6 +V +α +4 - +0.4 +2 - +0.2 +0.8 +0.5 +0 +0.0 +0 +2 +4 +6 +8 +p5 +stages is sufficient to model D[p] +j +as a uniform distribution +with minimal modeling penalty. As seen in Fig. 1, even when +only 10 independent Gaussian random variables are added to +the exponential random variable, the resulting sum modulo p +is already distributed almost uniformly. +Given the above results, we can conclude that as j gets large, +the distribution of D[p] +j +approaches the uniform distribution +on [0, p]. Now, to characterize L in eq. (8), let us define the +integral Ij as +Ij = +� D[p] +j +D[p] +j −Yj +U(t)dt. +(19) +As seen in eq. (8), the difficult part in characterizing L +comes from the fact that the terms Ij in the numerator are not +independent and do not share the same distribution. Hence, +providing a statistical convergence of the sum present in the +numerator is challenging. To address these challenges, we first +provide the following lemma for the convergence of the sum +of the expected values of Ij. +Proposition 2. Let Sn = +1 +n +�n +j=1 E[Ij] denote the partial +sum of the expected value of Ij. We have +Sn +n→+∞ +−−−−−→ I, +(20) +where I = E[Y ]U, and U = 1 +p +� p +0 U(t)dt is the average utility +in a period p. +Proof. Our proof consists of leveraging Lemma 2 and Proposi- +tion 1, the periodicity of U(t), and the Stolz–Ces`aro theorem. +The details of the proof are reported in Appendix C. +Given the above convergence results, the next step of the +analysis is to investigate the degree of dependence between +each of the integrals Ij and Ij+k for any j, k ∈ N∗. To do +so, we investigate the joint distribution function of D[p] +j +and +D[p] +j+k, as seen below. +Proposition 3. Let fD[p] +j ,D[p] +j+k(zj, zj+k) denote the joint dis- +tribution function of the RVs D[p] +j +and D[p] +j+k. There exists a +constant C′ > 0 such that +|fD[p] +j ,D[p] +j+k(zj, zj+k) − 1 +p2 | ≤ C′(Cαj +p ++ Cαk +p +), +(21) +for all zj, zj+k ∈ [0, p] and j, k ∈ N∗. +Proof. In this proof, we derive the joint probability distribution +and leverage the particularities of the modulo function along +with the independence between the RVs Xi and Yi involved. +For conciseness, the proof is reported in Appendix D. +Now that the convergence of Sn is proven and the joint +probability distribution fD[p] +j ,D[p] +j+k(·, ·) is characterized, we can +investigate the covariance of Ij and Ij+k. This will allow us +to measure the dependency between each of these integrals, +thus enabling us to derive L . +Proposition 4. The covariance Cov[Ij, Ij+k] for j, k ∈ N∗ +satisfies the following inequality +Cov[Ij, Ij+k] ≤I +2(1 + pC′C(αj + αk) − (1 − Cαj−1 +p +) +(1 − Cαj+k−1 +p +)). +(22) +Proof. The proof revolves around a derivation of the covari- +ance of Ij and Ij+k, and leveraging the propositions and +lemmas provided thus far, along with various algebraic ma- +nipulations. The details of the proof are reported in Appendix +E. +Interestingly, we can conclude from the results above that +when j is large, and the gap between j and j′ > j is also +large, then the two integrals Ij and Ij′ become uncorrelated. +This is going to be essential to establish the main results of +our mathematical analysis below. +Theorem 2. If (i) the anomalies inter-arrival times are +exponentially distributed, (ii) the maintenance times verify +Assumption 1, and (iii) U(t) satisfies Assumption 2, then the +expected loss L converges in probability to +L ∞ = +E[Y ]U +E[X] + E[Y ]. +(23) +Proof. The proof of the theorem amounts to characterizing +the probability that the difference between the numerator and +I exceeds a certain arbitrarily small ϵ. Using the triangular +inequality and the union bound, this probability is further +simplified. Then, by leveraging Chebyshev’s inequality and the +results presented in the paper thus far, we obtain the desired +characterization. Finally, using the weak law of large numbers +and the Mann-Wald theorem, we can conclude the convergence +in probability of L to L ∞. For conciseness, the details of +the proof are reported in Appendix F. +As can be seen above, the expression to which L converges +turns out to have a relatively simple formulation thanks to +the periodicity of the function U(t). Particularly, given the +periodic nature of U(t), the challenges in evaluating L due +to the dynamic nature of the utility function are overcome, +hence allowing us to evaluate L in a simplified manner. +Additionally, the expression has an intuitive meaning and +relationship with well-established metrics. In fact, considering +the system’s availability expression +Availability = +MTBF +MTBF + MTTR, +(24) +where MTBF= E[X] = 1 +λ and MTTR= E[Y ] denote the mean +time between failure and mean time to repair, respectively [5], +and by examining the expression of L ∞, we can deduce that +L ∞ = (1 − Availability +� +�� +� +(A) +) × U +���� +(B) +. +(25) +The term (A) can be seen as the probability that the system +suffers from an anomaly; the term (B) is the average utility + +6 +that the system delivers in a period p. Then, one can see that +the periodicity of the utility U(t), the exponential nature of +anomalies’ inter-arrival times, and the mathematical analysis +provided in the paper lead to the intuitive form of L ∞. +In a later section, we will consider a particular application +of interest and showcase the usefulness of the above-derived +analytical results for the analysis of complex networks. +Remark 3. It is worth noting that although in our analysis +we have focused on the case where one component makes up +the system, our analysis can be extended to the case where the +system is constituted of N components. Particularly, as long +as each component’s operation is independent of the others, +similar results to Theorem 2 can be obtained for the average +expected utility not satisfied +L = +lim +T →+∞ +1 +NT +N +� +i=1 +� T +0 +U i(t)W i(t)dt, +(26) +where U i(t) is the utility at time t of component i and W i(t) +is a binary random variable indicating if component i is +suffering from an anomaly. As will be seen in Section V-D, +the consideration of large-scale systems (i.e., N ≫ 1) will +have a beneficial effect on the convergence speed of L . +IV. GENERALIZATIONS OF THE ANALYSIS +The goal of this section is to provide extensions of our +theoretical analysis to more general scenarios that could arise +in practical situations. +A. Relaxing the Periodicity Assumption +In practice, the utility function U(t) does not exhibit perfect +periodicity but, rather, fluctuates slightly around a regular +period. These fluctuations are caused by unpredictable traffic +demand and user activity. To account for this, we model the +utility function U(t) as a stochastic process that randomly +fluctuates at each time instant t. Specifically, we write U(t) +as +U(t) = U ′(t) + B(t), +t ≥ 0, +(27) +where U ′(t) is a bounded, non-negative, periodic function +with period p, and B(t) is a stochastic process. The random +fluctuations in U(t) are determined by the variance of the +process B(t). In Section V-D, we will use data from a real +deployed network to show that these fluctuations are typically +small. Next, we suppose that B(t) verifies the following +assumption. +Assumption 3 (Zero-Mean Weakly Dependent Wide-Sense +Stationary Random Processes). We say that a stochastic +process B(t) is a zero-mean weakly dependent wide-sense +stationary random process if +• E[B(t)] = 0 for all t ∈ R. +• Cov[B(t), B(u)] = ρ(t − u) for all t, u ∈ R. +• +1 +T +� T +0 |ρ(τ)|dτ +T →∞ +−−−−→ 0. +There are many processes that satisfy this condition, in- +cluding the zero-mean Gaussian white noise and the Ornstein- +Uhlenbeck process. For these particular processes, we extend +the results of Theorem 2 below. +Theorem 3. If (i) the anomalies’ inter-arrival times are +exponentially distributed, (ii) the maintenance times verify +Assumption 1, and (iii) B(t) verifies Assumption 3, then the +expected loss L converges in probability to +L ∞ = +E[Y ]U +E[X] + E[Y ], +(28) +where U = 1 +p +� p +0 U ′(t)dt. +Proof. The proof involves using Assumption 3, integral ma- +nipulations, and Chebyshev inequality to show that B(t) +exhibits ergodic properties. See Appendix H for details. +B. General Inter-Arrival Distributions +In our system model reported in Section II, we have as- +sumed that the anomalies’ inter-arrival times are exponentially +distributed. The reasons behind this assumption are twofold: 1) +the data that we gathered from a large-scale deployed cellular +network suggest that the exponential distribution is a good +fit, as will be illustrated in Section V-B, and 2) the vast +literature on reliability confirms that exponential distribution +is found to be a good representative in many cases (e.g., [21, +Chapter 3], [22]). Additionally, this assumption has allowed us +to further understand the convergence rate reported in Theorem +1 and interpret the effect of the period p and the anomalies’ +rate λ on the convergence. Nevertheless, in this section, we +aim to generalize our analytical results to a large family of +distributions, particularly the family of bounded distributions. +To that end, we present our results below. +Theorem 4. Let fX(x) denote the probability distribution +function of the anomaly inter-arrival time Xj for j ∈ N∗. +If there exists a finite M > 0 such that +fX(x) ≤ M, +x ∈ [0, ∞), +(29) +then fX[p] +j (x) converges uniformly to the uniform distribution +on [0, p]. Precisely, there exists two constants α < 1 and C > +0 such that +sup +x∈[0,p] +|fX[p] +j (x) − 1 +p| ≤ Cαj +p , +(30) +where +α = sup +n∈N∗ |ˆg(n)|, +(31) +C = +�∞ +n=1 |ˆg(n)|2 +α2 +, +(32) +and +ˆg(n) = +� ∞ +0 +fX(x)e−2πi n +p xdx +∆= FX(n +p ), +(33) +where FX(·) is the Fourier transform of fX(·). + +7 +Proof. The proof follows steps similar to those provided in +Theorem 1’s proof reported in Appendix A. For completeness, +the details of the proof are reported in Appendix G. +Knowing that the rest of the mathematical analysis reported +in Section III holds given the above convergence results, we +can conclude that the results of Theorem 2 also hold for this +general family of probability distributions. Similar conclusions +can be made for Theorem 3. In essence, if the anomalies’ +inter-arrival time distribution is bounded, and the remaining +conditions of Theorem 2 and 3 are verified, then the expected +loss L converges in probability to +L ∞ = +E[Y ]U +E[X] + E[Y ]. +(34) +V. USE CASE: A LARGE CELLULAR NETWORK +In this section, our aim is to verify the key assumptions +adopted in our analysis, corroborate our theoretical findings, +and shed light on various challenges found in our imple- +mentations. To do so, we leverage the data that we have +gathered from a large long term evolution (LTE) cellular +network consisting of 660 cellular base stations and serving +approximately 22k users. The data gathered span two months +and consist of cell-level key performance indicators (KPIs), +along with troubleshooting tickets generated by the network’s +operator and alarms information registered by the base sta- +tions. For confidentiality reasons, the data have been scaled +when necessary. +A. Periodicity of Utility Functions +The first assumption we verify in this section, given its +importance to our theoretical analysis, concerns the periodicity +of the utility function of the cellular sites. Typically, the utility +function of a base station is taken as the amount of traffic that +it provides to the customers. In other words, the utility function +at time t is nothing but the user demand typically served by +the base station at this time instant. In Fig. 3, we illustrate the +average traffic served by a particular base station at each hour +of the week. As seen in the figure, a noticeable periodic trend +can be witnessed, although minor violations of the periodicity +can be observed. However, given that these violations remain +minor, and by recalling Theorem 3, we can conclude that the +periodicity assumption comes at a minor penalty. +B. Anomalies’ Key Distributions +Next, our goal is to investigate the distribution of both +the anomalies’ inter-arrival and maintenance times. To do so, +we leverage the troubleshooting tickets data provided by the +network’s operator along with alarms information registered +by the base stations. In essence, when an anomaly takes place +at any base station in the network, the base station raises +an alarm, and, consequently, a trouble ticket is issued by the +operator. This ticket contains details about the anomaly (e.g., +anomaly ID), its location, its occurrence time, and (eventually) +its resolution time. As seen in Fig 4, the distribution of the +anomalies’ inter-arrival times in the overall network is very +Fig. 3: LTE traffic demand. +close to an exponential distribution of rate λ = 12.6 anoma- +lies/hour. Supposing that the base stations are all identical, +and given the splitting property of Poisson processes [25], we +can conclude that the anomalies’ rate for each base station is +λ = 0.019 anomalies/hour. +Fig. 4: Anomalies’ inter-arrival times distribution. +On another note, we report in Fig. 5 an extract of the +anomalies’ maintenance time distribution. As one can see, this +distribution is far from regular. Specifically, we can see that a +part of the anomalies are resolved almost instantaneously by +the network itself. On the other hand, other anomalies require +either remote or on-site interventions that take longer time +(hours, days, and sometimes weeks). Modeling such a distri- +bution is a challenging task. However, we recall that the results +reported in Section III hold for any general maintenance time +distribution. In fact, all we need to characterize the expected +utility not satisfied is the average maintenance duration, which +puts into perspective the generality of our results and their +practical usefulness. To that end, using the troubleshooting +tickets data, we can conclude that the average maintenance +time is 2 hours and 8 minutes. +With the above results in mind, along with the periodic nature +of users’ traffic as was illustrated in the previous section, we + +2.0 +1.8 +_DL Traffic Volume(GB) +1.6 +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0 +24 +72 +96 +120 +144 +168 +Time(hours)0.0040 +Exponential distribution +Anomalies inter-arrival distribution +0.0035 +Probability density function +0.0030 +0.0025 +0.0020 +0.0015 +0.0010 +0.0005 +0.0000 +0 +1000 +2000 +3000 +4000 +5000 +Time (s)8 +Fig. 5: Extract of the maintenance time distribution. +can conclude that the results of Theorem 2 can be used to find +the expected utility not satisfied in the network. Particularly, +knowing that the average traffic per hour for each base station +is U = 1.55 GBs/hour, the operator can, then, conclude that +the expected traffic demand not satisfied in the entire network +is +L ∞ = 660 × +2.13 +1 +0.019 + 2.13 × 1.55 ≃ 39 GBs/hour +(35) +All in all, we can conclude that the network loses on average +around +2.13 +1 +0.019 +2.13 ≃ 3.8% of its traffic due to the various +anomalies that may occur. This reliability score can be used +by the operator to assess the network’s performance, with +respect to its business objective. Further planning and network +upgrades can, then, take place if necessary, and a re-evaluation +of the score can be done to conclude on the efficacy of the +proposed upgrades. +C. Convergence Speed: Single Cell +Thus far, we have been interested in verifying the key +assumptions adopted in our theoretical analysis and character- +izing the distributions needed to evaluate the theoretical limit +of the expected demand not satisfied L , denoted by L∞. +However, investigating how fast we converge to this limit L∞ +is of paramount interest. To that end, let +L T = 1 +T +� T +0 +U(t)W(t)dt +(36) +denote the average traffic demand not satisfied in the time +interval [0, T]. Equivalently, as we have done in eq. (8), we +can define the average demand not satisfied in the anomaly- +repair cycles [0, n] as +L n = +1 +n +�n +j=1 +� D[p] +j +D[p] +j −Yj U(t)dt +1 +n +�n +j=1(Xj + Yj) +. +(37) +With these definitions in mind, our aim is to find at what stage +n∗ the relative error +|L n−L∞| +L n +becomes smaller than 10% +for n ≥ n∗. To do so, we first consider that the inter-arrival +time between anomalies follows an exponential distribution +with rate λ = 0.019 hours−1. As for the maintenance time, +to facilitate our convergence speed study, we consider that +the maintenance time is exponentially distributed with rate +µ = 0.47 hours−1, which is in accordance with the average +maintenance time duration observed in the data. Note that the +insights we are about to provide would still hold for other +general maintenance time distributions. Finally, concerning the +utility function, we suppose that the utility function is perfectly +periodic and can be written as +U(t) = 1.75 sin(2πt +p ) + 3, +t ≥ 0, +(38) +where p = 24 hours. The above function mimics very well +the utility function illustrated in Fig. 3. We report our results +in Fig. 6. As can be seen, the loss function L n converges to +L ∞, as Theorem 2 predicts. Particularly, it takes on average +n∗ = 350 anomaly-repair cycles for the relative error to +become smaller than 10%. Given the average time of each +cycle, we can conclude that the convergence time can be quite +slow. The reason behind this is that although the convergence +of D[p] +j +to a uniform distribution happens exponentially fast, as +shown in Proposition 1, the overall convergence of L n to L∞ +reported in Theorem 2 is not exponentially fast. Particularly, as +can be seen in Appendix F, the convergence speed is inversely +proportional to n and depends heavily on the correlation +between the different quantities reported in Proposition 4 and +the variance of the random variables involved. However, as we +will show in the next section, this convergence speed becomes +substantially faster when a large-scale network is considered, +rather than an individual base station. +Fig. 6: Illustration of the convergence of the cellular network +with one cell +D. Convergence Speed: Multiple Cells +In practice, the network’s operator is interested in evaluating +the expected utility loss of the whole network rather than that + +Maintenance time +0.007 +Probability density function +0.006 +0.005 +0.004 +0.003 +0.002 +0.001 +0.000 +0 +200 +400 +600 +800 +1000 +Time (s)Expected loss L. +Loss En +0.20 +Average Loss +0.15 +0.10 +0.05 +0 +200 +400 +600 +800 +1000 +Cycle number n9 +of a particular cell. Specifically, if the network is made of N +cells, then we are interested in the speed of convergence of +L T = +1 +NT +N +� +i=1 +� T +0 +U i(t)W i(t)dt, +(39) +to its limit L ∞, where U i(t) is the utility at time t of cell +i and W i(t) is a binary random variable indicating if cell i +is suffering from an anomaly. To investigate this convergence +speed, we adopt the same settings of the previous subsection +and compare the multi-cell case to the single-cell one. Note +that since anomalies may occur at different instants at the +various cells, we report the average convergence speed in +time units rather than anomaly-repair cycles. The results are +reported in Fig. 7. +Fig. 7: Comparison of the convergence time in single and +multi-cell scenarios. +As can be seen, the average convergence speed for the whole +network’s average utility loss is around 41 hours, which is +significantly faster than the single-cell one. In other words, by +letting the network run for a couple of days, our evaluation of +the expected utility not satisfied through the simple theoretical +formula we provided in Theorem 2 would already be very +accurate. Given that networks are typically constituted of a +large number of sub-systems (e.g., cells, routers, etc.), we can +conclude that this convergence speed is particularly appealing. +To understand this trend, we note that the large number of +cells leads to a spatial averaging effect. Particularly, as more +cells are added, the variance reported in Appendix F will +be reduced, thus greatly accelerating the convergence speed. +The question that remains is the following: how big does +the network need to be to obtain such fast convergence? To +answer that question, we simulate 100 different realizations +of the network’s operation for a variety of network sizes. The +results are reported in Fig. 8. As can be observed, the average +convergence time decreases with the number of cells, reaching +as low as 1 week for a network size of 100 cells. +Finally, we examine the impact of violating the assumption of +perfect periodicity on the convergence speed of L T . To do +this, we compare the perfect periodicity case to the scenario +where U(t) in eq. (38) is corrupted by an Ornstein-Uhlenbeck +Fig. 8: Evolution of the convergence time with the number of +cells. +process B(t). Specifically, B(t) satisfies the following stochas- +tic differential equation: +dB(t) = −θB(t)dt + σdR(t), +(40) +where R(t) is the Wiener process. By setting θ = 1 and σ = +0.01, we plot the relative error |L T −L ∞| +L T +as a function of T. +As shown in Fig. 9, the relative error between the expected +loss and its limit calculated by our formula quickly converges +to zero. In fact, the convergence speed of the two cases is +similar, with both reaching the 10% mark at around 41 hours. +This confirms the results of Theorem 3, which states that even +with minor, random fluctuations from the perfect periodicity +regime, our convergence results still hold. +Fig. 9: Evolution of the relative error between the loss and its +expected limit as a function of T. +E. Non-Stationarity +Thus far, we have assumed in our analysis that the inter- +arrival times and maintenance times of the anomalies follow +stationary distributions. This means that the statistics of these +distributions, such as the expected values E[X] and E[Y ], do + +12 +Average convergence time +11 +10 +6 +8 +7 +0 +200 +400 +600 +800 +1000 +1200 +1400 +Number of cells0.40 +Relative error without an OU Process +0.35 +Relative error with an OU Process +0.30 +error +0.25 +0.20 +lative +0.15 +Rel +0.10 +0.05 +0.00 +0.05 +0 +20000 40000 60000 80000 100000120000140000160000 +Time(minutes)0.175 +Expected loss L. +Average Loss - one cell +0.150 +Average Loss - 660 cells +0.125 +0.100 +LOSS +0.075 +0.050 +0.025 +0.000 +0 +50000 100000 150000 200000 250000 300000 350000 400000 +Time(minutes)10 +not change over time. However, it is important to verify the +accuracy of this assumption. To do so, we can use our real +data to plot +∆ = +E[Y ] +E[X] + E[Y ] = 1 − Availability +(41) +to see if it remains constant over time. Our attention is on ∆, +as the theoretical limit L∞ depends on the aforementioned +distributions through ∆. Fig. 10 demonstrates that ∆ has +minimal fluctuations, causing minor violations of the stationar- +ity assumption. Nevertheless, these fluctuations have minimal +impact on the convergence time, as can be seen in Fig. 11. +The relative error |L T −L ∞| +L T +fluctuates minimally and remains +close to zero. +Fig. 10: Variation of ∆ as a function of time. +Fig. 11: Variation of the relative error as a function of time. +VI. CONCLUSIONS AND FUTURE WORK +In this paper, we have considered the evaluation of the re- +liability of large networked systems with respect to a periodic +time-dependent utility function related to the system’s service +performance over time. Under the assumption of exponential +anomalies’ inter-arrival times and general distributions of +maintenance time duration, we have leveraged the periodicity +of the utility function to derive the expected utility loss due to +the system’s anomalies. 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Jin, “Understanding mobile +traffic patterns of large scale cellular towers in urban environment,” +IEEE/ACM Transactions on Networking, vol. 25, no. 2, pp. 1147–1161, +2017. +[24] E. Yukseltan, A. Yucekaya, and A. H. Bilge, “Forecasting electricity +demand for turkey: Modeling periodic variations and demand segrega- +tion,” Applied Energy, vol. 193, pp. 287–296, 2017. [Online]. Available: +https://www.sciencedirect.com/science/article/pii/S0306261917301848 +[25] D. P. Bertsekas and J. N. Tsitsiklis, Introduction to probability: Dimitri +P. Bertsekas and John N. Tsitsikis. +Athena Scientific, 2008. +[26] P. Schatte, “On the asymptotic uniform distribution of sums reduced mod +1,” Mathematische Nachrichten, vol. 115, no. 1, pp. 275–281, 1984. +APPENDIX A +PROOF OF THEOREM 1 +Our proof revolves around a Fourier analysis of the distri- +bution of X[p] +j . To proceed in this direction, let us first note +that by definition +j +� +k=1 +Xk = mp + X[p] +j += p(m + +X[p] +j +p ), +(42) +where m ∈ N. By dividing both sides by 1 +p, we can conclude +that +X[p] +j += pR[1] +j , +(43) +where R[1] +j +is the remainder of the Euclidean division of Rj = +�j +k=1 +Xk +p +by 1. To that end, to characterize the distribution +of X[p] +j , we can start by studying that of R[1] +j . With this in +mind, the first step of our analysis in this direction consists of +showing the existence of an order j∗ such that +sup +r∈[0,1] +fR[1] +j∗ (r) < ∞, +(44) +where fR[1] +j∗ (r) is the probability density function of R[1] +j∗ . The +above condition is essential to our Fourier analysis as will be +seen shortly. To find j∗, we first note that thanks to the scaling +property of the exponential distribution, R1 is exponentially +distributed with rate λp. Therefore, we have +sup +r∈[0,∞) +fR1(r) = fR1(0) = λp. +(45) +Given that fR1(r) is bounded, we can assert that the same +holds for fR[1] +1 (r). Consequently, we can conclude that j∗ = 1. +Next, let +ˆg(n) = +� ∞ +0 +fR1(r)e−2πinrdr +(46) +denote the nth Fourier coefficient of the probability density +function of R1. Knowing that R1 is exponentially distributed, +we obtain +ˆg(n) = +� ∞ +0 +λpe−λpre−2πinrdr = +λp +λp + 2πin. +(47) +Given the above expression, we can conclude that |ˆg(n)| is +decreasing with n. Therefore, we have +α = sup +n∈N∗ |ˆg(n)| = |ˆg(1)| = +λp +� +λ2p2 + 4π2 . +(48) +Given that α < 1 and that condition (44) holds, we can +leverage [26, Theorem 1] to deduce that +sup +r∈[0,1] +|fR[1] +j (r) − 1| ≤ Cαj, +(49) +where +C = α−2 +∞ +� +n=1 +|ˆg(n)|2j∗ = +∞ +� +n=1 +λ2p2 + 4π2 +λ2p2 + 4π2n2 < ∞. +(50) +Note that the convergence of C is due to the fact that the term +n2 exists in the denominator. Lastly, given that X[p] +j += pR[1] +j , +we can conclude that +fX[p] +j (x) = 1 +pfR[1] +j (x +p ). +(51) +With this in mind, and using the results of eq. (49), we can +conclude our proof. +APPENDIX B +PROOF OF LEMMA 2 +Due to the independence between A and B, we can derive +the probability density function fZ(z) for z ∈ [0, p] as follows +fZ(z) = (fA ⊛p fB)(z) = +� z +0 +fA(x)fB(z − x)dx ++ +� p +z +fA(x)fB(z + p − x)dx, +(52) + +12 +where ⊛p indicates that the convolution is done modulo p. +Next, we investigate the gap between the density function +fZ(·) and the uniform distribution on [0, p]. Precisely, +|fZ(z) − 1 +p| +(a) += | +� z +0 +(fA(x) − 1 +p)fB(z − x)dx ++ +� p +z +(fA(x) − 1 +p)fB(z + p − x)dx| +(b) +≤ +� z +0 +|fA(x) − 1 +p|fB(z − x)dx ++ +� p +z +|fA(x) − 1 +p|fB(z + p − x)dx +(c) +≤ +sup +x∈[0,p] +|fA(x) − 1 +p|( +� z +0 +fB(z − x)dx ++ +� p +z +fB(z + p − x)dx) +(d) += +sup +x∈[0,p] +|fA(x) − 1 +p|, +(53) +where (a) is the result of writing 1 +p as an integral over the +density function fB(·), (b) is deduced from the triangular +inequality, (c) can be concluded from the definition of the +supremum, and (d) comes from the fact that fB(·) is a density +function. By taking the supremum on the left hand side, we +can conclude the lemma. +APPENDIX C +PROOF OF PROPOSITION 2 +To obtain our results, we leverage the Stolz–Ces`aro theorem. +To that end, let us define the sequences an = �n +j=1 E[Ij] and +bn = n. We have +lim +n→+∞ +an+1 − an +bn+1 − bn += +lim +n→+∞ +E[In+1] +1 +. +(54) +Let us now define I as follows +I = EY,V [ +� V +V −Y +U(t)dt], +(55) +where V is a RV uniformly distributed on [0, p]. In essence, I +is the expectation of Ij if D[p] +j +is uniformly distributed. Next, +we develop the expression above as follows +I = +� ∞ +0 +� p +0 +� v +v−y +U(t)1 +pfY (y)dtdvdy. +(56) +where fY (y) is the probability density function of the RVs +Yj for j ∈ N∗. To further simplify the expression above, we +proceed with a change of variable. Precisely, we let s = t−v. +Consequently, we end up with +I = +� ∞ +0 +� p +0 +� 0 +−y +U(s + v)1 +pfY (y)dsdvdy +(a) += +� ∞ +0 +� 0 +−y +� p +0 +U(s + v)1 +pfY (y)dvdsdy +(b) += +� ∞ +0 +� 0 +−y +� p +0 +U(v)1 +pfY (y)dvdsdy += E[Y ]1 +p +� p +0 +U(v)dv = E[Y ]U, +(57) +where (a) is the result of changing the order of integration, +and (b) is due to the periodicity of U(t). Now, let us write +In+1 in a more convenient form as follows +In+1 = +� D[p] +n+1 +D[p] +n+1−Yj +U(t)dt = +� [Dn+Xn+1] mod p+Yn+1 +[Dn+Xn+1] mod p +U(t)dt. +(58) +Then, we can proceed as done in eq. (57) to obtain +E[In+1] = +� ∞ +0 +� 0 +−y +� p +0 +U(s + y + v)fV (v)fY (y)dvdsdy, +(59) +where fV (v) is the probability density function of [Dn + +Xn+1] mod p. By leveraging Lemma 2 and Proposition 1, we +can conclude that +sup +v∈[0,p] +|fV (v) − 1 +p| ≤ Cαn +p +. +(60) +Consequently, we have fV (v) +≤ +Cαn +p ++ 1 +p. Knowing +this, we can now investigate the difference between I and +limn→+∞ E[In+1]. Given all what was presented above, we +obtain +|E[In+1] − I| ≤ Cαn +p +E[Y ]U +n→+∞ +−−−−−→ 0, +(61) +given that E[Y ], U are finite and α < 1. Finally, given that +bn is a strictly monotone and divergent sequence, we can +conclude from the Stolz–Ces`aro theorem that Sn converges +to I for large n. +APPENDIX D +PROOF OF PROPOSITION 3 +To start our proof, we first note that by the definition of +D[p] +j +reported in eq. (11) and the independence of Xi and Yi, +we have +fD[p] +j ,D[p] +j+k(zj, zj+k) = fD[p] +j (zj)fD[p] +k ([zj+k − zj] mod p). +(62) +Next, thanks to Lemma 2, and by leveraging the triangular +inequality, we can conclude that +sup +z∈[0,p] +|fD[p] +j (z) − 1 +p| ≤ sup +x∈[0,p] +|fX[p] +1 (x) − 1 +p| +≤ sup +x∈[0,p] +fX[p] +1 (x) + 1 +p. +(63) + +13 +We recall that X1 is an exponential random variable of rate +λ. Hence, the supremum of its probability density function +is equal to λ. Given that the probability density function of +X1 is bounded, we can conclude that there exists M such +that fX[p] +1 (x) ≤ M. With this in mind, and by leveraging the +triangular inequality, we can deduce that +|fD[p] +j (·)fD[p] +k (·) − 1 +p2 | = |(fD[p] +j (·) − 1 +p)fD[p] +k (·)+ +1 +p(fD[p] +k (·) − 1 +p)| ≤ (M + 1 +p)Cαj +p ++ 1 +p +Cαk +p +. +(64) +Finally, by letting C′ += M + 1 +p, we can conclude the +proposition. +APPENDIX E +PROOF OF PROPOSITION 4 +By definition, we have +Cov[Ij, Ij+k] = E[IjIj+k] − E[Ij]E[Ij+k]. +(65) +Let us first investigate the term E[IjIj+k]. As was done in eq. +(58), we can rewrite E[IjIj+k] as follows +E[IjIj+k] =EYj,Yj+k[ +� p +0 +� p +0 +� zj+Yj +zj +U(t)dt × +� zj+k+Yj+k +zj+k +U(t′)dt′fZ[p] +j +,Z[p] +j+k(zj, zj+k)dzjdzj+k], +(66) +where Z[p] +j +and Z[p] +j+k denote [Dj +Xj+1] mod p and [Dj+k + +Xj+k+1] mod p respectively. By leveraging Lemma 2 and +Proposition 3, we obtain +E[IjIj+k] ≤EYj,Yj+k[ +� p +0 +� p +0 +� zj+Yj +zj +U(t)dt × +� zj+k+Yj+k +zj+k +U(t′)dt′( 1 +p2 + C′C(αj +p + αk +p )dzjdzj+k]. (67) +Given that the RVs Yj and Yj+k are i.i.d., we obtain +E[IjIj+k] ≤ I +2(1 + pC′C(αj + αk)). +(68) +Next, we need to investigate the second term E[Ij]E[Ij+k]. To +do so, we recall from eq. (61) that +|E[Ij] − I| ≤ Cαj−1 +p +I. +(69) +With this in mind, we can obtain the results of the proposition, +which concludes our proof. +APPENDIX F +PROOF OF THEOREM 2 +From the expression in eq. (8), we can rewrite L as +L = +lim +n→+∞ +1 +n +�n +j=1 Ij +1 +n +�n +j=1(Xj + Yj). +(70) +Let us start by investigating the numerator Nn = 1 +n +�n +j=1 Ij. +To that end, let us consider the event +Pr(|Nn − I| ≥ ϵ) +(a) +≤ Pr(|Nn − Sn| ≥ ϵ +2 ∪ |Sn − I| ≥ ϵ +2) +(b) +≤ Pr(|Nn − Sn| ≥ ϵ +2) + Pr(|Sn − I| ≥ ϵ +2), +(71) +where (a) and (b) are the results of the triangular inequality +and the union bound, respectively. Note that the term |Sn −I| +is deterministic, and from Proposition 2, we know that this +term tends to zero when n is large. Therefore, what remains +is to examine the first term Pr(|Nn − Sn| ≥ +ϵ +2). To do so, +we leverage Chebyshev’s inequality to obtain the following +upperbound +Pr(|Nn − Sn| ≥ ϵ +2) ≤ 4Var[Nn] +ϵ2 +. +(72) +However, we know that +Var[Nn] = +n +� +j=1 +Var[Ij] + 2 +n−1 +� +j=1 +� +j 0 +such that +sup +r∈[0,1] +|fR[1] +j (r) − 1| ≤ Cαj, +(81) +In particular, these constants are equal to +α = sup +n∈N∗ |ˆg(n)|, +(82) +C = +�∞ +n=1 |ˆg(n)|2 +α2 +, +(83) +where +ˆg(n) = +� ∞ +0 +fR1(r)e−2πinrdr. +(84) +Lastly, by using the Fourier scaling property and the fact that +fX[p] +j (x) = 1 +pfR[1] +j ( x +p), we can conclude our proof. +APPENDIX H +PROOF OF THEOREM 3 +Let us start our proof by recalling the definition of the +average utility loss in any interval [0, T]. Particularly, we have +L T = 1 +T +� T +0 +U(t)W(t)dt. +(85) +Then, given eq. (27), we can rewrite L T as follows +L T = 1 +T +� T +0 +U ′(t)W(t)dt +� +�� +� +L +′ +T ++ 1 +T +� T +0 +B(t)W(t)dt +� +�� +� +L +′′ +T +. +(86) +With the above in mind, we recall that our goal is to show +that for every ϵ > 0, we have +Pr(|L T − L ∞| ≥ ϵ) ≤ g(T, ϵ), +(87) +where g(T, ϵ) +T →∞ +−−−−→ 0 for any positive ϵ. To prove this, +we leverage the triangular inequality and the union bound to +conclude that +Pr(|L T −L ∞| ≥ ϵ) ≤ Pr(|L ′T −L ∞| ≥ ϵ +2)+Pr(|L +′′ +T | ≥ ϵ +2). +(88) +Given that U ′(t) is a periodic function, we can conclude from +Theorem 2 that there exists a function g1(T, ϵ) such that +Pr(|L ′T − L ∞| ≥ ϵ +2) ≤ g1(T, ϵ), +(89) +and g1(T, ϵ) +T →∞ +−−−−→ 0 for any positive ϵ. What remains is to +show similar results for the second term of eq. (88). To do so, +we first note that +E[L +′′ +T ] = 1 +T +� T +0 +E[B(t)W(t)] +(a) += 1 +T +� T +0 +E[B(t)]E[W(t)] = 0, +(90) +where (a) is due to the independence between B(t) and W(t). +Next, we investigate the variance of the term L +′′ +T . To that +end, given that E[L +′′ +T ] = 0, we have Var[L +′′ +T ] = E[(L +′′ +T )2]. +Consequently, we derive the second order moment of L +′′ +T +below: +E[(L +′′ +T )2] = 1 +T 2 E[( +� T +0 +B(t)W(t)dt)( +� T +0 +B(s)W(s)ds)] +(a) += +1 +T 2 E[ +� T +0 +� T +0 +B(t)W(t)B(s)W(s)dtds] +(b) += +1 +T 2 +� T +0 +� T +0 +E[B(t)B(s)]E[W(t)W(s)]dtds +(c) +≤ +1 +T 2 +� T +0 +� T +0 +E[B(t)B(s)]dtds] +(d) += +1 +T 2 +� T +0 +� T +0 +Cov[B(t), B(s)]dtds +(91) +where (a) is the result of iterated integrals, (b) follows from +the independence between B(t) and W(t), (c) springs from +the fact that 0 ≤ W(·) ≤ 1, and (d) is due to B(·) being +of zero mean. Now, using the wide sense stationary property +along with a change of integration variable, we can rewrite +E[(L +′′ +T )2] as follows +E[(L +′′ +T )2] = 1 +T 2 +� T +0 +� T −s +−s +ρ(τ)dsdτ. +(92) +An illustration of the integration domain is reported in Fig. +12. Given the symmetry of the domain D with respect to +the horizontal axis, along with the symmetry of ρ(·), we can +manipulate the integration variables to obtain the following: +E[(L +′′ +T )2] = 2 +T 2 +� T +0 +(T − τ)ρ(τ)dτ. +(93) +Then, by introducing the absolute value, we get +E[(L +′′ +T )2] ≤ 2 +T 2 +� T +0 +|T − τ||ρ(τ)|dτ. +(94) + +15 +Now, given the fact that 0 ≤ τ ≤ T, we can deduce that +E[(L +′′ +T )2] ≤ 2 +T +� T +0 +|ρ(τ)|dτ. +(95) +With the above in mind, and given Assumption 3 and Cheby- +shev’s inequality, we can conclude that there exists a function +g2(T, ϵ) such that +Pr(|L ′′T | ≥ ϵ +2) ≤ g2(T, ϵ), +(96) +and g2(T, ϵ) +T →∞ +−−−−→ 0 for any positive ϵ. This concludes our +proof. +Fig. 12: Illustration of the integration region related to +E[(L +′′ +T )2]. + +D \ No newline at end of file diff --git a/S9E5T4oBgHgl3EQfag-J/content/tmp_files/load_file.txt b/S9E5T4oBgHgl3EQfag-J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce885cc08deab5e494d8b7e2173124bc590e5152 --- /dev/null +++ b/S9E5T4oBgHgl3EQfag-J/content/tmp_files/load_file.txt @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf,len=858 +page_content='1 A Framework for the Evaluation of Network Reliability Under Periodic Demand Ali Maatouk*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fadhel Ayed*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Shi Biao†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Wenjie Li*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Harvey Bao*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' and Enrico Zio¶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' ‡ Paris Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Huawei Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Boulogne-Billancourt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' France † ´Ecole polytechnique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Palaiseau,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' France ¶Centre de recherche sur les Risques et les crises,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' MINES Paris-PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' France ‡Energy Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Politecnico di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Milan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Italy Abstract—In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' we study network reliability in relation to a periodic time-dependent utility function that reflects the system’s functional performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' When an anomaly occurs, the system incurs a loss of utility that depends on the anomaly’s timing and duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We analyze the long-term average utility loss by considering exponential anomalies’ inter-arrival times and general distributions of maintenance duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We show that the expected utility loss converges in probability to a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We then extend our convergence results to more general distributions of anomalies’ inter-arrival times and to particular families of non- periodic utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To validate our results, we use data gathered from a cellular network consisting of 660 base stations and serving over 20k users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We demonstrate the quasi-periodic nature of users’ traffic and the exponential distribution of the anomalies’ inter-arrival times, allowing us to apply our results and provide reliability scores for the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We also discuss the convergence speed of the long-term average utility loss, the interplay between the different network’s parameters, and the impact of non-stationarity on our convergence results Keywords—Cellular networks, network reliability, expected utility not satisfied, periodic utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' INTRODUCTION The evaluation of the reliability of any large-scale service system aims to assess its ability to provide services, taking into account the various hazardous events and anomalous conditions that can occur and impact the functioning of its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' These events, such as component failures, can affect the overall service reliability and availability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In the literature on critical infrastructures, such as electric power grids, several measures are commonly used to evaluate reliability, including the loss of load probability (LOLP), expected frequency of load curtailment (EFLC), ex- pected duration of load curtailment (EDLC), expected duration of curtailment (EDC), and expected demand not satisfied (EDNS) [2]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In recent years, the extent to which we rely on networked data systems as part of our critical national infrastructure has become increasingly apparent [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As a result, the character- ization of reliability metrics in networked systems, such as cellular networks, has become crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Traditionally, the liter- ature on networked data systems has focused on maximizing This work has been presented in part at the 32nd European Safety and Reliability Conference (ESREL 2022) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' link-level reliability to ensure the best possible service for con- nected devices (see references [7], [8], and [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Most research on network-wide reliability has consisted of evaluating graph- based connectivity metrics in light of possible failures that may occur in the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', k-terminal network reliability, as described in reference [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, there has recently been a shift towards strict network-level reliability that takes into account user data traffic demand, particularly for next- generation cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For example, 5G networks are expected to provide 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='999% (or ”five nines”) of data avail- ability annually [11], with plans to improve to a seven-nines standard in 6G [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the nature of these requirements, the characterization of the expected demand not satisfied metric in such systems becomes especially important [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, evaluating this measure is challenging due to the complexity and randomness of anomalies, failures, outages, and maintenance in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In this paper, we focus on quantitatively characterizing the expected utility not satisfied of a networked system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, we characterize reliability with respect to a time- dependent utility function U(t) related to the system’s func- tional performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The utility function U(t) can represent, for example, the data traffic typically transmitted at time t, the number of users served, or other similar quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Evaluating the system’s reliability, then, involves deriving the long-term expected utility loss L of the system, taking into account the failures that may occur at its components and external factors that influence the impact of such failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the dependence of U(t) on time, the evaluation of L requires the formulation of the stochastic differential equations governing its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, tools such as stochastic hybrid systems and Dynkin’s formula are leveraged to analyze L [14]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, obtaining a closed-form expression of L is heavily contingent on the complexity of the differential equations involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Generally, only approximations can be obtained by such analytical frameworks [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Another approach to characterize L consists of running Monte Carlo simulations of the system [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, Monte Carlo simulations can be computationally costly, especially when a large number of components interact with one another to provide the system’s functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, the absence of closed-form expressions reduces the interpretability of L and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='05589v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='IT] 13 Jan 2023 2 hinders the optimization process of the system’s parameters in the design stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Hence, the goal of our paper is to address these challenges and provide a theoretical framework to obtain an expression of L under a periodicity assumption on U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, the following are the key contributions of this paper: We begin our stochastic analysis by formulating the expected utility not satisfied L as a function of various elements, such as the inter-arrival times of anomalies and their repair times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Through Fourier analysis of the anomalies’ inter-arrival time distribution, we show that a key stochastic process converges to a uniform distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We then leverage this convergence to provide limiting distributions for several quantities that impact the system’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we combine our earlier results and the periodicity of the utility function to show that the expected utility not satisfied converges in probability to a simple and intuitive form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We also establish connections between this expression and standard availability metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Afterward, we extend our theoretical results to the case where the periodicity assumption is slightly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, we show that our convergence results still hold when the periodic utility is corrupted by a random stochastic process that satisfies predefined conditions on its first and second-order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, through Fourier analysis, we also demonstrate that our assumption of exponential distribution for the anomalies’ inter-arrival times can be relaxed, and our theoretical findings hold for any bounded probability density function of the anomalies’ inter-arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Finally, we consider a large-scale cellular network with 660 cells serving over 20,000 users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Using data we obtained from the network operator, we demonstrate the quasi-periodic nature of user traffic and the exponential distribution of anomalies’ inter-arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We, then, use our theoretical results to characterize the expected data traffic not satisfied for this specific network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Addi- tionally, we show that the expected utility not satisfied converges quickly to its theoretical limit, and we inves- tigate the impact of non-stationarity on the convergence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The rest of the paper is organized as follows: Section II introduces the system model adopted in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In Section III, we present the mathematical analysis of our system, and we provide our main theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Afterward, we extend our theoretical analysis to more general settings in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, a use case scenario consisting of a large- scale cellular network is considered in Section V, and our theoretical findings are then corroborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Lastly, Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' SYSTEM MODEL We consider a system operating in its useful-life phase, during which independent anomalies occur randomly with a Poisson rate λ [21, Chapter 3], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Thus, the inter-arrival time Xj between anomalies j −1 and j is exponentially distributed with rate λ Pr(Xj < t) = 1 − e−λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (1) When an anomaly takes place, the system operator triggers a maintenance procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We let Yj denote the maintenance time associated with anomaly j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We make the assumption that the variables Yj are independent of Xj and are independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=') with a cumulative distribution function described by Pr(Yj < t) = � FY (t), for t ≥ 0, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (2) This means that for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', the variable Yj follows the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We also make the following assumption on the distribution of Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The first and second order moments E[Yj] and E[Y 2 j ] are finite for j ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Furthermore, we consider that when an anomaly takes place at time t0 and the problem is resolved at time t1, a utility loss � t1 t0 U(t)dt is incurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Letting W(t) be a binary random variable that is equal to 1 when the system is suffering from an anomaly and 0 otherwise, we can define the expected utility not satisfied as follows L = lim T →+∞ 1 T � T 0 U(t)W(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (3) In practice, the consideration of large T amounts to the system being operated long enough before the expected utility loss assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Note that the function U(t) can represent a large variety of system quantities depending on the system’s operator priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For example: U(t) can denote the customer demand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', electricity, communication traffic) served by the system at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' in this scenario, the expected loss score coincides with the notion of EDNS [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' U(t) can represent the number of users served by the system at time t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' thus, the expected utility loss in this case represents the expected number of users affected by the anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' U(t) = β(t)U ′(t), where U ′(t) denotes the utility of the system at time t (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', customer demand) and β(t) is a factor between 0 and 1 that indicates the fraction of utility lost depending on the occurrence time of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' One can also define U(t) as a combination of system quan- tities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This shows the generality of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In the next section, we will illustrate the mathematical framework to characterize L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' But, first, let us consider the following assumption for U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The utility function is a non-negative bounded periodic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely, there exists a constant K > 0 and a period p > 0 such that 0 ≤ U(t) ≤ K, U(t + p) = U(t), (4) 3 for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The interest in the above assumption is that, as we will show in later sections, when U(t) verifies the above assumption, L ends up converging in probability to a relatively simple form, which relaxes the difficulties of the quantitative char- acterization of the reliability of complex networked systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Note that the periodicity of system utility has been observed in various practical applications due to the nature of human behavior with respect to service demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For example, in cellular networks, it was shown that user traffic typically exhibits a periodical pattern [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Such patterns have been also found in data that we gathered from a large-scale cellular network, as will be illustrated in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' It is worth noting that such trends are not exclusive to cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For instance, the periodic behavior has also been witnessed in the electricity demands in power grid networks [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Accord- ingly, our theoretical derivations are not constrained to cellular networks settings, but rather can be leveraged for various other application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' MATHEMATICAL ANALYSIS We decompose the time horizon T in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (3) into multiple stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, we let Dn = �n j=1(Xj + Yj) and we rewrite the expected utility loss of the system as L = lim n→+∞ � Dn 0 U(t)W(t)dt Dn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (5) Next, by multiplying by 1 n both the numerator and denomina- tor, we end up with L = lim n→+∞ 1 n �n j=1 � Dj Dj−1 U(t)W(t)dt 1 n �n j=1(Xj + Yj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (6) Noting that W(t) is equal to 0 by definition in every interval [Dj, Dj + Xj], we can rewrite the expected loss as L = lim n→+∞ 1 n �n j=1 � Dj Dj−Yj U(t)dt 1 n �n j=1(Xj + Yj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (7) Clearly, the challenging part of the evaluation of the expected loss is the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To deal with this, we leverage the periodicity of the function U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' If U(t) is a periodic function of period p, then the expected loss L can be rewritten as L = lim n→+∞ 1 n �n j=1 � D[p] j D[p] j −Yj U(t)dt 1 n �n j=1(Xj + Yj) , (8) where D[p] j = Dj mod p is the remainder of the Euclidean division of Dj by p (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', the least positive residue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To prove this result, we first note that Dj = kp + (Dj mod p), (9) where k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we apply a change of variable to the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (7), letting t′ = t − kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By doing so, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (7) can be rewritten as follows L = lim n→+∞ 1 n �n j=1 � D[p] j D[p] j −Yj U(t′ + kp)dt′ 1 n �n j=1(Xj + Yj) (a) = lim n→+∞ 1 n �n j=1 � D[p] j D[p] j −Yj U(t′)dt′ 1 n �n j=1(Xj + Yj) , (10) where (a) results from the periodicity of U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, by interchanging t′ and t, we can confirm the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The next step in the analysis consists of finding the distri- bution of D[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we first rewrite D[p] j as follows D[p] j = (X[p] j + Y [p] j ) mod p, (11) where X[p] j = ( j � k=1 Xk) mod p, Y [p] j = ( j � k=1 Yk) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (12) Next, we investigate the distribution of X[p] j more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let fX[p] j (x) denote the probability distribution function of X[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We have that fX[p] j (x) converges uniformly to the uniform distribution on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely, sup x∈[0,p] |fX[p] j (x) − 1 p| ≤ Cαj p , (13) where α = λp � λ2p2 + 4π2 , (14) C = ∞ � n=1 λ2p2 + 4π2 λ2p2 + 4π2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The proof revolves around a Fourier analysis of the distribution of X[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In essence, we first analyze the behavior of the Fourier coefficients of the random variables making up X[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, by leveraging the scalability property of the exponential distribution, we can derive the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For conciseness, the details of the proof are reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In this theorem, we showed that X[p] j converges uniformly to a uniform distribution as j becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, the bounds we derived imply that ∇α(λ, p) ≥ 0 for λ, p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This means that the speed of convergence to a uniform distribution decreases as either p or λ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Figure 2 illustrates this trend, where α varies and its contour levels are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To understand this trend, we can examine how the cumulative density function of the anomalies’ inter-arrival times changes 4 (a) Zero Gaussian added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (b) One Gaussian added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (c) Ten Gaussians added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1: Illustrations of the smoothing phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' with increasing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For any interval I = [0, a] where a is any positive real number, Pr(Xk ∈ I ) = 1−exp(−λa) increases as λ grows for k ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Based on this and the expression for X[p] j given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (12), we can conclude that more realizations of Xk are needed to span all the possible values from [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Furthermore, as p increases, the range of values where X[p] j can fall also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Hence, using the expression for X[p] j , we can see that more realizations of Xk are also needed in this case to span all the possible values of X[p] j from [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In both cases, larger values of j are needed in order to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 2: Illustration of α in function of λ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we recall that our goal is to characterize the distribution of D[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we need to take into account the distribution of Y [p] j , which can be quite general as we impose no restriction on FY (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To alleviate this difficulty, we provide below an essential lemma to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let A be a random variable defined on [0, p] satisfying sup x∈[0,p] |fA(x) − 1 p| ≤ C, (16) where fA(x) is the probability density function of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let B be a random variable of arbitrary distribution defined on [0, p] independent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, sup x∈[0,p] |fZ(z) − 1 p| ≤ sup x∈[0,p] |fA(x) − 1 p| ≤ C, (17) where Z = A + B mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the independence between A and B, the proof revolves around the notion of probability distributions’ convo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, by leveraging the particularity of the distribution of A along with the definition of the modulo function, we can derive the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For conciseness, the details of the proof are reported in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The above lemma has an important interpretation: when a random variable (RV) B independent of A is added to the random variable A, it results in a smoothing effect when the modulo p function is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Thus, the distribution of Z = A + B mod p becomes even closer to a uniform distribution than A mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To illustrate this, we consider an exponentially distributed RV A of rate λ = 10, and we add to this RV 0, 1, and 10 independent standard Gaussian RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1, where the distribution of the sum of the RVs modulo p is plotted in each case, and the smoothing effect is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' With the above lemma in mind, we can now tackle the char- acterization of the distribution of D[p] j in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let fD[p] j (z) denote the probability distribution function of D[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We have that fD[p] j (z) converges uniformly to the uniform distribution on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely, sup z∈[0,p] |fD[p] j (z) − 1 p| ≤ Cαj p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To prove the proposition, we recall from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (11) that D[p] j can be written as the sum of two RVs defined on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, due to the independence between X[p] j and Y [p] j , we can leverage Lemma 2 to conclude the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The trend seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1 is confirmed by Propo- sition 1, which shows that the convergence of the distribution D[p] j to a uniform distribution happens exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This suggests that a relatively small number of anomalies/repair 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='14 Probability density function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='00 0 2 4 6 8 10 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='12 Probability density function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='00 0 2 4 6 8 10 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='10 Probability density function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='00 0 2 4 9 8 10 X8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='8 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='6 V α 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='4 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0 0 2 4 6 8 p5 stages is sufficient to model D[p] j as a uniform distribution with minimal modeling penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1, even when only 10 independent Gaussian random variables are added to the exponential random variable, the resulting sum modulo p is already distributed almost uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the above results, we can conclude that as j gets large, the distribution of D[p] j approaches the uniform distribution on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Now, to characterize L in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (8), let us define the integral Ij as Ij = � D[p] j D[p] j −Yj U(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (19) As seen in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (8), the difficult part in characterizing L comes from the fact that the terms Ij in the numerator are not independent and do not share the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Hence, providing a statistical convergence of the sum present in the numerator is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To address these challenges, we first provide the following lemma for the convergence of the sum of the expected values of Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let Sn = 1 n �n j=1 E[Ij] denote the partial sum of the expected value of Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We have Sn n→+∞ −−−−−→ I, (20) where I = E[Y ]U, and U = 1 p � p 0 U(t)dt is the average utility in a period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Our proof consists of leveraging Lemma 2 and Proposi- tion 1, the periodicity of U(t), and the Stolz–Ces`aro theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The details of the proof are reported in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the above convergence results, the next step of the analysis is to investigate the degree of dependence between each of the integrals Ij and Ij+k for any j, k ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we investigate the joint distribution function of D[p] j and D[p] j+k, as seen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let fD[p] j ,D[p] j+k(zj, zj+k) denote the joint dis- tribution function of the RVs D[p] j and D[p] j+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' There exists a constant C′ > 0 such that |fD[p] j ,D[p] j+k(zj, zj+k) − 1 p2 | ≤ C′(Cαj p + Cαk p ), (21) for all zj, zj+k ∈ [0, p] and j, k ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In this proof, we derive the joint probability distribution and leverage the particularities of the modulo function along with the independence between the RVs Xi and Yi involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For conciseness, the proof is reported in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Now that the convergence of Sn is proven and the joint probability distribution fD[p] j ,D[p] j+k(·, ·) is characterized, we can investigate the covariance of Ij and Ij+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This will allow us to measure the dependency between each of these integrals, thus enabling us to derive L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The covariance Cov[Ij, Ij+k] for j, k ∈ N∗ satisfies the following inequality Cov[Ij, Ij+k] ≤I 2(1 + pC′C(αj + αk) − (1 − Cαj−1 p ) (1 − Cαj+k−1 p )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The proof revolves around a derivation of the covari- ance of Ij and Ij+k, and leveraging the propositions and lemmas provided thus far, along with various algebraic ma- nipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The details of the proof are reported in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Interestingly, we can conclude from the results above that when j is large, and the gap between j and j′ > j is also large, then the two integrals Ij and Ij′ become uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This is going to be essential to establish the main results of our mathematical analysis below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' If (i) the anomalies inter-arrival times are exponentially distributed, (ii) the maintenance times verify Assumption 1, and (iii) U(t) satisfies Assumption 2, then the expected loss L converges in probability to L ∞ = E[Y ]U E[X] + E[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (23) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The proof of the theorem amounts to characterizing the probability that the difference between the numerator and I exceeds a certain arbitrarily small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Using the triangular inequality and the union bound, this probability is further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, by leveraging Chebyshev’s inequality and the results presented in the paper thus far, we obtain the desired characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Finally, using the weak law of large numbers and the Mann-Wald theorem, we can conclude the convergence in probability of L to L ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For conciseness, the details of the proof are reported in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As can be seen above, the expression to which L converges turns out to have a relatively simple formulation thanks to the periodicity of the function U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, given the periodic nature of U(t), the challenges in evaluating L due to the dynamic nature of the utility function are overcome, hence allowing us to evaluate L in a simplified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, the expression has an intuitive meaning and relationship with well-established metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In fact, considering the system’s availability expression Availability = MTBF MTBF + MTTR, (24) where MTBF= E[X] = 1 λ and MTTR= E[Y ] denote the mean time between failure and mean time to repair, respectively [5], and by examining the expression of L ∞, we can deduce that L ∞ = (1 − Availability � �� � (A) ) × U ���� (B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (25) The term (A) can be seen as the probability that the system suffers from an anomaly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' the term (B) is the average utility 6 that the system delivers in a period p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Then, one can see that the periodicity of the utility U(t), the exponential nature of anomalies’ inter-arrival times, and the mathematical analysis provided in the paper lead to the intuitive form of L ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In a later section, we will consider a particular application of interest and showcase the usefulness of the above-derived analytical results for the analysis of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' It is worth noting that although in our analysis we have focused on the case where one component makes up the system, our analysis can be extended to the case where the system is constituted of N components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, as long as each component’s operation is independent of the others, similar results to Theorem 2 can be obtained for the average expected utility not satisfied L = lim T →+∞ 1 NT N � i=1 � T 0 U i(t)W i(t)dt, (26) where U i(t) is the utility at time t of component i and W i(t) is a binary random variable indicating if component i is suffering from an anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As will be seen in Section V-D, the consideration of large-scale systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', N ≫ 1) will have a beneficial effect on the convergence speed of L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' GENERALIZATIONS OF THE ANALYSIS The goal of this section is to provide extensions of our theoretical analysis to more general scenarios that could arise in practical situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Relaxing the Periodicity Assumption In practice, the utility function U(t) does not exhibit perfect periodicity but, rather, fluctuates slightly around a regular period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' These fluctuations are caused by unpredictable traffic demand and user activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To account for this, we model the utility function U(t) as a stochastic process that randomly fluctuates at each time instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, we write U(t) as U(t) = U ′(t) + B(t), t ≥ 0, (27) where U ′(t) is a bounded, non-negative, periodic function with period p, and B(t) is a stochastic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The random fluctuations in U(t) are determined by the variance of the process B(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In Section V-D, we will use data from a real deployed network to show that these fluctuations are typically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we suppose that B(t) verifies the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Assumption 3 (Zero-Mean Weakly Dependent Wide-Sense Stationary Random Processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We say that a stochastic process B(t) is a zero-mean weakly dependent wide-sense stationary random process if E[B(t)] = 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Cov[B(t), B(u)] = ρ(t − u) for all t, u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1 T � T 0 |ρ(τ)|dτ T →∞ −−−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' There are many processes that satisfy this condition, in- cluding the zero-mean Gaussian white noise and the Ornstein- Uhlenbeck process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For these particular processes, we extend the results of Theorem 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' If (i) the anomalies’ inter-arrival times are exponentially distributed, (ii) the maintenance times verify Assumption 1, and (iii) B(t) verifies Assumption 3, then the expected loss L converges in probability to L ∞ = E[Y ]U E[X] + E[Y ], (28) where U = 1 p � p 0 U ′(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The proof involves using Assumption 3, integral ma- nipulations, and Chebyshev inequality to show that B(t) exhibits ergodic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' See Appendix H for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' General Inter-Arrival Distributions In our system model reported in Section II, we have as- sumed that the anomalies’ inter-arrival times are exponentially distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The reasons behind this assumption are twofold: 1) the data that we gathered from a large-scale deployed cellular network suggest that the exponential distribution is a good fit, as will be illustrated in Section V-B, and 2) the vast literature on reliability confirms that exponential distribution is found to be a good representative in many cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', [21, Chapter 3], [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, this assumption has allowed us to further understand the convergence rate reported in Theorem 1 and interpret the effect of the period p and the anomalies’ rate λ on the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Nevertheless, in this section, we aim to generalize our analytical results to a large family of distributions, particularly the family of bounded distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, we present our results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Let fX(x) denote the probability distribution function of the anomaly inter-arrival time Xj for j ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' If there exists a finite M > 0 such that fX(x) ≤ M, x ∈ [0, ∞), (29) then fX[p] j (x) converges uniformly to the uniform distribution on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely, there exists two constants α < 1 and C > 0 such that sup x∈[0,p] |fX[p] j (x) − 1 p| ≤ Cαj p , (30) where α = sup n∈N∗ |ˆg(n)|, (31) C = �∞ n=1 |ˆg(n)|2 α2 , (32) and ˆg(n) = � ∞ 0 fX(x)e−2πi n p xdx ∆= FX(n p ), (33) where FX(·) is the Fourier transform of fX(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The proof follows steps similar to those provided in Theorem 1’s proof reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For completeness, the details of the proof are reported in Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Knowing that the rest of the mathematical analysis reported in Section III holds given the above convergence results, we can conclude that the results of Theorem 2 also hold for this general family of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Similar conclusions can be made for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In essence, if the anomalies’ inter-arrival time distribution is bounded, and the remaining conditions of Theorem 2 and 3 are verified, then the expected loss L converges in probability to L ∞ = E[Y ]U E[X] + E[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (34) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' USE CASE: A LARGE CELLULAR NETWORK In this section, our aim is to verify the key assumptions adopted in our analysis, corroborate our theoretical findings, and shed light on various challenges found in our imple- mentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we leverage the data that we have gathered from a large long term evolution (LTE) cellular network consisting of 660 cellular base stations and serving approximately 22k users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The data gathered span two months and consist of cell-level key performance indicators (KPIs), along with troubleshooting tickets generated by the network’s operator and alarms information registered by the base sta- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' For confidentiality reasons, the data have been scaled when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Periodicity of Utility Functions The first assumption we verify in this section, given its importance to our theoretical analysis, concerns the periodicity of the utility function of the cellular sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Typically, the utility function of a base station is taken as the amount of traffic that it provides to the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In other words, the utility function at time t is nothing but the user demand typically served by the base station at this time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 3, we illustrate the average traffic served by a particular base station at each hour of the week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As seen in the figure, a noticeable periodic trend can be witnessed, although minor violations of the periodicity can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, given that these violations remain minor, and by recalling Theorem 3, we can conclude that the periodicity assumption comes at a minor penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Anomalies’ Key Distributions Next, our goal is to investigate the distribution of both the anomalies’ inter-arrival and maintenance times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we leverage the troubleshooting tickets data provided by the network’s operator along with alarms information registered by the base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In essence, when an anomaly takes place at any base station in the network, the base station raises an alarm, and, consequently, a trouble ticket is issued by the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This ticket contains details about the anomaly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', anomaly ID), its location, its occurrence time, and (eventually) its resolution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As seen in Fig 4, the distribution of the anomalies’ inter-arrival times in the overall network is very Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 3: LTE traffic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' close to an exponential distribution of rate λ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='6 anoma- lies/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Supposing that the base stations are all identical, and given the splitting property of Poisson processes [25], we can conclude that the anomalies’ rate for each base station is λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='019 anomalies/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 4: Anomalies’ inter-arrival times distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' On another note, we report in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 5 an extract of the anomalies’ maintenance time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As one can see, this distribution is far from regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, we can see that a part of the anomalies are resolved almost instantaneously by the network itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' On the other hand, other anomalies require either remote or on-site interventions that take longer time (hours, days, and sometimes weeks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Modeling such a distri- bution is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, we recall that the results reported in Section III hold for any general maintenance time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In fact, all we need to characterize the expected utility not satisfied is the average maintenance duration, which puts into perspective the generality of our results and their practical usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, using the troubleshooting tickets data, we can conclude that the average maintenance time is 2 hours and 8 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' With the above results in mind, along with the periodic nature of users’ traffic as was illustrated in the previous section, we 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='8 _DL Traffic Volume(GB) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='4 0 24 72 96 120 144 168 Time(hours)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0040 Exponential distribution Anomalies inter-arrival distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0035 Probability density function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='0000 0 1000 2000 3000 4000 5000 Time (s)8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 5: Extract of the maintenance time distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' can conclude that the results of Theorem 2 can be used to find the expected utility not satisfied in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, knowing that the average traffic per hour for each base station is U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='55 GBs/hour, the operator can, then, conclude that the expected traffic demand not satisfied in the entire network is L ∞ = 660 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='13 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='019 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='13 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='55 ≃ 39 GBs/hour (35) All in all, we can conclude that the network loses on average around 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='13 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='019 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='13 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='8% of its traffic due to the various anomalies that may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This reliability score can be used by the operator to assess the network’s performance, with respect to its business objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Further planning and network upgrades can, then, take place if necessary, and a re-evaluation of the score can be done to conclude on the efficacy of the proposed upgrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Convergence Speed: Single Cell Thus far, we have been interested in verifying the key assumptions adopted in our theoretical analysis and character- izing the distributions needed to evaluate the theoretical limit of the expected demand not satisfied L , denoted by L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, investigating how fast we converge to this limit L∞ is of paramount interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, let L T = 1 T � T 0 U(t)W(t)dt (36) denote the average traffic demand not satisfied in the time interval [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Equivalently, as we have done in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (8), we can define the average demand not satisfied in the anomaly- repair cycles [0, n] as L n = 1 n �n j=1 � D[p] j D[p] j −Yj U(t)dt 1 n �n j=1(Xj + Yj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (37) With these definitions in mind, our aim is to find at what stage n∗ the relative error |L n−L∞| L n becomes smaller than 10% for n ≥ n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we first consider that the inter-arrival time between anomalies follows an exponential distribution with rate λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='019 hours−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As for the maintenance time, to facilitate our convergence speed study, we consider that the maintenance time is exponentially distributed with rate µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='47 hours−1, which is in accordance with the average maintenance time duration observed in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Note that the insights we are about to provide would still hold for other general maintenance time distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Finally, concerning the utility function, we suppose that the utility function is perfectly periodic and can be written as U(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='75 sin(2πt p ) + 3, t ≥ 0, (38) where p = 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The above function mimics very well the utility function illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We report our results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As can be seen, the loss function L n converges to L ∞, as Theorem 2 predicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, it takes on average n∗ = 350 anomaly-repair cycles for the relative error to become smaller than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given the average time of each cycle, we can conclude that the convergence time can be quite slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The reason behind this is that although the convergence of D[p] j to a uniform distribution happens exponentially fast, as shown in Proposition 1, the overall convergence of L n to L∞ reported in Theorem 2 is not exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, as can be seen in Appendix F, the convergence speed is inversely proportional to n and depends heavily on the correlation between the different quantities reported in Proposition 4 and the variance of the random variables involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, as we will show in the next section, this convergence speed becomes substantially faster when a large-scale network is considered, rather than an individual base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 6: Illustration of the convergence of the cellular network with one cell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Convergence Speed: Multiple Cells In practice, the network’s operator is interested in evaluating the expected utility loss of the whole network rather than that Maintenance time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='007 Probability density function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='000 0 200 400 600 800 1000 Time (s)Expected loss L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Loss En 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='20 Average Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='05 0 200 400 600 800 1000 Cycle number n9 of a particular cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, if the network is made of N cells, then we are interested in the speed of convergence of L T = 1 NT N � i=1 � T 0 U i(t)W i(t)dt, (39) to its limit L ∞, where U i(t) is the utility at time t of cell i and W i(t) is a binary random variable indicating if cell i is suffering from an anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To investigate this convergence speed, we adopt the same settings of the previous subsection and compare the multi-cell case to the single-cell one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Note that since anomalies may occur at different instants at the various cells, we report the average convergence speed in time units rather than anomaly-repair cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 7: Comparison of the convergence time in single and multi-cell scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As can be seen, the average convergence speed for the whole network’s average utility loss is around 41 hours, which is significantly faster than the single-cell one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In other words, by letting the network run for a couple of days, our evaluation of the expected utility not satisfied through the simple theoretical formula we provided in Theorem 2 would already be very accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given that networks are typically constituted of a large number of sub-systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', cells, routers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' ), we can conclude that this convergence speed is particularly appealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To understand this trend, we note that the large number of cells leads to a spatial averaging effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Particularly, as more cells are added, the variance reported in Appendix F will be reduced, thus greatly accelerating the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The question that remains is the following: how big does the network need to be to obtain such fast convergence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To answer that question, we simulate 100 different realizations of the network’s operation for a variety of network sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The results are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As can be observed, the average convergence time decreases with the number of cells, reaching as low as 1 week for a network size of 100 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Finally, we examine the impact of violating the assumption of perfect periodicity on the convergence speed of L T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do this, we compare the perfect periodicity case to the scenario where U(t) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (38) is corrupted by an Ornstein-Uhlenbeck Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 8: Evolution of the convergence time with the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' process B(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Specifically, B(t) satisfies the following stochas- tic differential equation: dB(t) = −θB(t)dt + σdR(t), (40) where R(t) is the Wiener process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By setting θ = 1 and σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='01, we plot the relative error |L T −L ∞| L T as a function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 9, the relative error between the expected loss and its limit calculated by our formula quickly converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In fact, the convergence speed of the two cases is similar, with both reaching the 10% mark at around 41 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This confirms the results of Theorem 3, which states that even with minor, random fluctuations from the perfect periodicity regime, our convergence results still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 9: Evolution of the relative error between the loss and its expected limit as a function of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Non-Stationarity Thus far, we have assumed in our analysis that the inter- arrival times and maintenance times of the anomalies follow stationary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' This means that the statistics of these distributions, such as the expected values E[X] and E[Y ], do 12 Average convergence time 11 10 6 8 7 0 200 400 600 800 1000 1200 1400 Number of cells0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='40 Relative error without an OU Process 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='35 Relative error with an OU Process 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='30 error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='20 lative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='15 Rel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='05 0 20000 40000 60000 80000 100000120000140000160000 Time(minutes)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='175 Expected loss L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Average Loss - one cell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='150 Average Loss - 660 cells 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='100 LOSS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='000 0 50000 100000 150000 200000 250000 300000 350000 400000 Time(minutes)10 not change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' However, it is important to verify the accuracy of this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we can use our real data to plot ∆ = E[Y ] E[X] + E[Y ] = 1 − Availability (41) to see if it remains constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Our attention is on ∆, as the theoretical limit L∞ depends on the aforementioned distributions through ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 10 demonstrates that ∆ has minimal fluctuations, causing minor violations of the stationar- ity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Nevertheless, these fluctuations have minimal impact on the convergence time, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The relative error |L T −L ∞| L T fluctuates minimally and remains close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 10: Variation of ∆ as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 11: Variation of the relative error as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this paper, we have considered the evaluation of the re- liability of large networked systems with respect to a periodic time-dependent utility function related to the system’s service performance over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Under the assumption of exponential anomalies’ inter-arrival times and general distributions of maintenance time duration, we have leveraged the periodicity of the utility function to derive the expected utility loss due to the system’s anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In these settings, we have shown that the expected utility loss converges in probability to a simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We have also extended our results to the case where the periodicity of the utility is slightly violated and the distribution of the inter-arrival times of anomalies is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We applied our analysis to a cellular network use case with real field data gathered from a deployed cellular network and demonstrated its usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Additionally, we verified our key assumptions using the available data and investigated the interplay between various network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Maatouk, F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Athena Scientific, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' [26] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Schatte, “On the asymptotic uniform distribution of sums reduced mod 1,” Mathematische Nachrichten, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 115, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' 275–281, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX A PROOF OF THEOREM 1 Our proof revolves around a Fourier analysis of the distri- bution of X[p] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To proceed in this direction, let us first note that by definition j � k=1 Xk = mp + X[p] j = p(m + X[p] j p ), (42) where m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By dividing both sides by 1 p, we can conclude that X[p] j = pR[1] j , (43) where R[1] j is the remainder of the Euclidean division of Rj = �j k=1 Xk p by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, to characterize the distribution of X[p] j , we can start by studying that of R[1] j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' With this in mind, the first step of our analysis in this direction consists of showing the existence of an order j∗ such that sup r∈[0,1] fR[1] j∗ (r) < ∞, (44) where fR[1] j∗ (r) is the probability density function of R[1] j∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' The above condition is essential to our Fourier analysis as will be seen shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To find j∗, we first note that thanks to the scaling property of the exponential distribution, R1 is exponentially distributed with rate λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Therefore, we have sup r∈[0,∞) fR1(r) = fR1(0) = λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (45) Given that fR1(r) is bounded, we can assert that the same holds for fR[1] 1 (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Consequently, we can conclude that j∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, let ˆg(n) = � ∞ 0 fR1(r)e−2πinrdr (46) denote the nth Fourier coefficient of the probability density function of R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Knowing that R1 is exponentially distributed, we obtain ˆg(n) = � ∞ 0 λpe−λpre−2πinrdr = λp λp + 2πin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (47) Given the above expression, we can conclude that |ˆg(n)| is decreasing with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Therefore, we have α = sup n∈N∗ |ˆg(n)| = |ˆg(1)| = λp � λ2p2 + 4π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (48) Given that α < 1 and that condition (44) holds, we can leverage [26, Theorem 1] to deduce that sup r∈[0,1] |fR[1] j (r) − 1| ≤ Cαj, (49) where C = α−2 ∞ � n=1 |ˆg(n)|2j∗ = ∞ � n=1 λ2p2 + 4π2 λ2p2 + 4π2n2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (50) Note that the convergence of C is due to the fact that the term n2 exists in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Lastly, given that X[p] j = pR[1] j , we can conclude that fX[p] j (x) = 1 pfR[1] j (x p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (51) With this in mind, and using the results of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (49), we can conclude our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX B PROOF OF LEMMA 2 Due to the independence between A and B, we can derive the probability density function fZ(z) for z ∈ [0, p] as follows fZ(z) = (fA ⊛p fB)(z) = � z 0 fA(x)fB(z − x)dx + � p z fA(x)fB(z + p − x)dx, (52) 12 where ⊛p indicates that the convolution is done modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we investigate the gap between the density function fZ(·) and the uniform distribution on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' |fZ(z) − 1 p| (a) = | � z 0 (fA(x) − 1 p)fB(z − x)dx + � p z (fA(x) − 1 p)fB(z + p − x)dx| (b) ≤ � z 0 |fA(x) − 1 p|fB(z − x)dx + � p z |fA(x) − 1 p|fB(z + p − x)dx (c) ≤ sup x∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='p] |fA(x) − 1 p|( � z 0 fB(z − x)dx + � p z fB(z + p − x)dx) (d) = sup x∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='p] |fA(x) − 1 p|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (53) where (a) is the result of writing 1 p as an integral over the density function fB(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (b) is deduced from the triangular inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (c) can be concluded from the definition of the supremum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' and (d) comes from the fact that fB(·) is a density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By taking the supremum on the left hand side, we can conclude the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX C PROOF OF PROPOSITION 2 To obtain our results, we leverage the Stolz–Ces`aro theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, let us define the sequences an = �n j=1 E[Ij] and bn = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' We have lim n→+∞ an+1 − an bn+1 − bn = lim n→+∞ E[In+1] 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (54) Let us now define I as follows I = EY,V [ � V V −Y U(t)dt], (55) where V is a RV uniformly distributed on [0, p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' In essence, I is the expectation of Ij if D[p] j is uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Next, we develop the expression above as follows I = � ∞ 0 � p 0 � v v−y U(t)1 pfY (y)dtdvdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (56) where fY (y) is the probability density function of the RVs Yj for j ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To further simplify the expression above, we proceed with a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Precisely, we let s = t−v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Consequently, we end up with I = � ∞ 0 � p 0 � 0 −y U(s + v)1 pfY (y)dsdvdy (a) = � ∞ 0 � 0 −y � p 0 U(s + v)1 pfY (y)dvdsdy (b) = � ∞ 0 � 0 −y � p 0 U(v)1 pfY (y)dvdsdy = E[Y ]1 p � p 0 U(v)dv = E[Y ]U, (57) where (a) is the result of changing the order of integration, and (b) is due to the periodicity of U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Now, let us write In+1 in a more convenient form as follows In+1 = � D[p] n+1 D[p] n+1−Yj U(t)dt = � [Dn+Xn+1] mod p+Yn+1 [Dn+Xn+1] mod p U(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (58) Then, we can proceed as done in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (57) to obtain E[In+1] = � ∞ 0 � 0 −y � p 0 U(s + y + v)fV (v)fY (y)dvdsdy, (59) where fV (v) is the probability density function of [Dn + Xn+1] mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By leveraging Lemma 2 and Proposition 1, we can conclude that sup v∈[0,p] |fV (v) − 1 p| ≤ Cαn p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (60) Consequently, we have fV (v) ≤ Cαn p + 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Knowing this, we can now investigate the difference between I and limn→+∞ E[In+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given all what was presented above, we obtain |E[In+1] − I| ≤ Cαn p E[Y ]U n→+∞ −−−−−→ 0, (61) given that E[Y ], U are finite and α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Finally, given that bn is a strictly monotone and divergent sequence, we can conclude from the Stolz–Ces`aro theorem that Sn converges to I for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX D PROOF OF PROPOSITION 3 To start our proof, we first note that by the definition of D[p] j reported in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (11) and the independence of Xi and Yi, we have fD[p] j ,D[p] j+k(zj, zj+k) = fD[p] j (zj)fD[p] k ([zj+k − zj] mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (62) Next, thanks to Lemma 2, and by leveraging the triangular inequality, we can conclude that sup z∈[0,p] |fD[p] j (z) − 1 p| ≤ sup x∈[0,p] |fX[p] 1 (x) − 1 p| ≤ sup x∈[0,p] fX[p] 1 (x) + 1 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (63) 13 We recall that X1 is an exponential random variable of rate λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Hence, the supremum of its probability density function is equal to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Given that the probability density function of X1 is bounded, we can conclude that there exists M such that fX[p] 1 (x) ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' With this in mind, and by leveraging the triangular inequality, we can deduce that |fD[p] j (·)fD[p] k (·) − 1 p2 | = |(fD[p] j (·) − 1 p)fD[p] k (·)+ 1 p(fD[p] k (·) − 1 p)| ≤ (M + 1 p)Cαj p + 1 p Cαk p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (64) Finally, by letting C′ = M + 1 p, we can conclude the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX E PROOF OF PROPOSITION 4 By definition, we have Cov[Ij, Ij+k] = E[IjIj+k] − E[Ij]E[Ij+k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (65) Let us first investigate the term E[IjIj+k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' As was done in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (58), we can rewrite E[IjIj+k] as follows E[IjIj+k] =EYj,Yj+k[ � p 0 � p 0 � zj+Yj zj U(t)dt × � zj+k+Yj+k zj+k U(t′)dt′fZ[p] j ,Z[p] j+k(zj, zj+k)dzjdzj+k], (66) where Z[p] j and Z[p] j+k denote [Dj +Xj+1] mod p and [Dj+k + Xj+k+1] mod p respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' By leveraging Lemma 2 and Proposition 3, we obtain E[IjIj+k] ≤EYj,Yj+k[ � p 0 � p 0 � zj+Yj zj U(t)dt × � zj+k+Yj+k zj+k U(t′)dt′( 1 p2 + C′C(αj p + αk p )dzjdzj+k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (67) Given that the RVs Yj and Yj+k are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=', we obtain E[IjIj+k] ≤ I 2(1 + pC′C(αj + αk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (68) Next, we need to investigate the second term E[Ij]E[Ij+k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we recall from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (61) that |E[Ij] − I| ≤ Cαj−1 p I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (69) With this in mind, we can obtain the results of the proposition, which concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' APPENDIX F PROOF OF THEOREM 2 From the expression in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (8), we can rewrite L as L = lim n→+∞ 1 n �n j=1 Ij 1 n �n j=1(Xj + Yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (70) Let us start by investigating the numerator Nn = 1 n �n j=1 Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To that end, let us consider the event Pr(|Nn − I| ≥ ϵ) (a) ≤ Pr(|Nn − Sn| ≥ ϵ 2 ∪ |Sn − I| ≥ ϵ 2) (b) ≤ Pr(|Nn − Sn| ≥ ϵ 2) + Pr(|Sn − I| ≥ ϵ 2), (71) where (a) and (b) are the results of the triangular inequality and the union bound, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Note that the term |Sn −I| is deterministic, and from Proposition 2, we know that this term tends to zero when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' Therefore, what remains is to examine the first term Pr(|Nn − Sn| ≥ ϵ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' To do so, we leverage Chebyshev’s inequality to obtain the following upperbound Pr(|Nn − Sn| ≥ ϵ 2) ≤ 4Var[Nn] ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E5T4oBgHgl3EQfag-J/content/2301.05589v1.pdf'} +page_content=' (72) However, we know that Var[Nn] = n � j=1 Var[Ij] + 2 n−1 � j=1 � j 0.7 and RT) yielded a +sub-stoichiometric Ga2O3-x film with increased absorptance (Fig. S4). +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +0 +20 +40 +60 +80 +100 +200 +250 +300 +350 +0 +20 +40 +60 +80 +100 +b +R and T (%) +Wavelength (nm) +Ga2O3 thin films deposited on: + f-quartz, RT + f-quartz, 700 +oC + c-sapphire, RT + c-sapphire, 800 +oC +a +Absorbance (%) +Wavelength (nm) +Ga2O3 thin films deposited on: + f-quartz, RT + f-quartz, 700 +oC + c-sapphire, RT + c-sapphire, 800 +oC + + +a +Ga LMM +Oxygen +Gallium +1000 +LMM +Aluminum + LMM +800- +lodine'(probe beam) +Ga +Intensity (a.u.) +Ga +channel + LMM +600 +Ga +ToF +400 +3 +Ga +Ga +200- +0- +800 +700 +600 +500 +400 +300 +200 +100 +0 +0 +400 +800 +1200 +1600 +Binding energy, eV +Energy channel12 + +Fig. 6. (a) Transmittance (T) and specular reflectance (R) spectra (solid and dashed lines, respectively) +in the range 200–2000 nm for Ga2O3 films deposited on f-quartz and c-sapphire substrates at different +temperatures; (b) absorbance spectra for the same films at short wavelengths. The films were deposited +at Iproc/Imet ≈ 0.5. + +The sharp increase in absorptance below 300 nm is due to the onset of the fundamental absorption of +Ga2O3. The optical band gap (Eg) of ∼5.0–5.1 eV for both the crystalline and X-ray amorphous films +was estimated by a Tauc plot assuming direct band gap transitions (Fig. S6(a)), and is in good agreement +with values obtained by applying the same procedure to magnetron sputtered films [29,34,35]. The +changes in the phase and structure of the film can often be indirectly evaluated by observing a +fundamental absorption edge shift. A slight increase in Eg can be observed for both types of substrates +when the deposition temperature is increased (Fig. S6(b)). The band gap of Ga2O3 can be affected by +deviation from stoichiometry, i.e., excess gallium or a deficiency of oxygen [1], or by anisotropy of +crystalline orientation [36]. + +Conclusions +We have demonstrated the feasibility of depositing stoichiometric Ga2O3 thin films by reactive pulsed- +DCMS from a liquid Ga metal target. The practical problem with metallic Ga, in which it contracts into +islands leaving parts of the target container uncovered and exposed to sputtering, was solved by pre- +coating the stainless steel container by a wettability-promoting layer of carbon. No difficulties were +encountered when conducting the sputtering process. The deposition rates of up to 37 nm/min at room +temperature on f-quartz and 5 nm/min at 800°C on c-sapphire were higher than the data in the literature +for RF sputtering of 22 and 1 nm/min, respectively. In line with the results in the literature for RF +sputtering, the deposition rate was found to depend on the temperature of the substrate, with the atom’s +probability of sticking to the substrate decreasing significantly with temperature, and the factor limiting +the deposition rate shifting gradually from the rate of arrival to the probability of sticking of the atoms +to the surface as the substrate temperature was increased from RT to 800°C. We believe that there is +still room for a further increase in the deposition rate, for instance by increasing the sputtering power +and optimizing the process parameters. +At substrate temperatures ranging from RT to 500°C, a feature of the X-ray diffractograms is observed +that may be indicative of the presence of small crystallites of β-Ga2O3 in the predominantly amorphous +films. Pronounced XRD maxima start to appear at 500°C. Under the same sputtering conditions, the +crystal structure depends on the substrate. For both f-quartz and c-sapphire, this is the β-Ga2O3 phase, +but whereas differently oriented crystallites are detected for f-quartz, epitaxial films with a single + +13 + +orientation grow on c-sapphire above 600°C. These films are dense, with the surface becoming +smoother as the substrate temperature increases. The dependence of the structure on the temperature is +consistent with the data in the literature for other methods of deposition. Our results also indicate that +post-annealing may be an alternative to heating the substrate during deposition to obtain crystalline +films. +The films exhibit a low absorbance of less than 1% in the visible range. There is a slight blue-shift of +the fundamental absorption edge with an increase in the substrate temperature, apparently because there +are fewer defects in the films. No differences in the optical band gap (∼5.0–5.1 eV, estimated from +Tauc plots) were observed between the films on f-quartz and c-sapphire. The optical properties are +consistent with the data in the literature for other deposition methods. + +Declaration of competing interest +The authors declare the following financial interests/personal relationships which may be considered as +potential competing interests: A. Azens, M. Zubkins, E. Butanovs, and J. Purans (applicant to the +Institute of Solid State Physics University of Latvia) have a national patent pending (No. +LVP2021000105) and a European patent (No. EP22195507.3) pending. + +Acknowledgements +This study was financially supported via ERDF project No. 1.1.1.1/20/A/057 “Functional ultrawide +bandgap gallium oxide and zinc gallate thin films and novel deposition technologies”. The Institute of +Solid State Physics, University of Latvia, as a Center of Excellence, has received funding from the +European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017- +TeamingPhase2 under grant agreement No. 739508, project CAMART². Support for the ERDA +measurements from the Ion Technology Centre (ITC) at Uppsala University is gratefully acknowledged. + +References +[1] Y. Yuan, W. Hao, W. Mu, Z. Wang, X. Chen, Q. Liu, G. Xu, C. Wang, H. Zhou, Y. Zou, X. +Zhao, Z. Jia, J. Ye, J. Zhang, S. Long, X. Tao, R. Zhang, and Y. Hao, Fundam. Res. 1, 697 +(2021). +[2] J. Shi, J. Zhang, L. Yang, M. Qu, D. C. Qi, and K. H. Zhang, Adv. Mater. 33, 2006230 (2021). + +14 + +[3] S. J. Pearton, J. Yang, P. H. Cary, F. Ren, J. Kim, M. J. Tadjer, and M. A. Mastro, Appl. Phys. +Rev. 5, 011301 (2018). +[4] M. Razeghi, J. Park, R. McClintock, D. Pavlidis, F. H. Teherani, D. J. Rogers, B. A. Magill, G. +A. Khodaparast, Y. Xu, J. Wu, and V. P. Dravid, Proc. SPIE 10533, 105330R (2018). +[5] E. Ahmadi and Y. Oshima, J. Appl. Phys. 126, 160901 (2019). +[6] M. Bosi, P. Mazzolini, L. Seravalli, and R. Fornari, J. Mater. Chem. C 8, 10975 (2020). +[7] D. Yang, B. Kim, T. H. Eom, Y. Park, and H. W. Jang, Electron. Mater. 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Li, J. Wang, D. Wang, S. Gao, Q. Yu, and H. Li, Cryst. Eng. Comm. 20, 133 +(2018). +[36] N. Ueda, H. Hosono, R. Waseda, and H. Kawazoe, Appl. Phys. Lett. 71, 933 (1997). + diff --git a/U9E2T4oBgHgl3EQfXgeS/content/tmp_files/load_file.txt b/U9E2T4oBgHgl3EQfXgeS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..416a541b4a9e1df56e5a574fb5fcd5736ff3b70f --- /dev/null +++ b/U9E2T4oBgHgl3EQfXgeS/content/tmp_files/load_file.txt @@ -0,0 +1,693 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf,len=692 +page_content='1 Deposition of Ga2O3 thin films by liquid metal target sputtering Martins Zubkins1*, Viktors Vibornijs1, Edvards Strods1, Edgars Butanovs1, Liga Bikse1, Mikael Ottosson2, Anders Hallén3, Jevgenijs Gabrusenoks4, Juris Purans1, Andris Azens4 1Institute of Solid State Physics, University of Latvia, Kengaraga 8, LV-1063 Riga, Latvia 2Department of Chemistry, Ångström Laboratory, Lägerhyddsvägen 1, SE 75120 Uppsala, Sweden 3KTH Royal Institute of Technology, School of EECS, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Box Electrum 229, SE 16440, Kista- Stockholm, Sweden 4AGL Technologies SIA, Smerla 3, LV-1006 Riga, Latvia Corresponding author: martins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='zubkins@cfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='lv Abstract This paper reports on the deposition of amorphous and crystalline thin films of Ga2O3 by reactive pulsed direct current magnetron sputtering from a liquid gallium target onto fused (f-) quartz and c plane (c-) sapphire substrates, where the temperature of the substrate is varied from room temperature (RT) to 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The deposition rate (up to 37 nm/min at RT on f-quartz and 5 nm/min at 800°C on c-sapphire) is two to five times higher than the data given in the literature for radio frequency sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Deposited onto unheated substrates, the films are X-ray amorphous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Well-defined X-ray diffraction peaks of β- Ga2O3 start to appear at a substrate temperature of 500°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Films grown on c-sapphire at temperatures above 600°C are epitaxial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' However, the high rocking curve full width at half maximum values of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5° are indicative of the presence of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A dense and void-free microstructure is observed in electron microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Composition analysis show stoichiometry close to Ga2O3 and no traces of impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The optical properties of low absorptance (<1%) in the visible range and an optical band gap of approximately 5 eV are consistent with the data in the literature for Ga2O3 films produced by other deposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Key words: gallium oxide, thin films, magnetron sputtering, liquid metal target 2 Introduction Gallium oxide (Ga2O3) has attracted a lot of attention as an ultra-wide bandgap semiconductor [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' It can crystallize in five different phases, of which the most thermodynamically stable and technologically relevant is the monoclinic β phase [3,4], although there is also increasing interest in the other phases [5–7] and amorphous coatings [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The properties of greatest technological importance are transparency in the UV part of the spectrum due to its wide band gap of approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='9 eV, high carrier mobility (up to 200 cm2/Vs), high breakdown field (8 MV/cm), and high thermal and chemical stability [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Practical applications include solar-blind photodetectors, high-voltage transistors, high- power Schottky diodes, high-temperature chemical sensors, and transparent electrical conductors in optoelectronic devices [11–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The known methods of Ga2O3 thin film deposition include metal-organic chemical vapor deposition [16], plasma-enhanced chemical vapor deposition [17], molecular beam epitaxy [18], atomic layer deposition [19], pulsed laser deposition [20], e-beam evaporation [9], and radio-frequency (RF) magnetron sputtering (MS) from ceramic Ga2O3 targets [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' While MS is generally known for high- quality coatings produced under well controlled and reproducible conditions [22–24], the drawback of the RF mode of sputtering is its impractically low sputtering rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Moreover, this technique has scalability issues, as RF power supplies have a limited maximum power (on the order of a few kW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Although the usual source of the sputtered particles is a solid target, it has been demonstrated that gallium nitride films can also be deposited by reactive MS from a liquid gallium target [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' In this study, we used reactive pulsed DCMS from a liquid gallium target for the deposition of Ga2O3 thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The goals of the study were twofold: (i) to establish whether a robust sputtering process could be set up and optimized with high deposition rates compared to RF sputtering from ceramic targets, and (ii) to verify that the film composition, structure and optical properties were those of stoichiometric Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' We found that highly transparent amorphous and crystalline β-Ga2O3 coatings could be created at deposition rates two to five times higher than those typical for RF sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Experimental details Ga2O3 thin films were deposited by reactive pulsed DCMS from a gallium metal target in an Ar/O2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' As the melting temperature of gallium is 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='8°C, the target was in a liquid state during sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' To prevent the melted metal from flowing off the magnetron surface, a box-shaped stainless steel target container was used with 3-mm-high walls and a machined recess in the base plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S1 in the supplementary material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' To prevent the liquid gallium from contracting into islands and leaving parts of the container base plate exposed to sputtering, the container surface was pre-coated with a wettability-promoting layer of carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The target was prepared by melting metallic gallium 3 pellets (PI-KEM, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='999%) and then cooling the container to allow the target to solidify prior to being placed onto the magnetron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Film deposition was performed using a G500M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2 PVD coater equipped with a planar balanced magnetron (Sidrabe Vacuum, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The magnetron was placed under the substrate holder at a distance of 11 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The same cooling system was used as for ordinary solid targets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', the magnetron was cooled by a flow of water (≈2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5 l/min) at 20°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The base pressure was ≤7×10-4 Pa in a turbomolecular pumped chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The residual gas composition at the base pressure of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='4×10-4 Pa is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S2, and consisted of approximately 92% (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='9×10-4 Pa) H2O, 2% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='3×10-5 Pa) N2, 2% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2×10-5 Pa) CO2, 1% (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='0×10-5 Pa) O2, 1% (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='0×10-6 Pa) H2, and total contamination of other gases <2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The process pressure was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='4 Pa by feeding 30 sccm (standard cubic centimeters per minute) of argon and partly closing the throttle valve between the chamber and the pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Sputtering was carried out in pulsed DC (80 kHz, trev = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5 µs) mode in an O2 (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='999% pure)/Ar (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='9999% pure) atmosphere at a power of 150 W (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='3 W/cm2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The amount of residual gases in the sputtering atmosphere was less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Fused (f-) quartz (SPI Supplies) and c-plane (c-) sapphire (Biotain Crystal) substrates were used, and the substrate temperature was varied between room temperature (RT) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', no intentional heating provided) and 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A few selected samples deposited at RT and 500°C were post-annealed for 5 h in air at 800°C and 700°C, respectively, to determine whether there were differences in the properties of films heated during or after deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Control of the sputtering process was achieved by optical emission spectroscopy (OES), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', the oxygen flow was controlled by a feedback loop keeping the ratio Iproc/Imet constant, where Iproc and Imet are the intensities of the gallium 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2 nm emission line in the process and metal modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' An example of a typical deposition cycle is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The process was started in the metal mode (feeding argon only) with the shutter closed in front of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The target was conditioned until Imet became constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Oxygen was added and the flow was increased manually to reduce the intensity of the emission line to the desired value, Iproc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The process was locked into PID (proportional-integral- derivative) control mode, to keep the Iproc/Imet ratio constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The shutter was opened, and the film was deposited onto the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' In the specific example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 1(b), the oxygen flow during the deposition was approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='8 sccm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The reason for using the ratio rather than Iproc was to achieve better process reproducibility over a long period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' While a few consecutive deposition cycles could be successfully carried out by using the absolute value of Iproc, taking into account Imet compensates for any changes in the light collection efficiency due to instrumental factors, for example the protective window of the optical channel gradually becoming coated over the course of many cycles, or minor changes in the interior geometry of the chamber introduced by service and maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' This makes the ratio Iproc/Imet a more robust parameter than Iproc alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 4 The film thickness was measured using a stylus profilometer (Veeco Dektak 150) and ellipsometry (Woollam RC2-XI), and both techniques gave similar values for the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The thickness of all films was in the range 60–780 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The crystallographic structure of the films was examined by X-ray diffraction (XRD), using a Rigaku MiniFlex600 X-ray powder diffractometer with Bragg-Brentano θ-2θ geometry and a 600 W Cu anode (Cu Kα radiation, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5406 Å) X-ray tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The pole figures and the rocking curves were measured with a Philips MRD Pro Diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The system was set up with copper Cu Kα radiation using point focus and a capillary lens, with a beam size of 2 × 2 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='18 parallel plate collimator and graphite monochromator were used as primary and secondary optics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The pole figures for the (– 401) plane were measured for PSI = 0–80°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The optical transmittance and reflectance of the films in the range 200–2000 nm were measured by an Agilent Cary 7000 spectrophotometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The sample was placed at an angle of 6° against the incident beam of P-polarized light, and the detector was placed at an angle of 180° behind the sample to measure the transmittance and at 12° in front of the sample to measure the specular reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' X-ray photoelectron spectrometry (XPS) measurements were performed by an ESCALAB 250Xi (ThermoFisher) instrument to determine the film composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' An Al Kα X-ray tube with energy 1486 eV was used as an excitation source, the size of the analyzed sample area was 650 × 100 μm, and the angle between the analyzer and the sample surface was 90°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' An electron gun was used to perform charge compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The base pressure during spectra acquisition was better than 10-5 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Prior to analysis, the samples were sputter-cleaned for 60 s with an argon gas cluster ion beam (atom cluster size 150, energy 4000 eV, area 2 × 2 mm) at an incidence angle of 30° with respect to the normal of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Elastic recoil detection analysis (ERDA) utilizing 36 MeV 127I incident ions and time-of-flight detection was performed to determine the chemical purity of the films [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The surfaces of the films were characterized by scanning electron microscopy (SEM), using a Thermo Scientific Helios 5 UX dual-beam microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Cross-section studies were performed using a transmission electron microscope (TEM, Tecnai G20, FEI) operating at 200 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Lamellae were prepared by a focused ion beam (FIB) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' To protect the films during preparation of the lamellae, the surface was coated with a 30-nm-thick Au layer and a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5-µm-thick Pt layer before FIB exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (a) Schematic diagram of the target preparation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (b) example of control of the deposition process by optical emission spectroscopy, where the intensity of the gallium 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2 nm emission line was controlled by a PID loop to adjust the oxygen flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The cycle was initiated by sputtering in argon (line intensity Imet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Oxygen was added and the flow increased until the process line intensity Iproc reached the Iproc/Imet level chosen for deposition, and was then kept at Iproc/Imet by the PID for film deposition onto the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Results and discussion The deposition rates as function of Iproc/Imet for the films deposited on f-quartz and c-sapphire for varying substrate temperatures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2, where larger values of Iproc/Imet correspond to less oxygen in the sputtering atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For the data points shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2, the films deposited at rates of up to 37 nm/min (Iproc/Imet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='69) were highly transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Higher Iproc/Imet values (and hence higher rates) yielded apparently underoxidized (optically absorbing) films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' a Stainless steel container Coating with carbon Container filled with liquid gallium b OEs line intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=') met Deposition proc 0 300 600 900 1200 1500 Time (s)6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Deposition rate of Ga2O3 film as a function of normalized Ga 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2 nm emission line intensity (Iproc/Imet) for OES-controlled sputtering from metallic gallium target in Ar/O2 atmosphere for substrates at room temperature (squares), 600°C (triangles), 700°C (diamonds), 800°C (circles) onto f-quartz (open symbols) or c-sapphire (solid symbols) substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Larger values of Iproc/Imet correspond to less oxygen in the sputtering atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The lines between the deposition rate points on quartz and sapphire substrates at room temperature (black line) and on sapphire substrates at elevated temperatures (green lines) are drawn to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The orange symbols represent underoxidized (optically absorbing) films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The main takeaways from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2 are as follows: (i) For unheated substrates, the trend of the deposition rate with Iproc/Imet is what one would expect for a typical oxide coating deposited in a reactive process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', the rate increases upon reducing the oxygen content in the sputtering atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The linearity between the deposition rate and Iproc/Imet suggests that (a) the amount of light emitted by the sputtered atoms and detected by OES is directly proportional to the number of sputtered atoms and hence to the number of atoms arriving at the substrate, and (b) the higher order effects that could potentially disrupt the linearity (for instance, the optical excitation yield changing with Iproc/Imet) are neglectable within the studied range of deposition conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (ii) Within the present statistics, there is no difference in the deposition rates for unheated f-quartz and c-sapphire substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (iii) For a given Iproc/Imet, the deposition rate decreases with the substrate temperature, and the decrease is substrate-dependent: while the trend is the same for both substrates, the effect is stronger for c- sapphire than for f-quartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Since the sputtering conditions are unchanged and the rate of arrival of atoms at the substrate is the same for both substrates at any temperature, the apparent reason for 50 RT 口 600°C 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Deposition rate (nm/min) 700°C 800°℃ 30 20 10 口 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='9 Iproc/ Imet7 the decrease in deposition rate is the decrease in an atom’s probability of sticking to the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The exact reason for the dependence of the rate on the substrate is a matter for future study, but we note that one difference detected in this study is that the films grown on c-sapphire are epitaxial, while the films on f-quartz consist of differently oriented crystallites (see the XRD data below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The remarkably strong effect of the substrate temperature is consistent with the data in the literature for RF sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Ga2O3 films have been RF sputter-deposited on 800°C c-sapphire substrates at 1 nm/min [27–30], in sharp contrast with the rates of 15–22 nm/min reported for films deposited onto unheated glass substrates [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A decrease in the deposition rate from approximately 20 nm/min to less than 10 nm/min by increasing the silicon substrate temperature from 100°C to 600°C was reported in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (iv) The slope of the deposition rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Iproc/Imet changes with temperature, from its steepest value at a substrate temperature of RT (black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2), to less steep at 600°C, to practically independent of Iproc/Imet at 700°C and 800°C (green lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Again, since the sputtering conditions are unchanged with temperature, the change in slope reflects a shift in the factor limiting the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' At RT, the deposition rate is limited by the rate of arrival of the atoms at the substrate, and a higher rate of arrival at larger Iproc/Imet results in a higher deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' As the substrate temperature increases, the increase in the rate of arrival of the atoms does not result in an equally high increase (at 600°C), or any increase at all (at 700°C and 800°C) in the deposition rate with Iproc/Imet, suggesting that the factor limiting the deposition rate is gradually shifting from the rate of arrival to the rate at which the atoms stick to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (v) There is a sharp drop in the deposition rate to virtually zero once Iproc/Imet exceeds a certain value (approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='7) for a substrate temperature of 800°C for c-sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Since the sputtering yield, and hence the arrival rate of gallium atoms at the substrate, increases with Iproc/Imet irrespective of the substrate temperature, this drop in the deposition rate is indicative of a certain minimum oxidation rate required for a film to be formed on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' If the oxidation rate is below this minimum level, the arriving gallium atoms appear to be re-evaporated from the surface at a rate exceeding their rate of arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (vi) The deposition rate achieved for optically transparent films (up to 37 nm/min at RT on f-quartz and 5 nm/min at 800°C on c-sapphire) is higher than the data in the literature for RF sputtering (22 and 1 nm/min, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' In addition to the reduced process time, another benefit from the higher deposition rate is the higher purity of the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The main source of contamination for sputter deposited films (in the context of gallium oxide, this is often referred to as unintentional doping [1]) are the residual gas molecules present in the chamber at the base pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For a constant arrival rate of these molecules at the substrate at a given base pressure, the higher arrival rate of gallium atoms means fewer contaminants per incoming gallium atom in the growing film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 8 XRD patterns for the as-deposited and post-annealed films are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films deposited on f-quartz and c-sapphire substrates at RT were X-ray amorphous (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(a,b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' however, the films on f- quartz showed a broad, relatively weak yet detectable XRD band between 29° and 39° (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S3 in the Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Similar XRD features reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' [29,31] were attributed to the onset of crystallization of β-Ga2O3, suggesting that crystallites of very small size may be present even in the predominantly amorphous films deposited at RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For the films deposited on f-quartz, clearly detectable crystallization starts between 500°C and 600°C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The maximum at ≈30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5° can be ascribed to either the (400) or (−401) plane of the β-Ga2O3 phase, according to ICDD card 01-087-1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' When the deposition temperature is raised to 700°C and above, the maximum becomes slightly asymmetric (with a shoulder towards smaller angles), and other low-intensity maxima (−201) and (002) appear at 19° and 32°, respectively, indicating a slight change in the texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The origin of the asymmetric shape is most likely a superposition of signals coming from several planes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', (400), (−401), or (110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The X-ray amorphous films deposited at RT can be crystallised by post-annealing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(a));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' however, the texture of these films differs from that of those deposited at high (≥ 600°C) temperatures, in that the (002) maximum in the annealed films is more intense and comparable to the maximum at ≈30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5°, while the (−201) maximum is not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' X-ray diffractograms of as-deposited and post-annealed Ga2O3 films on (a) f-quartz and (b) c- sapphire substrates at different temperatures, including the signals from the substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The angles of the Bragg peaks for monoclinic β-Ga2O3 (ICDD card 01-087-1901) are shown by vertical red lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The a b (-201) XRD intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=') (-603) XRD intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=") RT, annealed at 800 °c 500 °C, annealed at 700 'C RT, 800 annealed 700°0 at 800 °C 600°C 800°℃ 500°C 700°℃ 600°C- RT 500°℃ f-Sio, RT c-Al,O, B-Ga." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='O,: Card 01-087-1901 β-Ga,O,: Card 01-087-1901 10 20 30 40 50 60 70 10 20 30 40 50 60 70 2Theta (degrees) 2Theta (degrees) c d Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=') Aw=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='9° 800°C 区 Aw=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5° 700°℃ Aw=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5° 600°C 6 8 10 12 14 600°C 700°C 800°C w (degrees)9 maximum (006) for the c-sapphire substrate is denoted by *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (c) Pole figures for the (−401) plane and (d) ω scans of (–201) reflections are shown for films deposited on c-sapphire at substrate temperatures of 600°C, 700°C, and 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films were deposited at Iproc/Imet ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For the films deposited on c-sapphire, the onset of crystallization is similar to those deposited on f- quartz, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', clearly detectable XRD maxima for the β-Ga2O3 phase appear at substrate temperatures of 500°C and higher (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The pole figures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(c) show a six-fold symmetry indicative of epitaxial growth of the films deposited at 600°C, 700°C, and 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' However, the rocking curve full width at half maximum (FWHM) values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 3(d) of approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5° suggest that defects and twins are present in the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' No clear correlation was observed between the FWHM values and the deposition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Surface morphology and cross-sectional images of the crystalline β-Ga2O3 films obtained by electron microscopy are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The surface of the film deposited on f-quartz at 700°C exhibits grainy features with sharp edges, random shapes, and an average size of less than 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The surface becomes visibly smoother when the films are grown at 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films grown on c-sapphire show a smooth and void-free surface morphology, with some flat, randomly shaped features still present in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Dense structures are in evidence in the cross-sectional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The β-Ga2O3 films prepared at 700°C on both f-quartz and c-sapphire are highly crystalline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' however, while the grains in the film on f-quartz are slightly misoriented, a specific crystallographic direction is clearly visible in the film on c-sapphire, indicating an oriented growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' In this case, the interplanar distance was measured to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='7 Å, which matches well with the [−201] growth direction of β-Ga2O3 on a (0001) sapphire substrate [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' SEM images of the β-Ga2O3 films on (a, b) f-quartz and (e, f) c-sapphire prepared at (a, e) 700°C and (b, f) 800°C, and of the films’ cross-sectional lamellae on (c) f-quartz and (g) c-sapphire prepared a C β-Ga2O3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5μm T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='-700°C T=800°C 100nm f-quartz e f g 2011 β-Ga2O3 T=700°C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5μm T,=800°C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5μm 100nm 5hm C-sapphire10 at 700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' HR-TEM images of the cross-sections (lamellae) of the β-Ga2O3 films, prepared at 700°C, on (d) f-quartz and (h) c-sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The insets in (h) show the direction of growth and the measured lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films were deposited at Iproc/Imet ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The composition of the films was verified with XPS and ERDA measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A survey scan of Ga2O3 film deposited on f-quartz at RT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 5(a)) indicated the presence of Ga and O only, with the Ga/O atom ratio closely matching that of the Ga2O3 compound, and no peaks of possible contaminants were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Similar XPS results were also obtained for films prepared at higher temperatures on the f- quartz and c-sapphire substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 5(b) shows the ERDA results for a sample grown on c-sapphire at 700 ºC and Iproc/Imet ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The figure displays the counts for coincidently detected particles at the energy (x-axis) and time-of-flight (y-axis) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For practical reasons, the y-axis is inverted, meaning that ions with low mass, such as oxygen, will have the longest flight times in the ERDA plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The particles ejected from the surface of the film will have the highest energy, while particles coming from deeper layers will have lower energy, due to the energy loss in passing through the film, and the energy of the incident primary iodine ions will also be reduced, since they had to penetrate deeper into the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Since the energy is proportional to the square of the inverse flight time, plotting the data in this manner produces square, root-shaped traces for the different elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The particles emanating from the surface are seen in the upper right-hand side of each trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Only four elements are shown in the figure: oxygen, aluminum, gallium and iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The iodine counts build up in the spectra during the measurement, since previously incident iodine projectiles become recoils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The iodine therefore does not originate from the as-deposited films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Both O and Ga are present at the surface, while the Al trace starts deeper in the sample, since it originates from the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The ratio Ga/O suggests a 1–2 atomic% oxygen deficiency in the film, although this is within the range of potential calibration errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The film thickness is estimated at about 200 nm, which is in good agreement with the results from the profilometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (a) XPS survey scan of Ga2O3 film deposited on f-quartz at room temperature, showing the elemental composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (b) ERDA spectrum of β-Ga2O3 films deposited on c-sapphire at 700°C for Iproc/Imet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 6(a) shows the transmittance and specular reflectance spectra of selected samples, as-deposited Ga2O3 films on f-quartz and c-sapphire substrates, for wavelengths from 200 to 2000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The average transmittance is from 80% to 85%, and is mainly limited by the reflectance of ~15–20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The low amplitude of the interference fringes for the films on c-sapphire is most likely to be due to the apparently similar refractive indices of the film and the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The absorptance spectra of the films in the narrow wavelength range (200–350 nm) can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A slight blue-shift with the deposition temperature is observed for both types of substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The absorptance spectra of other films produced in this study can be found in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S4 and S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The absorptance in the visible range stays low (< 1%) and is typical for stoichiometric Ga2O3 films for all samples deposited at Iproc/Imet ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' An attempt to increase the deposition rate (more specifically, depositing the films at Iproc/Imet > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='7 and RT) yielded a sub-stoichiometric Ga2O3-x film with increased absorptance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=" 200 400 600 800 1000 1200 1400 1600 1800 2000 0 20 40 60 80 100 200 250 300 350 0 20 40 60 80 100 b R and T (%) Wavelength (nm) Ga2O3 thin films deposited on: f-quartz, RT f-quartz, 700 oC c-sapphire, RT c-sapphire, 800 oC a Absorbance (%) Wavelength (nm) Ga2O3 thin films deposited on: f-quartz, RT f-quartz, 700 oC c-sapphire, RT c-sapphire, 800 oC a Ga LMM Oxygen Gallium 1000 LMM Aluminum LMM 800- lodine'(probe beam) Ga Intensity (a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=') Ga channel LMM 600 Ga ToF 400 3 Ga Ga 200- 0- 800 700 600 500 400 300 200 100 0 0 400 800 1200 1600 Binding energy, eV Energy channel12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (a) Transmittance (T) and specular reflectance (R) spectra (solid and dashed lines, respectively) in the range 200–2000 nm for Ga2O3 films deposited on f-quartz and c-sapphire substrates at different temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' (b) absorbance spectra for the same films at short wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films were deposited at Iproc/Imet ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The sharp increase in absorptance below 300 nm is due to the onset of the fundamental absorption of Ga2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The optical band gap (Eg) of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='0–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='1 eV for both the crystalline and X-ray amorphous films was estimated by a Tauc plot assuming direct band gap transitions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S6(a)), and is in good agreement with values obtained by applying the same procedure to magnetron sputtered films [29,34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The changes in the phase and structure of the film can often be indirectly evaluated by observing a fundamental absorption edge shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' A slight increase in Eg can be observed for both types of substrates when the deposition temperature is increased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' S6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The band gap of Ga2O3 can be affected by deviation from stoichiometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=', excess gallium or a deficiency of oxygen [1], or by anisotropy of crystalline orientation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Conclusions We have demonstrated the feasibility of depositing stoichiometric Ga2O3 thin films by reactive pulsed- DCMS from a liquid Ga metal target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The practical problem with metallic Ga, in which it contracts into islands leaving parts of the target container uncovered and exposed to sputtering, was solved by pre- coating the stainless steel container by a wettability-promoting layer of carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' No difficulties were encountered when conducting the sputtering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The deposition rates of up to 37 nm/min at room temperature on f-quartz and 5 nm/min at 800°C on c-sapphire were higher than the data in the literature for RF sputtering of 22 and 1 nm/min, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' In line with the results in the literature for RF sputtering, the deposition rate was found to depend on the temperature of the substrate, with the atom’s probability of sticking to the substrate decreasing significantly with temperature, and the factor limiting the deposition rate shifting gradually from the rate of arrival to the probability of sticking of the atoms to the surface as the substrate temperature was increased from RT to 800°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' We believe that there is still room for a further increase in the deposition rate, for instance by increasing the sputtering power and optimizing the process parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' At substrate temperatures ranging from RT to 500°C, a feature of the X-ray diffractograms is observed that may be indicative of the presence of small crystallites of β-Ga2O3 in the predominantly amorphous films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Pronounced XRD maxima start to appear at 500°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Under the same sputtering conditions, the crystal structure depends on the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' For both f-quartz and c-sapphire, this is the β-Ga2O3 phase, but whereas differently oriented crystallites are detected for f-quartz, epitaxial films with a single 13 orientation grow on c-sapphire above 600°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' These films are dense, with the surface becoming smoother as the substrate temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The dependence of the structure on the temperature is consistent with the data in the literature for other methods of deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Our results also indicate that post-annealing may be an alternative to heating the substrate during deposition to obtain crystalline films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The films exhibit a low absorbance of less than 1% in the visible range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' There is a slight blue-shift of the fundamental absorption edge with an increase in the substrate temperature, apparently because there are fewer defects in the films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' No differences in the optical band gap (∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='0–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='1 eV, estimated from Tauc plots) were observed between the films on f-quartz and c-sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The optical properties are consistent with the data in the literature for other deposition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Azens, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Zubkins, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Butanovs, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Purans (applicant to the Institute of Solid State Physics University of Latvia) have a national patent pending (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' LVP2021000105) and a European patent (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' EP22195507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='3) pending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Acknowledgements This study was financially supported via ERDF project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content='1/20/A/057 “Functional ultrawide bandgap gallium oxide and zinc gallate thin films and novel deposition technologies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' The Institute of Solid State Physics, University of Latvia, as a Center of Excellence, has received funding from the European Union’s Horizon 2020 Framework Programme H2020-WIDESPREAD-01-2016-2017- TeamingPhase2 under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 739508, project CAMART².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Support for the ERDA measurements from the Ion Technology Centre (ITC) at Uppsala University is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' References [1] Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Zhang, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 33, 2006230 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 14 [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Pearton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' 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Kumar, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Kumar, ACS Photonics 5, 2391 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Pérez-Tomás, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Chikoidze, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' Dumont, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfXgeS/content/2301.03845v1.pdf'} +page_content=' R.' metadata={'source': 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--git a/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/2301.04306v1.pdf.txt b/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/2301.04306v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a97f688fccb9260e645735c57dbaeb5b8695f23f --- /dev/null +++ b/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/2301.04306v1.pdf.txt @@ -0,0 +1,858 @@ +Prepared for submission to JINST +Implementation of ACTS for STCF track reconstruction +Xiaocong Ai,𝑎 Xingtao Huang,𝑏 Yi Liu𝑎 +𝑎School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan, 450001, China +𝑏Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisci- +plinary Science, Shandong University, Qingdao, Shandong, 266327, China +E-mail: yiliu@zzu.edu.cn +Abstract: With an electron-positron collider operating at center-of-mass-energy 2-7 GeV and +a peak luminosity above 0.5 × 1035𝑐𝑚−2𝑠−1, the STCF physics program will provide an unique +platform for in-depth studies of hadron structure and non-perturbative strong interaction as well +as probing new physics beyond the Standard Model in the 𝜏-Charm sector, succeeding the present +Beijing Electron-Positron Collider. To fulfill the physics targets and further maximize the physics +potential at STCF, the STCF tracking software should have capability to reconstruct charged particles +with high efficiency and excellent momentum resolution, especially for the charged particles with +low transverse momentum down to 50 MeV. A Common Tracking Software (ACTS) providing a set +of detector-independent tracking algorithms is adopted for reconstructing charged tracks with the +information of two sub-detectors, a 𝜇RWELL-based inner tracker and a drift chamber, at STCF. This +is the first demonstration of ACTS for a drift chamber. The implementation details and performance +of track reconstruction are presented. +Keywords: STCF, Track reconstruction, ACTS, Drift chamber +1Corresponding author +arXiv:2301.04306v1 [hep-ex] 11 Jan 2023 + +Contents +1 +Introduction +1 +2 +The STCF detector +2 +3 +The STCF software framework and ACTS +3 +4 +STCF track reconstruction using ACTS +4 +4.1 +Extension of interface for STCF +4 +4.2 +Seed finding +5 +4.3 +Simultaneous track finding and track fitting using CKF +5 +4.4 +Ambiguity solving +6 +5 +Track reconstruction performance +7 +5.1 +Monte-Carlo samples +7 +5.2 +Track parameters resolution +7 +5.3 +Track finding performance +7 +6 +Conclusion +9 +1 +Introduction +Hadron physics plays an important role in studying Quantum Chromodynamics (QCD) in the low +energy region, where perturbative QCD is not applicable due to color confinement. +A multi- +GeV 𝑒+𝑒− collider operating in the 𝜏-charm sector provides an unique platform for studying non- +perturbative QCD and strong interactions of the Standard Model (SM). Currently, the Beijing +Electron Positron Collider (BEPCII)- Beijing Spectrometer (BESIII) [1] is the only facility at such +energy region in the world. With a luminosity of two-orders higher (peak luminosity is above +0.5 × 1035𝑐𝑚−2𝑠−1) and the energy region (2-7 GeV) wider than those at BEPCII, the future 𝜏- +charm factory, Super Tau-Charm Facility (STCF) [2], aims to continue and extend the physics +programs at BESIII in the post-BEPCII era. The physics goals of STCF include in-depth studies of +hadron structure and the nature of non-perturbative strong interactions, exploring the asymmetry of +matter-antimatter and searches for particles and physics beyond the SM. +To fullfill the physics goals of STCF, the charged tracks must be reconstructed with both +high efficiency and high precision, as the capability of charged tracks reconstruction, i.e. tracking, +has significant impact on the performance of vertex reconstruction, particle identification and +background suppression. In particular, among the final state particles of many important processes +for studying CP Violation, CKM elements, 𝐷0 − ¯𝐷0 mixing and so on at STCF, there are a +considerable number of particles with momentum lower than 400 MeV. Therefore, good tracking +– 1 – + +efficiency for particles with momentum below a few hundreds of MeV down to 50 MeV is very +important and also very challenging. +A Common Tracking Software (ACTS) [3] is a tracking toolkit with a set of detector- +independent and framework-independent modular tools dedicated to track reconstruction and vertex +reconstruction for High Energy Physics (HEP) experiments. It has been used by a variety of HEP +experiments, e.g. ATLAS [4] and sPHENIX [5]. However, its application on 𝑒+𝑒− colliders at +the precision frontier is very limited so far. +In particular, ACTS has not been used for track +reconstruction with a gaseous drift chamber yet. +In this study, the tracking performance of STCF with a fully gaseous tracking system consisting +of a 𝜇RWELL [6]-based inner tracker and a drift chamber is studied using the Kalman Filter [7] +based tracking algorithms of ACTS. The manuscript is organized as follows. In Section 2, the STCF +detector is introduced. The STCF offline software framework and ACTS are described in Section 3. +Section 4 focuses on implementation of ACTS for track reconstruction and the performance is +presented in Section 5. A brief conclusion is given in Section 6. +2 +The STCF detector +The STCF detector is designed to provide a coverage of almost the entire solid angle around the +collision point. The baseline layout of the STCF detector is shown in Figure 1. It consists of a +tracking system which includes an inner tracker (ITK) and a Main Drift Chamber (MDC), a Ring +Imaging Cherenkov (RICH) detector and a DIRC [8]-like time-of-flight (DTOF) detector for particle +identification in the barrel and endcap, respectively, a homogeneous Electro-magnetic Calorimeter +(EMC), a superconducting solenoid magnetic producing a 1 Tesla axial magnetic field, and a Muon +Detector (MUD) at the outermost of the detector system. +The tracking system provides position measurement for charged particles in the range |cos𝜃| < +0.94. To achieve the physics goals of STCF, it’s required to provide a tracking efficiency above 99% +(90%) for charged tracks with 𝑝𝑇 > 300 (100) MeV, and a momentum resolution of 𝜎𝑝𝑇 /𝑝𝑇 < 0.5% +for charged tracks with 𝑝𝑇 = 1 GeV. There are two options for the ITK, the MAPS-based ITK and +𝜇RWELL-based ITK. This study is based on the 𝜇RWELL-based ITK, which consists of three +layers of light-material 𝜇RWELL-based gaseous detectors around the beam pipe with the inner +radii of 60 mm, 110 mm and 160 mm, respectively. It provides a spatial resolution around 100 +𝜇m in the 𝑟-𝜙 direction and around 400 𝜇m in the 𝑧 direction. The MDC adopts a square cell +and a superlayer wire configuration, and uses He/C3H8 (60/40) as the working gas. There are +eight superlayers and each superlayer contains six layers of drift cells. The superlayers alternate +between axial ("A") orientation, aligned with the direction of the beam line, and stereo ("U", "V") +orientation. The eight superlayers are arranged in AUVAUVAA. The inner radius and outer radius +of MDC is 200 mm and 850 mm, respectively. The MDC is designed to provide a spatial resolution +between 120 𝜇m and 130 𝜇m, and a dE/dx resolution around 6%. +By measuring the characteristic radiation angle or spatial-time hit pattern of Cherenkov photons, +the RICH and DTOF are designed to facilitate identification of charged hadrons (𝜋, 𝐾, proton) with +momentum from 700 MeV to 2 GeV. The scintillating crystals based EMC provides measurement +of energy and direction for photons as well as identification for photons, electrons and charged +– 2 – + +11(9.8) cm +16 cm +20 cm +85 cm +105 cm +149 cm +185 cm +291 cm +140 cm +160 cm +190 cm +240 cm +347 cm +20° +MDC +ITK +PID (RICH) +PID (DTOF) +EMC +Superconducting magnet (1 T) +Iron York/MUD +IP +RPC +Iron York/MUD +Iron +Scintillator +40 cm +6(3.6) cm +Figure 1. Schematic layout of the STCF detector. The number in brackets indicate the radii of the MAPS- +based ITK. +hadrons. +The MUD consists of both plastic scintillator strips and resistive plate chambers to +provide information for identification between 𝜋 and 𝜇. +3 +The STCF software framework and ACTS +The Offline Software System of Super Tau-Charm Facility (OSCAR) [9] is the offline event process- +ing framework for STCF. It consists of an interface for external third-party software, a framework +providing common functionalities for data processing and a set of application tools for event gener- +ation, simulation, reconstruction and physics analysis. The physics generators with high precision, +e.g. KKMC [10], EVTGEN [11], are integrated into OSCAR for simulating the 𝜏-charm physics +processes. The Detector Description Toolkit, DD4Hep [12], is adopted to describe STCF detector +geometry with all geometric parameters stored in the compact files with eXtensible Markup Lan- +guage (XML) [13]. Geant4 [14] is integrated into OSCAR for full simulation of the interaction of +particles with detector. +To make a common tracking toolkit, the geometry and navigation model of ACTS, and its +Event Data for describing measurements, track parameters and vertex parameters, are designed to +be independent on the details of specific experiments. ACTS is written in modern C++17 and +features stateless modules to facilitate multi-threaded event reconstruction, in compliance with +modern multi-core CPU architectures. +In ACTS, the track parameters can be described with either bound (also called local) track +parameters or free (also called global) track parameters. With the bound track parameters, the +position of the track is represented in local coordinates of a detector surface, and the track direction +is described by azimuthal and polar angles. With thefreetrack parameters, the positionisrepresented +– 3 – + +in global coordinates of the detector and the track direction is represented by a unit vector. Both +the bound track parameters and free track parameters include a curvature parameter and the flight +time of the particle. +The bound track parameters are represented as: +𝑏 = (𝑙0, 𝑙1, 𝜙, 𝜃, 𝑞/𝑝, 𝑡), +(3.1) +where the first two track parameters are the coordinates of the track in the local coordinates of a +reference surface, 𝜙 and 𝜃 are the azimuthal and polar angles of the track direction, 𝑞/𝑝 is the ratio +of charge 𝑞 and momentum 𝑝, and 𝑡 is time coordinate of a particle in space-time. +The measurement is designed to be a subset of the bound track parameters. +Therefore, +projection from the bound track parameters to the measurement is realized using a projection +matrix. +Depending on the detector readout geometry, surface of dedicated type is used as the reference +surface of the bound track parameters and measurement. For STCF, the cylinder surface and line +surface are used for the ITK and the MDC, respectively. The line surface in ACTS is also used +for representing the track parameters near the interaction point, i.e. perigee track parameters. In +such case, the 𝑙0 and 𝑙1 represent the transverse impact parameter 𝑑0 and longitudinal impact track +parameter 𝑧0, respectively. +4 +STCF track reconstruction using ACTS +Figure 2 shows the workflow of applying the ACTS tracking toolkit for track reconstruction at +STCF. The interface of ACTS with experiments is extended for STCF. The ACTS Combinatorial +Kalman Filter (CKF) is used to find the tracks based on the seeds provided by the ACTS seed finding +algorithm, and then it is followed by a subsequent ambiguity solving step to remove incomplete or +duplicate tracks. +4.1 +Extension of interface for STCF +Based on the interfaces provided by ACTS, several geometry plugins are developed to facili- +tate transformation of an experiment’s geometry in an existing representation, e.g. DD4hep or +TGeo [15], into an internal geometry description of ACTS. The ACTS TGeo plugin is extended +for both ITK and MDC. For ITK, the tube containing the signal readout unit in each 𝜇RWELL +layer is transformed into a sensitive cylinder surface. For MDC, each sense wire in a drift cell is +transformed into a line surface. The dedicated material mapping tools [3] in ACTS are used to +project the detailed material description into internal auxiliary surfaces of ACTS geometry. +A ROOT [16]-based reader is implemented to read the simulated hits from full simulation and +convert them into ACTS measurements after taking into account the resolution of detectors. The +ITK measurement is two dimensional, and the 𝑙0 and 𝑙1 represent 𝑟 · 𝜑 and 𝑙𝑧, respectively, with 𝑟 +being the radius of the cylinder, 𝜑 being the azimuthal angle of the track position on the local 𝑥-𝑦 +plane of the cylinder frame, and 𝑙𝑧 being the local 𝑧 coordinate of the track position in the cylinder +frame. The MDC measurement is one dimensional with 𝑙0 representing the drift distance of the +ionized electrons to the anode wire of the MDC cell. +– 4 – + + + +Extended ACTS +TGeo Plugin +Root file for sim hits&particles +from full simulation +ACTS seeding algorithm +TGeo based STCF geometry +from DD4hep +ACTS tracking geometry +Root reader&converter for +STCF sim hits +ACTS Measurement +ACTS CKF algorithm +Ambiguity solving +ACTS space point converter +Reconstructed tracks +Figure 2. The workflow of applying ACTS for track reconstruction at STCF. +4.2 +Seed finding +Track seeds, i.e. triplets of measurements from increasing radii which are likely to belong to the +same track, are created from the measurements of the ITK detector using the ACTS track seed +finding algorithm. Using the coordinates of the measurements in global coordinate frame, the +algorithm groups the measurements based on a number of criteria related with the fiducial coverage +of the detector and the present magnetic field. Under the assumption of a homogeneous magnetic +field along global 𝑧 axis, a track follows a helix trajectory, i.e. a circle on the 𝑥-𝑦 plane and a straight +line in the 𝑠1-𝑧 plane. For each seed candidate, the curvature and center of the circle on the 𝑥-𝑦 +plane can be obtained using Conformal transform [? ], and then used to derive the transverse impact +parameter on the 𝑥-𝑦 plane. Filtering of the seeds is performed based on the estimated curvature, +transverse impact parameter and the polar angle. The filtering criteria is to achieve the maximum +efficiency of seed finding while keeping the rate of fake or duplicate seeds at a low level. +4.3 +Simultaneous track finding and track fitting using CKF +The CKF performs track finding through track fitting. Starting from track parameters estimated +from a seed, it searches and associates the related measurements to the track iteratively during track +propagation, as shown in Fig. 3. For each measurement candidate considered in a propagation step, +a prediction 𝜒2 is calculated using the predicted track parameters and the measurement, +𝜒2 = 𝑟𝑇 (𝐻𝐶𝐻𝑇 + 𝑉)−1𝑟, +(4.1) +1The 𝑠 is the path length of the track on the 𝑥 − 𝑦 plane +– 5 – + +where 𝐻 is the projection matrix from the bound track parameters to the measurement, 𝐶 is the +covariance of the bound track parameters, 𝑉 is the covariance of the measurement and 𝑟 is the +residual, +𝑟 = 𝑚 − 𝐻𝑏, +(4.2) +where 𝑚 and 𝑏 are the measurement vector and bound track parameters vector, respectively. For +a MDC measurement, the left/right sign of the drift distance is taken to be the same as that of the +predicted track parameters. The measurement with the minimum 𝜒2, 𝜒2 +min, is associated to the track +if 𝜒2 +min is below a threshold. +Upon the ending of forward track propagation, the Kalman smoothing of the track parameters +is performed for each found track candidate. The perigee track parameters for each track candidate +is obtained by propagating the smoothed track parameters at the first measurement plane to the +perigee plane. +Figure 3. +Illustration of track finding using CKF with ITK (purple) and MDC (blue) of STCF. Only two +MDC layers are shown here. +4.4 +Ambiguity solving +A threshold is set on the minimum number of measurements associated to a track to remove +incomplete tracks. +Because the charged track multiplicity is low at STCF and beam induced +background is not considered in this study, the fake tracks among the found track candidates, +i.e. tracks cannot be associated to any simulated particle, are negligible. +However, it can happen that multiple tracks are associated to the same simulated particle, +i.e. duplicate tracks are present. This is mainly due to the presence of duplicate seeds, in particular +in events with low momentum looping tracks. If two track candidates have at least two shared +– 6 – + +measurements, the one with less measurements is removed from the track candidates. This pro- +cedure is repeated iteratively until any two of the track candidates have no more than one shared +measurement. +5 +Track reconstruction performance +The track reconstruction performance of the STCF tracking system following the track reconstruction +procedure in Section 4 is discussed in this section. +5.1 +Monte-Carlo samples +Both single particle (𝜇− and 𝜋−) events generated using particle gun, and 𝑒+𝑒− → 𝜓(3686) → +𝜋+𝜋−𝐽/𝜓, 𝐽/𝜓 → 𝜇+𝜇− events generated with KKMC generator, are used for the tracking perfor- +mance studies. Each single particle sample is generated with polar angle and transverse momentum +of the particle fixed, and azimuthal angle uniformly distributed from [-𝜋, 𝜋]. The Geant4 in OSCAR +is used to simulate the hits of the generated final state particles with STCF tracking system immersed +in a uniform magnetic field of 1 T. +The detector measurements are created by smearing the position of the simulated hits with +Gaussian functions with zero means and widths equivalent to the resolutions of the detectors. For +ITK, the hit resolution of 100 𝜇m for 𝑙0 and 400 𝜇m for 𝑙1 are assumed. For MDC, the resolution +of 125 𝜇m for 𝑙0 is assumed. +The tracks are reconstructed as described in Section 4. For a track seed, its transverse momen- +tum is required to be larger than 40 MeV and its transverse impact track parameter is required to be +no larger than 10 mm. The 𝜒2 +𝑚𝑖𝑛 = 30 for the CKF is used to associate the measurements to tracks. +The reconstructed tracks are required to have at least 5 measurements on the track. The resolution +of the fitted perigee track parameters is studied using the single particle samples. The track finding +performance is studied using the 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓 sample. +5.2 +Track parameters resolution +The resolution of a track parameter is obtained by fitting the distribution of residuals of the track +parameter with a Gaussian function. +Figure 4 shows the resolution of impact track parameters 𝑑0, 𝑧0 and transverse momentum 𝑝𝑇 +as a function of particle 𝑝𝑇 at three different polar angles, |cos𝜃| = 0.0, 0.5 and 0.8, for single 𝜇− +and 𝜋− events. For 𝜇− and 𝜋− tracks with 𝑝𝑇 = 1 GeV and cos𝜃 = 0.0, the resolution of 𝑑0, 𝑧0 and +relative resolution of 𝑝𝑇 are about 150 𝜇m, 400 𝜇m and 0.45% respectively. +Due to more material effects at lower momentum and larger |cos𝜃|, the 𝑑0 and 𝑧0 have a worse +resolution at lower 𝑝𝑇 and larger |cos𝜃|. The relative resolution of 𝑝𝑇 is also dependent on 𝑝𝑇 +since the curvature of the track on 𝑥-𝑦 plane is dependent on 𝑝𝑇 as well as the material effects. At +low momentum range, the material effects are dominant hence the resolution is worse at lower 𝑝𝑇 . +At high momentum range, the resolution is worse with larger 𝑝𝑇 . +5.3 +Track finding performance +The merits often used to characterize the performance of track finding are track reconstruction +efficiency, the rate of fake tracks and the rate of duplicate tracks. The classification of a reconstructed +– 7 – + +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +) [mm] +0 +(d +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +- +µ +single +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +) [mm] +0 +(d +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +-π +single +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +) [mm] +0 +(z +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +- +µ +single +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +) [mm] +0 +(z +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +-π +single +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 + [%] +T +)/p +T +(p +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +- +µ +single +0.2 0.4 0.6 0.8 +1 +1.2 1.4 1.6 1.8 + [GeV] +T +True p +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 + [%] +T +)/p +T +(p +σ +|=0.8 +θ +|cos +|=0.5 +θ +|cos +|=0.0 +θ +|cos +-π +single +Figure 4. The resolution of 𝑑0 (top panels), 𝑧0 (middle panels) and relative resolution of 𝑝𝑇 (bottom panels) +for single 𝜇− (left panels) and single 𝜋− (right panels) as a function of particle 𝑝𝑇 . The blue dot, yellow +triangle and green circle represent the results with |cos𝜃| = 0.0, 0.5 and 0.8, respectively. For each 𝑝𝑇 and +|cos𝜃|, a sample of 5k events is generated for the study. +– 8 – + +track to be fake or duplicate is performed by identifying the particle which has the most number +of simulated hits contributing to the track. In this study, a track is classified as a fake track if the +ratio of the hits from this particle is smaller than 0.5. The track reconstruction efficiency is defined +by the ratio of particles which have matched reconstructed tracks among the particles which have +at least 5 simulated hits. The fake/duplicate rate are defined by the ratio of fake/duplicate tracks +among the reconstructed tracks. +Figure 5 shows the tracking efficiency, fake rate and duplicate rate of 𝜇 and 𝜋 in 𝜓(3686) → +𝜋+𝜋−𝐽/𝜓 events. The 𝑝𝑇 of 𝜇 is in the range of [0.4, 1.8] GeV while that of 𝜋 is in the range of +[50, 450] MeV. For 𝜇 and 𝜋 with 𝑝𝑇 above 150 MeV, the tracking efficiency is above 99%. For 𝜋 +with momentum in the range of [50, 100] MeV, a tracking efficiency of 95% is achieved. The fake +rate is found to be negligible. For 𝜋 track with 𝑝𝑇 below 150 MeV, duplicate tracks due to looping +trajectories of the particles are found with the duplicate rate below 0.4%. +6 +Conclusion +To achieve the physics goals of the STCF physics program, which is dedicated to studies of +hadron physics, the asymmetry of matter-antimatter and searches for physics beyond SM in the +𝜏-charm sector in the post-BEPCII era, the tracking system of STCF is required to provide sufficient +performance for charged tracks reconstruction. +We implemented the common tracking toolkit ACTS for track reconstruction at STCF based +on a three-layer 𝜇RWELL-based ITK and a 48-layer MDC, and demonstrated the performance of +ACTS for a drift chamber for the first time. The performance of track reconstruction was studied +on top of Geant4-based full simulation of interaction of particles with detectors. The track finding +and track fitting was performed based on the Combinatorial Kalman Filter in ACTS. The tracking +efficiency is above 95% for tracks with 𝑝𝑇 above 50 MeV. For tracks with 𝑝𝑇 > 150 MeV, a tracking +efficiency above 99% is achieved. The rate of fake tracks is negligible and the rate of duplicate +tracks, which arise in the range of 𝑝𝑇 < 150 MeV, is below 0.4%. The 𝜎(𝑝𝑇 )/𝑝𝑇 is below 0.5% +when 𝑝𝑇 = 1 GeV and |cos𝜃| ≤ 0.5. Those excellent performance shows that ACTS is a very +promising tracking toolkit to be used for further optimization of the design and geometry layout +of the STCF tracking system. Meanwhile, track finding using Hough Transform [17] for STCF is +being studied. The tracking performance considering beam induced background will be explored +in future studies. +Acknowledgments +The authors are grateful to the STCF group for the profitable discussions and express gratitude +to the Hefei Comprehensive National Science Center for their strong support. +This work is +supported by the international partnership program of the Chinese Academy of Sciences (Grant +No. 211134KYSB20200057) and the National Natural Science Foundation of China (Grant No. +12025502). +– 9 – + +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 + [GeV] +T +Truth p +0.94 +0.96 +0.98 +1 +1.02 +1.04 +1.06 +Efficiency +- +µ ++ +µ +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 + [GeV] +T +Truth p +0.92 +0.94 +0.96 +0.98 +1 +1.02 +1.04 +1.06 +1.08 +Efficiency +-π ++ +π +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 + [GeV] +T +p +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +Fake rate +- +µ ++ +µ +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 + [GeV] +T +p +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +Fake rate +-π ++ +π +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 + [GeV] +T +p +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +Duplicate rate +- +µ ++ +µ +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 + [GeV] +T +p +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +0.014 +0.016 +0.018 +0.02 +Duplicate rate +-π ++ +π +) +- +µ ++ +µ + +→ +( +ψ +J/ +-π ++ +π + +→ +(3686) +ψ +Figure 5. The tracking efficiency (top panels), fake rate (middle panels) and duplicate rate (bottom panels) +for 𝜇 (left panels) and 𝜋 (right panels) with 100k 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓, 𝐽/𝜓 → 𝜇+𝜇− events as a function +of 𝑝𝑇 . The blue dot, and yellow circle represent the results for positive charge particles and negative charge +particles, respectively. +– 10 – + +References +[1] M. Ablikim et al., Design and construction of the besiii detector, Nucl. Instrum. Meth. A 614 (2010) +345. +[2] Q. Luo, W. Gao, J. Lan, W. Li and D. Xu, Progress of Conceptual Study for the Accelerators of a 2-7 +GeV Super Tau Charm Facility at China, in Proc. 10th International Particle Accelerator Conference +(IPAC’19), Melbourne, Australia, 19-24 May 2019, no. 10 in International Particle Accelerator +Conference, (Geneva, Switzerland), pp. 643–645, JACoW Publishing, Jun., 2019, DOI. +[3] X. Ai, C. Allaire, N. Calace, A. Czirkos, M. Elsing, I. Ene et al., A common tracking software project, +Computing and Software for Big Science 6 (2022) 8. +[4] ATLAS Collaboration, Software Performance of the ATLAS Track Reconstruction for LHC Run 3, +Tech. Rep. ATL-PHYS-PUB-2021-012, CERN, Geneva (May, 2021). +[5] J.D. Osborn, A.D. Frawley, J. Huang, S. Lee, H.P.D. Costa, M. Peters et al., Implementation of ACTS +into sPHENIX Track Reconstruction, Computing and Software for Big Science 5 (2021) 23. +[6] G. Bencivenni, L. Benussi, L. Borgonovi, R. de Oliveira, P.D. Simone, G. Felici et al., The 𝜇-rwell +detector, Journal of Instrumentation 12 (2017) C06027. +[7] R.E. Kalman, A New Approach to Linear Filtering and Prediction Problems, Journal of Basic +Engineering 82 (1960) 35. +[8] I. Adam et al., The dirc particle identification system for the babar experiment, Nucl. Instrum. Meth. A +538 (2005) 281. +[9] W.H. Huang, H. Li, H. Zhou, T. Li, Q.Y. Li and X.T. Huang, Design and development of the core +software for stcf offline data processing, 2022. 10.48550/ARXIV.2211.03137. +[10] S. Jadach, B.F.L. Ward and Z. Wa¸s, Coherent exclusive exponentiation for precision monte carlo +calculations, Phys. Rev. D 63 (2001) 113009. +[11] D.J. Lange, The evtgen particle decay simulation package, Nucl. Instrum. Meth. A 462 (2001) 152. +[12] M. Frank, F. Gaede, C. Grefe and P. Mato, Dd4hep: A detector description toolkit for high energy +physics experiments, Journal of Physics: Conference Series 513 (2014) 022010. +[13] “Extensible markup language (xml) webpage.” https://www.w3.org/XML. +[14] S. Agostinelli et al., Geant4—a simulation toolkit, Nucl. Instrum. Meth. A 506 (2003) 250. +[15] R. Brun, A. Gheata and M. Gheata, The root geometry package, Nucl. Instrum. Methods. Phys. Res. A +502 (2003) 676. +[16] R. Brun and F. Rademakers, ROOT: An object oriented data analysis framework, Nucl. Instrum. Meth. +A 389 (1997) 81. +[17] R. Duda and P. Hart, Use of the hough transformation to detect lines and curves in pictures, Commun. +ACM 15 (1972) 11. +– 11 – + diff --git a/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/load_file.txt b/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a219bc47a66a53f8df72bc02da361c92166f0fe --- /dev/null +++ b/W9E3T4oBgHgl3EQfFgmY/content/tmp_files/load_file.txt @@ -0,0 +1,544 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf,len=543 +page_content='Prepared for submission to JINST Implementation of ACTS for STCF track reconstruction Xiaocong Ai,𝑎 Xingtao Huang,𝑏 Yi Liu𝑎 𝑎School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan, 450001, China 𝑏Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisci- plinary Science, Shandong University, Qingdao, Shandong, 266327, China E-mail: yiliu@zzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='cn Abstract: With an electron-positron collider operating at center-of-mass-energy 2-7 GeV and a peak luminosity above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 × 1035𝑐𝑚−2𝑠−1, the STCF physics program will provide an unique platform for in-depth studies of hadron structure and non-perturbative strong interaction as well as probing new physics beyond the Standard Model in the 𝜏-Charm sector, succeeding the present Beijing Electron-Positron Collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' To fulfill the physics targets and further maximize the physics potential at STCF, the STCF tracking software should have capability to reconstruct charged particles with high efficiency and excellent momentum resolution, especially for the charged particles with low transverse momentum down to 50 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' A Common Tracking Software (ACTS) providing a set of detector-independent tracking algorithms is adopted for reconstructing charged tracks with the information of two sub-detectors, a 𝜇RWELL-based inner tracker and a drift chamber, at STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' This is the first demonstration of ACTS for a drift chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The implementation details and performance of track reconstruction are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Keywords: STCF, Track reconstruction, ACTS, Drift chamber 1Corresponding author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='04306v1 [hep-ex] 11 Jan 2023 Contents 1 Introduction 1 2 The STCF detector 2 3 The STCF software framework and ACTS 3 4 STCF track reconstruction using ACTS 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1 Extension of interface for STCF 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 Seed finding 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='3 Simultaneous track finding and track fitting using CKF 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 Ambiguity solving 6 5 Track reconstruction performance 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1 Monte-Carlo samples 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 Track parameters resolution 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='3 Track finding performance 7 6 Conclusion 9 1 Introduction Hadron physics plays an important role in studying Quantum Chromodynamics (QCD) in the low energy region, where perturbative QCD is not applicable due to color confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' A multi- GeV 𝑒+𝑒− collider operating in the 𝜏-charm sector provides an unique platform for studying non- perturbative QCD and strong interactions of the Standard Model (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Currently, the Beijing Electron Positron Collider (BEPCII)- Beijing Spectrometer (BESIII) [1] is the only facility at such energy region in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' With a luminosity of two-orders higher (peak luminosity is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 × 1035𝑐𝑚−2𝑠−1) and the energy region (2-7 GeV) wider than those at BEPCII, the future 𝜏- charm factory, Super Tau-Charm Facility (STCF) [2], aims to continue and extend the physics programs at BESIII in the post-BEPCII era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The physics goals of STCF include in-depth studies of hadron structure and the nature of non-perturbative strong interactions, exploring the asymmetry of matter-antimatter and searches for particles and physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' To fullfill the physics goals of STCF, the charged tracks must be reconstructed with both high efficiency and high precision, as the capability of charged tracks reconstruction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' tracking, has significant impact on the performance of vertex reconstruction, particle identification and background suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In particular, among the final state particles of many important processes for studying CP Violation, CKM elements, 𝐷0 − ¯𝐷0 mixing and so on at STCF, there are a considerable number of particles with momentum lower than 400 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Therefore, good tracking – 1 – efficiency for particles with momentum below a few hundreds of MeV down to 50 MeV is very important and also very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' A Common Tracking Software (ACTS) [3] is a tracking toolkit with a set of detector- independent and framework-independent modular tools dedicated to track reconstruction and vertex reconstruction for High Energy Physics (HEP) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' It has been used by a variety of HEP experiments, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' ATLAS [4] and sPHENIX [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' However, its application on 𝑒+𝑒− colliders at the precision frontier is very limited so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In particular, ACTS has not been used for track reconstruction with a gaseous drift chamber yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In this study, the tracking performance of STCF with a fully gaseous tracking system consisting of a 𝜇RWELL [6]-based inner tracker and a drift chamber is studied using the Kalman Filter [7] based tracking algorithms of ACTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In Section 2, the STCF detector is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The STCF offline software framework and ACTS are described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Section 4 focuses on implementation of ACTS for track reconstruction and the performance is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' A brief conclusion is given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 2 The STCF detector The STCF detector is designed to provide a coverage of almost the entire solid angle around the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The baseline layout of the STCF detector is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' It consists of a tracking system which includes an inner tracker (ITK) and a Main Drift Chamber (MDC), a Ring Imaging Cherenkov (RICH) detector and a DIRC [8]-like time-of-flight (DTOF) detector for particle identification in the barrel and endcap, respectively, a homogeneous Electro-magnetic Calorimeter (EMC), a superconducting solenoid magnetic producing a 1 Tesla axial magnetic field, and a Muon Detector (MUD) at the outermost of the detector system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The tracking system provides position measurement for charged particles in the range |cos𝜃| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' To achieve the physics goals of STCF, it’s required to provide a tracking efficiency above 99% (90%) for charged tracks with 𝑝𝑇 > 300 (100) MeV, and a momentum resolution of 𝜎𝑝𝑇 /𝑝𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5% for charged tracks with 𝑝𝑇 = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' There are two options for the ITK, the MAPS-based ITK and 𝜇RWELL-based ITK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' This study is based on the 𝜇RWELL-based ITK, which consists of three layers of light-material 𝜇RWELL-based gaseous detectors around the beam pipe with the inner radii of 60 mm, 110 mm and 160 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' It provides a spatial resolution around 100 𝜇m in the 𝑟-𝜙 direction and around 400 𝜇m in the 𝑧 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The MDC adopts a square cell and a superlayer wire configuration, and uses He/C3H8 (60/40) as the working gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' There are eight superlayers and each superlayer contains six layers of drift cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The superlayers alternate between axial ("A") orientation, aligned with the direction of the beam line, and stereo ("U", "V") orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The eight superlayers are arranged in AUVAUVAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The inner radius and outer radius of MDC is 200 mm and 850 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The MDC is designed to provide a spatial resolution between 120 𝜇m and 130 𝜇m, and a dE/dx resolution around 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' By measuring the characteristic radiation angle or spatial-time hit pattern of Cherenkov photons, the RICH and DTOF are designed to facilitate identification of charged hadrons (𝜋, 𝐾, proton) with momentum from 700 MeV to 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The scintillating crystals based EMC provides measurement of energy and direction for photons as well as identification for photons, electrons and charged – 2 – 11(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8) cm 16 cm 20 cm 85 cm 105 cm 149 cm 185 cm 291 cm 140 cm 160 cm 190 cm 240 cm 347 cm 20° MDC ITK PID (RICH) PID (DTOF) EMC Superconducting magnet (1 T) Iron York/MUD IP RPC Iron York/MUD Iron Scintillator 40 cm 6(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6) cm Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Schematic layout of the STCF detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The number in brackets indicate the radii of the MAPS- based ITK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The MUD consists of both plastic scintillator strips and resistive plate chambers to provide information for identification between 𝜋 and 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 3 The STCF software framework and ACTS The Offline Software System of Super Tau-Charm Facility (OSCAR) [9] is the offline event process- ing framework for STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' It consists of an interface for external third-party software, a framework providing common functionalities for data processing and a set of application tools for event gener- ation, simulation, reconstruction and physics analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The physics generators with high precision, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' KKMC [10], EVTGEN [11], are integrated into OSCAR for simulating the 𝜏-charm physics processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The Detector Description Toolkit, DD4Hep [12], is adopted to describe STCF detector geometry with all geometric parameters stored in the compact files with eXtensible Markup Lan- guage (XML) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Geant4 [14] is integrated into OSCAR for full simulation of the interaction of particles with detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' To make a common tracking toolkit, the geometry and navigation model of ACTS, and its Event Data for describing measurements, track parameters and vertex parameters, are designed to be independent on the details of specific experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' ACTS is written in modern C++17 and features stateless modules to facilitate multi-threaded event reconstruction, in compliance with modern multi-core CPU architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In ACTS, the track parameters can be described with either bound (also called local) track parameters or free (also called global) track parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' With the bound track parameters, the position of the track is represented in local coordinates of a detector surface, and the track direction is described by azimuthal and polar angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' With thefreetrack parameters, the positionisrepresented – 3 – in global coordinates of the detector and the track direction is represented by a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Both the bound track parameters and free track parameters include a curvature parameter and the flight time of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The bound track parameters are represented as: 𝑏 = (𝑙0, 𝑙1, 𝜙, 𝜃, 𝑞/𝑝, 𝑡), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1) where the first two track parameters are the coordinates of the track in the local coordinates of a reference surface, 𝜙 and 𝜃 are the azimuthal and polar angles of the track direction, 𝑞/𝑝 is the ratio of charge 𝑞 and momentum 𝑝, and 𝑡 is time coordinate of a particle in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The measurement is designed to be a subset of the bound track parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Therefore, projection from the bound track parameters to the measurement is realized using a projection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Depending on the detector readout geometry, surface of dedicated type is used as the reference surface of the bound track parameters and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For STCF, the cylinder surface and line surface are used for the ITK and the MDC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The line surface in ACTS is also used for representing the track parameters near the interaction point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' perigee track parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In such case, the 𝑙0 and 𝑙1 represent the transverse impact parameter 𝑑0 and longitudinal impact track parameter 𝑧0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 4 STCF track reconstruction using ACTS Figure 2 shows the workflow of applying the ACTS tracking toolkit for track reconstruction at STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The interface of ACTS with experiments is extended for STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The ACTS Combinatorial Kalman Filter (CKF) is used to find the tracks based on the seeds provided by the ACTS seed finding algorithm, and then it is followed by a subsequent ambiguity solving step to remove incomplete or duplicate tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1 Extension of interface for STCF Based on the interfaces provided by ACTS, several geometry plugins are developed to facili- tate transformation of an experiment’s geometry in an existing representation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' DD4hep or TGeo [15], into an internal geometry description of ACTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The ACTS TGeo plugin is extended for both ITK and MDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For ITK, the tube containing the signal readout unit in each 𝜇RWELL layer is transformed into a sensitive cylinder surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For MDC, each sense wire in a drift cell is transformed into a line surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The dedicated material mapping tools [3] in ACTS are used to project the detailed material description into internal auxiliary surfaces of ACTS geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' A ROOT [16]-based reader is implemented to read the simulated hits from full simulation and convert them into ACTS measurements after taking into account the resolution of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The ITK measurement is two dimensional, and the 𝑙0 and 𝑙1 represent 𝑟 · 𝜑 and 𝑙𝑧, respectively, with 𝑟 being the radius of the cylinder, 𝜑 being the azimuthal angle of the track position on the local 𝑥-𝑦 plane of the cylinder frame, and 𝑙𝑧 being the local 𝑧 coordinate of the track position in the cylinder frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The MDC measurement is one dimensional with 𝑙0 representing the drift distance of the ionized electrons to the anode wire of the MDC cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' – 4 – Extended ACTS TGeo Plugin Root file for sim hits&particles from full simulation ACTS seeding algorithm TGeo based STCF geometry from DD4hep ACTS tracking geometry Root reader&converter for STCF sim hits ACTS Measurement ACTS CKF algorithm Ambiguity solving ACTS space point converter Reconstructed tracks Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The workflow of applying ACTS for track reconstruction at STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 Seed finding Track seeds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' triplets of measurements from increasing radii which are likely to belong to the same track, are created from the measurements of the ITK detector using the ACTS track seed finding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Using the coordinates of the measurements in global coordinate frame, the algorithm groups the measurements based on a number of criteria related with the fiducial coverage of the detector and the present magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Under the assumption of a homogeneous magnetic field along global 𝑧 axis, a track follows a helix trajectory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' a circle on the 𝑥-𝑦 plane and a straight line in the 𝑠1-𝑧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For each seed candidate, the curvature and center of the circle on the 𝑥-𝑦 plane can be obtained using Conformal transform [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' ], and then used to derive the transverse impact parameter on the 𝑥-𝑦 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Filtering of the seeds is performed based on the estimated curvature, transverse impact parameter and the polar angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The filtering criteria is to achieve the maximum efficiency of seed finding while keeping the rate of fake or duplicate seeds at a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='3 Simultaneous track finding and track fitting using CKF The CKF performs track finding through track fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Starting from track parameters estimated from a seed, it searches and associates the related measurements to the track iteratively during track propagation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For each measurement candidate considered in a propagation step, a prediction 𝜒2 is calculated using the predicted track parameters and the measurement, 𝜒2 = 𝑟𝑇 (𝐻𝐶𝐻𝑇 + 𝑉)−1𝑟, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1) 1The 𝑠 is the path length of the track on the 𝑥 − 𝑦 plane – 5 – where 𝐻 is the projection matrix from the bound track parameters to the measurement, 𝐶 is the covariance of the bound track parameters, 𝑉 is the covariance of the measurement and 𝑟 is the residual, 𝑟 = 𝑚 − 𝐻𝑏, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2) where 𝑚 and 𝑏 are the measurement vector and bound track parameters vector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For a MDC measurement, the left/right sign of the drift distance is taken to be the same as that of the predicted track parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The measurement with the minimum 𝜒2, 𝜒2 min, is associated to the track if 𝜒2 min is below a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Upon the ending of forward track propagation, the Kalman smoothing of the track parameters is performed for each found track candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The perigee track parameters for each track candidate is obtained by propagating the smoothed track parameters at the first measurement plane to the perigee plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Illustration of track finding using CKF with ITK (purple) and MDC (blue) of STCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Only two MDC layers are shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 Ambiguity solving A threshold is set on the minimum number of measurements associated to a track to remove incomplete tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Because the charged track multiplicity is low at STCF and beam induced background is not considered in this study, the fake tracks among the found track candidates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' tracks cannot be associated to any simulated particle, are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' However, it can happen that multiple tracks are associated to the same simulated particle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' duplicate tracks are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' This is mainly due to the presence of duplicate seeds, in particular in events with low momentum looping tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' If two track candidates have at least two shared – 6 – measurements, the one with less measurements is removed from the track candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' This pro- cedure is repeated iteratively until any two of the track candidates have no more than one shared measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 5 Track reconstruction performance The track reconstruction performance of the STCF tracking system following the track reconstruction procedure in Section 4 is discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='1 Monte-Carlo samples Both single particle (𝜇− and 𝜋−) events generated using particle gun, and 𝑒+𝑒− → 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓, 𝐽/𝜓 → 𝜇+𝜇− events generated with KKMC generator, are used for the tracking perfor- mance studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Each single particle sample is generated with polar angle and transverse momentum of the particle fixed, and azimuthal angle uniformly distributed from [-𝜋, 𝜋].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The Geant4 in OSCAR is used to simulate the hits of the generated final state particles with STCF tracking system immersed in a uniform magnetic field of 1 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The detector measurements are created by smearing the position of the simulated hits with Gaussian functions with zero means and widths equivalent to the resolutions of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For ITK, the hit resolution of 100 𝜇m for 𝑙0 and 400 𝜇m for 𝑙1 are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For MDC, the resolution of 125 𝜇m for 𝑙0 is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The tracks are reconstructed as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For a track seed, its transverse momen- tum is required to be larger than 40 MeV and its transverse impact track parameter is required to be no larger than 10 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The 𝜒2 𝑚𝑖𝑛 = 30 for the CKF is used to associate the measurements to tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The reconstructed tracks are required to have at least 5 measurements on the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The resolution of the fitted perigee track parameters is studied using the single particle samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The track finding performance is studied using the 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 Track parameters resolution The resolution of a track parameter is obtained by fitting the distribution of residuals of the track parameter with a Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Figure 4 shows the resolution of impact track parameters 𝑑0, 𝑧0 and transverse momentum 𝑝𝑇 as a function of particle 𝑝𝑇 at three different polar angles, |cos𝜃| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8, for single 𝜇− and 𝜋− events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For 𝜇− and 𝜋− tracks with 𝑝𝑇 = 1 GeV and cos𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0, the resolution of 𝑑0, 𝑧0 and relative resolution of 𝑝𝑇 are about 150 𝜇m, 400 𝜇m and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='45% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Due to more material effects at lower momentum and larger |cos𝜃|, the 𝑑0 and 𝑧0 have a worse resolution at lower 𝑝𝑇 and larger |cos𝜃|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The relative resolution of 𝑝𝑇 is also dependent on 𝑝𝑇 since the curvature of the track on 𝑥-𝑦 plane is dependent on 𝑝𝑇 as well as the material effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' At low momentum range, the material effects are dominant hence the resolution is worse at lower 𝑝𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' At high momentum range, the resolution is worse with larger 𝑝𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='3 Track finding performance The merits often used to characterize the performance of track finding are track reconstruction efficiency, the rate of fake tracks and the rate of duplicate tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The classification of a reconstructed – 7 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 ) [mm] 0 (d σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos µ single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 [GeV] T True p 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 ) [mm] 0 (d σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos π single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 [GeV] T True p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 2 ) [mm] 0 (z σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos µ single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 [GeV] T True p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 2 ) [mm] 0 (z σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos π single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 [%] T )/p T (p σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos µ single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 [GeV] T True p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='2 [%] T )/p T (p σ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 θ |cos |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0 θ |cos π single Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The resolution of 𝑑0 (top panels), 𝑧0 (middle panels) and relative resolution of 𝑝𝑇 (bottom panels) for single 𝜇− (left panels) and single 𝜋− (right panels) as a function of particle 𝑝𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The blue dot, yellow triangle and green circle represent the results with |cos𝜃| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For each 𝑝𝑇 and |cos𝜃|, a sample of 5k events is generated for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' – 8 – track to be fake or duplicate is performed by identifying the particle which has the most number of simulated hits contributing to the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' In this study, a track is classified as a fake track if the ratio of the hits from this particle is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The track reconstruction efficiency is defined by the ratio of particles which have matched reconstructed tracks among the particles which have at least 5 simulated hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The fake/duplicate rate are defined by the ratio of fake/duplicate tracks among the reconstructed tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Figure 5 shows the tracking efficiency, fake rate and duplicate rate of 𝜇 and 𝜋 in 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The 𝑝𝑇 of 𝜇 is in the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='8] GeV while that of 𝜋 is in the range of [50, 450] MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For 𝜇 and 𝜋 with 𝑝𝑇 above 150 MeV, the tracking efficiency is above 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For 𝜋 with momentum in the range of [50, 100] MeV, a tracking efficiency of 95% is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The fake rate is found to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For 𝜋 track with 𝑝𝑇 below 150 MeV, duplicate tracks due to looping trajectories of the particles are found with the duplicate rate below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 6 Conclusion To achieve the physics goals of the STCF physics program, which is dedicated to studies of hadron physics, the asymmetry of matter-antimatter and searches for physics beyond SM in the 𝜏-charm sector in the post-BEPCII era, the tracking system of STCF is required to provide sufficient performance for charged tracks reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' We implemented the common tracking toolkit ACTS for track reconstruction at STCF based on a three-layer 𝜇RWELL-based ITK and a 48-layer MDC, and demonstrated the performance of ACTS for a drift chamber for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The performance of track reconstruction was studied on top of Geant4-based full simulation of interaction of particles with detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The track finding and track fitting was performed based on the Combinatorial Kalman Filter in ACTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The tracking efficiency is above 95% for tracks with 𝑝𝑇 above 50 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' For tracks with 𝑝𝑇 > 150 MeV, a tracking efficiency above 99% is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The rate of fake tracks is negligible and the rate of duplicate tracks, which arise in the range of 𝑝𝑇 < 150 MeV, is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The 𝜎(𝑝𝑇 )/𝑝𝑇 is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5% when 𝑝𝑇 = 1 GeV and |cos𝜃| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Those excellent performance shows that ACTS is a very promising tracking toolkit to be used for further optimization of the design and geometry layout of the STCF tracking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Meanwhile, track finding using Hough Transform [17] for STCF is being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The tracking performance considering beam induced background will be explored in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Acknowledgments The authors are grateful to the STCF group for the profitable discussions and express gratitude to the Hefei Comprehensive National Science Center for their strong support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' This work is supported by the international partnership program of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 211134KYSB20200057) and the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' 12025502).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The tracking efficiency (top panels), fake rate (middle panels) and duplicate rate (bottom panels) for 𝜇 (left panels) and 𝜋 (right panels) with 100k 𝜓(3686) → 𝜋+𝜋−𝐽/𝜓, 𝐽/𝜓 → 𝜇+𝜇− events as a function of 𝑝𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' The blue dot, and yellow circle represent the results for positive charge particles and negative charge particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' – 10 – References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} +page_content=' Ablikim et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E3T4oBgHgl3EQfFgmY/content/2301.04306v1.pdf'} diff --git a/WNAyT4oBgHgl3EQf8_py/content/tmp_files/2301.00866v1.pdf.txt b/WNAyT4oBgHgl3EQf8_py/content/tmp_files/2301.00866v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a49f736fdfe67059b5b1ca6920f1fb7e0eb45378 --- /dev/null +++ b/WNAyT4oBgHgl3EQf8_py/content/tmp_files/2301.00866v1.pdf.txt @@ -0,0 +1,990 @@ +3DSGrasp: 3D Shape-Completion for Robotic Grasp +Seyed S. Mohammadi2,3 Nuno F. Duarte1 Dimitris Dimou1 Yiming Wang3,4 Matteo Taiana3 Pietro Morerio3 +Atabak Dehban1 +Plinio Moreno1 +Alexandre Bernardino1 +Alessio Del Bue3 +Jos´e Santos-Victor1 +Abstract— Real-world robotic grasping can be done robustly +if a complete 3D Point Cloud Data (PCD) of an object is +available. However, in practice, PCDs are often incomplete +when objects are viewed from few and sparse viewpoints before +the grasping action, leading to the generation of wrong or +inaccurate grasp poses. We propose a novel grasping strategy, +named 3DSGrasp, that predicts the missing geometry from the +partial PCD to produce reliable grasp poses. Our proposed +PCD completion network is a Transformer-based encoder- +decoder network with an Offset-Attention layer. Our network is +inherently invariant to the object pose and point’s permutation, +which generates PCDs that are geometrically consistent and +completed properly. Experiments on a wide range of partial +PCD show that 3DSGrasp outperforms the best state-of-the- +art method on PCD completion tasks and largely improves the +grasping success rate in real-world scenarios. The code and +dataset will be made available upon acceptance. +I. INTRODUCTION +Robotic grasping has recently gained increasing attention +because of its essential role in many real-world applications, +such as domestic and collaborative robotics. The seminal +work of Pas et al. [1] uses 3D Point Cloud Data (PCD) +to generate grasp poses directly on the available 3D object +structure. However, in real practical scenarios we often have +to rely on incomplete geometric information acquired from +single or few viewpoints, which leads to drastic reduction of +grasping success rate. +Researchers bypassed this problem by acquiring complete +3D object scans [2] but this requires a feasible camera path +around the object, which is time consuming to obtain and +not always feasible. Another strategy is to place additional +sensors around the object of interest [3], but this is not cost- +effective and it requires careful calibration. +Instead, this paper aims at improving single-view grasping +by predicting the missing geometrical structure from a partial +PCD. 3D shape completion is an inherently ambiguous +*This work has partially received funding from the European Union’s +Horizon 2020 research and innovation programme under grant agreement +No 964854; the FCT funding to the ISR/LARSyS Associated Laboratory +UID/EEA/50009/2020 and LA/P/0083/2020 N. F. Duarte is supported by +FCT-IST fellowship grant PD/BD/135116/2017. +1Vislab, +Institute +for +Systems +and +Robotics—Lisboa, +Instituto +Superior +T´ecnico, +Universidade +de +Lisboa, +Portu- +gal. +Email:{nferreiraduarte, plinio, alex, +jasv}@isr.tecnico.ulisboa.pt +2Department of Marine, Electrical, Electronic and Telecommunications +Engineering, University of Genoa, Italy. +3Pattern +Analysis +& +Computer +Vision +(PAVIS), +Istituto +Italiano +di +Tecnologia +(IIT), +Genoa, +Italy. +Email:{seyed.mohammadi, +yiming.Wang, matteo.taiana, pietro.morerio, +alessio.delbue}@iit.it +4Deep Visual Learning (DVL), Fondazione Bruno Kessler, Trento, Italy. +Fig. 1. +Overall pipeline of the proposed 3D robotic grasping strategy. We +first capture a partial PCD from a single view of the object using a depth +sensor located on the Kinova robotic arm. We then feed the single-view +PCD to the completion network and produce a completed PCD. Finally, we +generate the grasp pose and execute the grasp with feasible trajectory for +the robot. +problem but recent learning-based approaches have provided +encouraging results on different classes of objects. Initial +shape completion solutions [4], [5] converted the 3D point +cloud to a voxel grid with the rendering of additional data +that increases processing time and memory requirements. +More efficient networks [6], [7] were inspired by the Point- +Net [8] architecture that directly processes unordered PCDs. +However, most of these methods have been evaluated on +synthetic, noise-free datasets, far from real-world scenarios. +Differently, this work proposes a new model for 3D point +completion that can operate in a realistic scenario for robot +grasping with arbitrary object classes. Our method adopts a +transformer-based network [9] and it proposes a modification +of an Offset-Attention layer [10], [11] with the introduction +of skip connections that is able to complete the partial PCD +as extracted from just a single depth camera frame. By +completing the point cloud, the computation of the grasp +poses can leverage the additional information of a full PCD. +Our proposed grasping pipeline is shown in Figure 1. With +the calibrated camera equipped on the robotic arm, we first +acquire the PCD and segment the background information +using PointNet++ [12]. The segmented partial PCD of the +object is then normalised, i.e. scaled and centered, and +fed to the PCD completion network to predict the missing +geometry of the object. We then map back the predicted +point cloud in the real-world scene reference in order to +merge the predicted missing PCD with the observed partial +input. Furthermore, we generate the grasp pose on the top of +arXiv:2301.00866v1 [cs.RO] 2 Jan 2023 + +Initialisation +Single-view +PCD completion +Completed +PCD +network +PCD +Encoder +Decoder +Grasp +Trajectory +generationvirtually completed point cloud using the method proposed +in Grasp Pose Detection in Point Clouds (GPD) [13]. Finally, +we utilise Moveit! [14] to plan the arm trajectory that moves +the gripper to the pose estimated by GPD. +We first evaluate our PCD completion method on a PCD +completion benchmark dataset [4] that has been generated +on the top of YCB dataset [15], by training all the state- +of-the-art methods (from scratch) using the same dataset +(and split), and outperform the reconstruction error of the +best state-of-the-art methods. Then, we test the proposed +grasping pipeline in a real scenario using a Kinova arm, +our completion network, and GPD. Our method provides +accurate completions for successful grasp poses, which en- +close the self-occluded parts of the object. Thus, the set of +promising grasp hypothesis is larger, which improves the +overall success rate score. +To summarise, these are our main contributions: +• We propose a novel partial PCD completion network +based on the Offset-Attention encoder-decoder Trans- +former, that achieves state-of-the-art PCD completion +performance when evaluated on the partial version of +the YCB dataset proposed in [4]. +• We integrate and test our grasping pipeline with a Ki- +nova arm, showing a significant improvement in robotic +grasping success rate. +• We present extensive ablation studies on the architecture +of our proposed completion network to best justify our +design choices. +II. RELATED WORK +We mainly cover related works addressing shape comple- +tion with 3D data and robotic grasping. +3D shape completion. In environments where objects are +not placed on top of others, such as cupboards and shelves, +object shape completion can provide additional grasp poses +that augment the selection range. +Given the incomplete partial 3D data as the input, the aim +is to predict an approximation of the complete shape. 3D +shape completion methods can be categorised into geometric +and data-driven approaches [16]. Geometry-based methods +[17], [18] assume the presence of shape priors, such as ge- +ometric primitives, symmetry and structural repetition [19]. +However, the application of these priors may lead to less ac- +curate reconstructions for large-scale datasets and real-world +3D data. Data-driven (i.e. learning-based) approaches rely +on deep neural networks that discover the shape completion +priors from the data both at local and global point cloud level +[20], [16]. +In earlier works, the irregular 3D data (i.e., raw point +cloud) is converted to a regular data representation (i.e.,voxel +grid), where 3D CNNs applied on voxelized inputs have been +widely adopted for the pure 3D shape completion task [16] +and for shape completion for improving grasp estimation [5], +[4]. However, the cost of memory usage and computational +time for such methods is very large [8]. +Instead, PCN [6] directly uses raw PCD for shape comple- +tion tasks, and it is based on an encoder-decoder architecture. +The encoder is a PointNet-based backbone network that +provides global features. The decoder has two stages, one +that estimates a coarse point cloud by applying an MLP. After +that, FoldingNet [21] is used to generate the detailed and +completed point cloud. Following PCN, a range of learning- +based methods for pure 3D shape completion tasks from PCD +were proposed [7], [22], [23], [24] to improve the resolution +and robustness of the reconstructed PCD, while others [25], +[26], [27], were proposed for improving the performance +of grasp success rate by directly processing 3D PCD for +completing the shape of the object. +PoinTr [24] was the first PCD completion system to +adopt the Transformer architecture [9], leading to a signifi- +cant improvement in performance. Later, [27] introduced a +transformer-based network for object completion that con- +sists of an encoder-decoder architecture, where the encoder +is a conventional Multi-Head Self-Attention module, and +the decoder is based on the AtlasNet [28]. Although the +authors improve the reconstruction result of the GRnet [29] +network that use 3D grids to regularize unordered PCD for +point cloud completion, they do not compare their results +with PoinTr [24], which consistently outperforms GRnet. In +addition, the alignment between the partial point cloud and +the reconstructed one requires a 6D pose estimation module. +In contrast, our method accurately aligns the observed point +cloud with the reconstructed one. Additionally, according +to our experiments, we improve the reconstruction results +compared to the state-of-the-art and provide more promising +grasp poses. +Vision based Robotic Grasping. Robotic Grasping aims to +find the optimal pose of the robot’s end-effector that leads +to a successful grasp of an object. In one way, model-based +methods consider contact points and exerted forces to select +the grasps that provide more stability, but the evaluation +is usually in simulation, which suffers from a large reality +gap [30]. In the other, data-driven approaches aim to map +directly perceptual input such as RGB [31], [32] and RGB-D +images [33], [34], [35], to the grasp success. Recent methods +take advantage of model-based and data-driven approaches +by generating data samples and labels from simulations using +domain randomization [33], [34], [35]. +Current approaches are able to map 6DoF pose candi- +dates to point clouds [36], [13], [35], [37], [12], address- +ing successful grasping in cluttered scenarios. From the +perceptual point of view, segmentation of the objects is +very challenging, so these approaches start by sampling +grasp poses, followed by grasp pose score computation and +finally a refinement pose procedure. Amongst the 6DoF +grasping approaches, we select Grasp Pose Detection in Point +Clouds (GPD) [13] to be used in our system, due to the +computational efficiency of the grasp sampling and score +computation [37]. The main steps of GPD are: (i) heuristics- +based grasp candidate sampling and (ii) binary classification +of candidates by a CNN. A detailed description of GPD is +in Section III-C. + +Fig. 2. +Architecture of our point cloud completion network. Given the partial PCD as the input, we first apply FPS to the subset of the points representing +the center point CR of each local region LR. Then we use KNN to gather the points around each CR and send them to DGCNN to extract embedding +feature FE. We then send the CR to a FC layer to learn the Positional Embedding PE. Furthermore, we concatenate PE and with the corresponding FE to +be the input of the Transformer. As the output we predict the shape feature for a missing PCD PM and fed to the FoldingNet to generate high resolution +PCD, then we merge the input PCD with the predicted output PCD to shape the completed PCD PC. +III. APPROACH +We assembled the setup as a Kinova robotic arm equipped +with a RealSense depth camera and Robotiq gripper; and an +object O to be grasped. We then utilise the depth camera +to capture a depth image from a single viewpoint of the +scene. Furthermore, we convert the depth image to PCD +using camera parameters. The reconstructed PCD contains +only the visible part of the object from the camera’s point +of view (i.e partial 3D scan). +Given a partial 3D scan, containing background infor- +mation and a colourless partial PCD, we first segment the +partial PCD PP, PP = +� +PPi | PPi ∈ R3,i = 1...N +� +N=2048, +using the PCD segmentation network presented in [12]. +Then, we use our proposed completion network for pro- +cessing PP to predict the missing PCD PM, PM = +� +PMi | PMi ∈ R3,i = 1...M +� +M=6144, representing the miss- +ing point cloud of the complete shape. Finally we map +back the predicted missing PCD to the real scene and +merge it with the partial PCD. Furthermore, we generate a +grasp candidate GK on the completed PCD using the Grasp +Pose Detection (GPD) network [13], which outputs a set of +grasp poses {Gk}, GK = {GK1,GK2,...GKV}V=5 with their +corresponding classification scores CS. Lastly, the grasp with +the best classification score GKBCS that is considered feasible +by MoveIt! [14], is executed on a real robot. +In Section III-A, we introduce a dataset pre-processing +step and address the PCD alignment problem for PCD +completion task. Section III-B describes the proposed point +cloud completion network in detail and the defined loss +functions used for training the network. Finally, Section III-C +describes the grasp pose generation and evaluation network. +A. PCD alignment pre-processing +Data normalisation is a primary stage for improving the +generalisation of deep models on the learning process [38]. +However, standard PCD completion approaches use a data +normalisation that is not applicable to grasping problems. +The centroid of each PCD in training is given by the +centroid of the completed (full PCD) object, either from +CAD model [39] or the GT PCD [6]. This is not an issue in +general, as PCD completion protocols during testing provide +the partial shape aligned with the centroid of GT. Differently, +in a real testing scenario, the shape centroid can only be +computed from partial PCDs and thus being different from +the GT one. For this reason, the pre-processing of PCD in +training has to take into account that the centroid available +is only related to the partial PCD. Otherwise the completed +PCD would be misaligned as shown in the ablation studies +in Section IV-A. +In this work, we have proposed a simple but an effective +technique to solve this problem without using GT informa- +tion. Given the partial PCD {PP}, we first calculate the +translational offset vector {Tp ∈ R3}, where Tp = 1 +N ∑N +i=1 PPi. +We then calculate the centered PCD {PP}c as; PPc = +PP−Tp. Furthermore, we normalise the scale {SP ∈ R} as; +SP = maxi ∥PPi −Tp∥2, where ∥·∥2 is norm-2 and the final +normalise PCD {PPn} will define as; PPn = PPc/SP. +Tp and SP are the normalisation parameters calculated from +the partial point cloud. However, to avoid the misalignment +phenomenon, instead of separately calculating the offset and +the scaling for ground-truth point cloud PGT, we simply +apply the same parameters (i.e., Tp , Sp ) achieved by the +normalisation of PP on Pgt such that: PGT c = PGT − Tp +and PGT n = PGT c/Sp. In this way, we precisely align the +ground-truth PCD with the partial one, and also we consider +the partial PCD as a reference PCD which is the application + +Folding +Net +Output +DGCNN +Transformer Decoder +Transformer Encoder +Input +FE +PE +F1 +FC +MP + Decoder layer +Extracted feature +Positional embedding +Encoder layer +Fully connected +Max pooling +Backbone +Sum +Concatenation +Input feature +Queryfor a real-world scenario. After normalising the dataset, we +apply FPS to sample 2048 points for a partial PCD and 8192 +points for the Ground-Truth PCD. +B. Point cloud completion network +This section illustrates in detail how the proposed Trans- +former completion network predicts the missing geometry of +the 3D data. The architecture is inspired from [24], but using +an Offset-Attention [11] instead of the usual Self-Attention +encoder-decoder block, which was shown be more suitable +to process PCD given its intrinsic invariance to rigid trans- +formation. Moreover, we propose Skip-Connections among +the layer of the encoder and decoder for the better gener- +alisation of the network. The network is composed of three +main blocks: The PCD embedding, the Transformer block +consisting of the Offset-Attention encoder-decoder layer, and +the block that generates the PCD for the missing part. +1) Point cloud embedding: The Transformer architecture +requires an ordered sequence of vectors (e.g. like words in a +sentence). However, PCD is invariant to permutations, (i.e., +by changing the point sequence order there should be no +difference in the description of the shape of the object). +To address this property of PCD, in this work, we follow +the pipeline as in [24]. We divide partial PCD PP into the set +of Local Regions LR, LR = {LR1,LR2,...LRR}R=128 by ap- +plying FarthestPointSampling (FPS) [12] and then we rep- +resent the centroid as CR, +� +CRi | CRi ∈ R3,i = 1...B +� +B=128, +of each LR. We then apply KNN [40] to find the points +around each CR. Furthermore, we feed the points in each LR +into the PCD-backbone network [40] (DGCNN) to compute +the embedding feature FE, FE = {FE1,FE2,...,FEB}B=128. +We also feed each CR to the fully−connected (FC) layer +to extract the positional embedding PE (i.e., describe the +location of each subset of the points in each LR [9]) for each +FE. Finally, we concatenate the PE with the corresponding +FE to be the input FI, FI = {FI1,FI2,...FIJ}J=128 of the +Offset-attention Transformer encoder network. +2) Transformer architecture: We propose to use a multi- +head Offset-Attention encoder-decoder Transformer layer +[11], [10] for PCD completion task, since Offset-Attention +layers have been shown to be advantageous over the usual +self-attention layer on point cloud segmentation and clas- +sification. This is specially important in robotic grasping +contexts where the relative pose between the object and end- +effector is arbitrary. The real-world point cloud completion +task must be independent from the initial pose of the object +as the camera can see the object from different positions. +By using the Offset-Attention layer, we take advantage of +its invariance to rigid transformations, resulting in a more +robust object completion. Fig. 3 shows the architecture of +the Offset-Attention layer, where the offset is calculated +by measuring the difference between the input features FI +and Self-attention features SA, SA = {SA1,SA2,...SAJ} by +subtracting one from the other FIJ −SAJ. +Given a sequence of the input features FI, we formulate +the encoder as: AE = E(FI), where E is the encoder and +AE = {AE1,AE2,...AEw}W=1024 is the output feature vector +Fig. 3. +The Offset-Attention layer measure the difference between the +Attention and the input feature. +of the encoder. The Offset-Attention in the encoder layer +first updates the input features FI. Then, we feed the output +of the encoder to the FC layer, followed by a Max − +Pooling (MP) operation. Moreover, to force the encoder +to learn and generalise better about the global complete +shape information, we predict the sparse PCD PS, PS = +� +PSi | PSi ∈ R3,i = 1...S +� +S=192, where PS is the predicted +PCD, representing the complete shape of the object with +lower number of the points. We predict PS by passing the +generated feature vector AE (i.e., the output of the encoder) +to the queries Q layer that containing FC. Then we reshape +the output of the FC layer to S×3 to create PS. +On the other hand, the Offset-Attention decoder layer +D shares the exact architecture of the encoder network +except having cross-attention mechanisms [41]. We for- +mulate the decoder architecture as AD = D(Q,H), where +Q = {Q1,Q2,...QX}X=192 is a set of queries and AD = +{AD1,AD2,...ADY}Y=512 are the predicted output features +representing the feature vector of the missing point cloud. +3) Point Cloud Generation: The main objective of our +proposed PCD completion network is to predict the missing +point cloud representing the unseen part of the object. To +do that, we feed the predicted feature vector AD (i.e., the +output of the decoder) to the FC layer, followed by Max- +pooling and another FC layer. Furthermore, the output of the +last FC layer will be concatenated with the predicted sparse +point cloud PS reconstructed by query Q, and pass through +another FC layer. Then, we utilise FoldingNet [21] FN +which is able to output a high resolution PCD by applying +Fold operation on the output predicted feature vector of the +missing PCD from decoder. We can define the point cloud +generation process as: PM = FN(AD) + PS, (symbol ’+’ +represents set concatenation) where PM is the prediction of +the missing parts of the point cloud. The predicted missing +point cloud PM will be merged with the partial input point +cloud PP to shape the final complete point cloud PC where +PC = +� +PCi ∈ R3,i = 1...Z +� +Z=8192. We also fed the output +feature vector of each encoder layer (by an element-wise +summation) to the corresponding decoder layer using Skip- +Connections (see Table II for our design choice). +4) Network training: Training is achieved by summation +of the Chamfer-Distance (CD) loss between the sparse and +completed point cloud and ground-truth point cloud: + +H +H +FC +Input +Key +Query Soft-Max Map Value Feature Output subtract multiplyTABLE I +COMPARISON OF L2 CD LOSS IN DIFFERENT POINT CLOUD COMPLETION MODELS ON YCB DATASET. WE REPORT THE RESULT +OF 14 SEEN CATEGORIES AND 4 UNSEEN CATEGORIES. +Method +Avg +Drill box +Mini Soccer ball +Tomato Soup +Cleanser +Comet Bleach +Box of Sugar +Mustard +Lemon +Morton Salt +Pringles +Pitcher +Sponge +Cup +Block +Cracker Box +Banana +Stack Blocks +TopNet [42] +2.51 +2.18 +2.24 +1.98 +1.84 +1.84 +1.79 +2.01 +2.51 +1.87 +1.70 +1.99 +2.24 +2.84 +3.32 +3.56 +3.90 +7.52 +FoldingNet [21] +2.28 +2.01 +2.19 +1.81 +1.66 +1.59 +1.48 +1.87 +2.29 +1.63 +1.55 +1.82 +2.18 +2.53 +3.10 +3.14 +3.33 +7.01 +PCN [6] +2.07 +1.86 +1.95 +1.59 +1.48 +1.41 +1.62 +1.98 +1.42 +1.50 +1.46 +1.69 +2.00 +2.05 +2.89 +2.93 +2.99 +6.53 +MSN [7] +1.98 +1.81 +2.0 +1.49 +1.39 +1.44 +1.51 +1.88 +1.25 +1.38 +1.52 +1.63 +1.84 +1.91 +2.92 +2.65 +2.78 +6.41 +PoinTr [24] +1.15 +0.83 +0.95 +0.99 +0.83 +0.68 +0.64 +0.79 +0.91 +0.83 +0.61 +0.72 +0.87 +0.98 +1.46 +1.32 +1.5 +5.91 +3DSGrasp +0.92 +0.64 +1.00 +0.81 +0.52 +0.49 +0.48 +0.53 +0.99 +0.65 +0.40 +0.68 +0.51 +0.70 +1.18 +1.15 +0.98 +4.89 +TABLE II +ABLATION STUDY ON THE NETWORK DESIGNS. +Model +Skip-connection +Offset-Attention +L2 CD +A +1.15 +B +✓ +1.02 +C +✓ +0.98 +D +✓ +✓ +0.92 +TABLE III +ABLATION STUDY ON THE NORMALISATION TECHNIQUE EFFECT IN +L2 CD LOSS +Method +Baseline norm. +Our norm. +PCN +2.59 +2.07 +PoinTr +1.66 +1.15 +Ours +1.44 +0.92 +L = Lcd (PS,PGT)+Lcd (PC,PGT), +where Lcd is the Chamfer-Distance loss [6], PGT is the +ground-truth PCD, PC is the the completed PCD, and PS +is the predicted sparse PCD. We supervise both PS and PC +using ground-truth completed point cloud during training to +force both encoder and decoder about the complete shape of +the GT PCD. +C. Grasp pose generation +To generate the 6DoF grasp pose candidate for the two- +fingered gripper, we use the Grasp Pose Detection (GPD) +network introduced in [13]. GPD uniformly samples points +in a Region of Interest (ROI) at random. ROIs are selected +from an image-based object detection algorithm, but the +algorithm can be tailored to the application’s constraints. On +the randomly sampled points of the ROIs, a local search +heuristic is applied to find suitable orientations in the vicinity +of each point, so a grasp candidate GK corresponds to the +sampled point and selected orientation. Then, the candidates +GK are classified as graspable by a four-layer CNN. The +input of the CNN is a multiple view representation of the +(clipped by the gripper) point cloud. To obtain the views, +the PCD is voxelized, and the voxels are projected onto +orthogonal axes. Finally, the grasps are ranked according to +the output of the last layer of the CNN before the application +of the Soft-max function. Thus, the classification score CS +provides the ranking of GK. In real-world experiments, +grasps are executed according to their ranks. Each grasp +candidate corresponds to the goal pose of the end-effector +of the robotic arm. We use MoveIt to compute a collision- +free trajectory for the arm to reach the target pose. +Fig. 4. +Qualitative results of generated grasp proposal on the top of partial +and completed PCD of 4 objects using our PCD completion network. The +partial PCDs are acquired by the real sensor on the Kinova robotic arm. +Each candidate grasp pose generated by GPD is color-coded with green to +red representing the score from high to low. +IV. EXPERIMENTS +The experiments are divided into two parts. In Section +IV-A, we first evaluate the performance of our proposed +PCD completion network against a range of state-of-the-art +methods on the partial YCB dataset [4]. We also perform +extensive ablation studies to justify the design choices of our +PCD completion network architecture in terms of the Offset- +Attention encoder-decoder and the skip-connection. We then +present real robotic experiments in Section IV-B utilising a +Kinova robotic arm equipped with an RGB-D sensor. We +evaluate the grasp success rate of 3DSGrasp +(i.e., GPD +with our PCD completion network) in comparison to the one +using only partial PCD. A grasp is considered successful if +the object is held (i.e. does not fall) after lifting it up. +A. Evaluation on 3D shape completion +Dataset. We use the partial version of the YCB dataset [4], +which is a popular choice in PCD completion for robotic +grasping [5]. We randomly sample 50 views from the training +set (Training Views), 50 views from the holdout view set +(Holdout Views), and 50 views from the holdout models +set (Holdout Models). We evaluate the completion network +on holdout views and holdout model sets and the training +is achieved with only the training set split. As the exact + +Grasps +Grasps +Input +bu +BLTABLE IV +REAL ROBOT EXPERIMENT RESULT. +Method +Avg +Pringles +Drill box +Mustard +Mug +Cleanser +Clamps (biggest) +Drill +Jell-o +Baseball +Pitcher with lid +GPD [13] +46% +50% +0% +50% +70% +60% +30% +40% +80% +60% +20% +GPD + ours +76% +80% +80% +80% +80% +70% +70% +80% +90% +90% +40% +train/test split is unspecified in [4], for fair comparison with +the state-of-the-art methods on PCD shape completion, we +train all compared methods using our own split dataset. All +PCDs are pre-processed as described in Section III-A for +normalisation and sampling. +Comparison. We compare our proposed PCD completion +network against a range of state-of-the-art methods for shape +completion in terms of the L2 Chamfer-Distance loss [6] +(multiplied by 1000) between the reconstructed, and the +ground truth PCD. For a fair comparison, we train from +scratch (all the mentioned methods using the same dataset +and split) and test against the existing PCD completion +networks such as FoldingNet [21], PCN [6], MSN [7], and +PoinTr [24] on the partial YCB dataset using their open- +source code with their best hyper-parameters. We are unable +to fairly compare against [27] as the code is unavailable. +As shown in Table I, on average, our completion network +achieves the lowest reconstruction loss among the competi- +tors, outperforming the state-of-the-art method (i.e., PoinTr) +by +0.23. +Ablation. We perform extensive experiments to justify our +network design choices in terms of the Offset-attention and +skip-connection using the partial YCB dataset. Moreover, +we evaluate the effect of our proposed PCD alignment +processing technique on PCD completion performance. +Do the Offset-attention and skip-connection improve the +PCD completion accuracy? We evaluate the impact of our +proposed Offset-Attention encoder-decoder layer and skip +connection on the PCD reconstruction error. A set of variant +models are studied: model A is the baseline Transformer +with Self-Attention encoder-decoder layer, model B adds +Skip-connection between the encoder and decoder to the +baseline model, model C replaces Self-Attention with Offset- +Attention layer and model D adds both skip-connection and +Offset-Attention layer to the transformer. As shown in Table +II, we observed that using skip-connection can improve the +performance of the baseline model by +0.13. When using +Offset-Attention layer (model C) instead of Offset-Attention +layer (model A), we observe an improvement to 0.98. The +best result is achieved by model D when adding both skip- +connection and Offset-Attention layer to the transformer. +Does the PCD pre-processing help with PCD completion? +We evaluate the effect of our proposed PCD pre-processing +technique on point cloud completion using our completion +network, PCN [6] and PoinTr [24]. The Baseline norm. +stands for normalising the GT and partial PCD with same +formula but different parameters and Our norm. use the +parameters of partial PCD to normalise GT PCD (See Section +III-A). As shown in Table III, our network achieved the low- +est reconstruction error compared to the other two methods +using both PCD processing techniques. Moreover, there is a +large reduction in PCD reconstruction error when applying +our proposed pre-processing technique to all methods. +B. Evaluation on robotic grasping +We perform the real-world experiments utilising a Kinova +Gen3 robot equipped with a Robotiq 2F-85 gripper for +grasping and an Intel RealSense D430 depth camera to +capture the point cloud. During the experiments, an object is +placed on the table and the robot starts at a predefined initial +pose facing the object as shown in Figure 1. For each test, +the partial PCD of the object is extracted by removing the +background information using a PCD segmentation network +[12]. Then, the segmented point cloud is fed to our comple- +tion network. Finally, the GPD [13] network generates and +ranks grasp candidates for both the completed and the partial +PCD. The final grasp is chosen as the best ranked and with +a feasible solution when sent to the MoveIt! motion planner. +Each object is grasped 10 times at different poses w.r.t. +the arm in its workspace. We compare GPD vs. GPD with +3DSGrasp in Table IV, where GPD with 3DSGrasp con- +sistently outperform GPD without object PCD completion. +For example, the drill box dimensions (specially width, See +figure 4) are not captured by the partial PCD, which results +in a low success rate due to collisions with the object. With +our method, the completed PCD can better address this issue +after correctly reconstructing its shape. Another hard case +is the Pitcher with a lid given the size of the object and +the gripper’s maximum aperture. Since the lid reduces the +available grasp poses from the top, only grasps from the +handle are feasible. Nevertheless, our 3DSGrasp doubles the +successful grasps compared to the baseline with partial PCD. +V. CONCLUSIONS +In this work, we propose a new system called 3DS- +Grasp for improving the robotic grasp success rate in a +real-world experiment. The central core of the proposed +system is the PCD completion head with the ability to +complete accurately the missing geometry of the 3D objects +that have not been observed before and without moving the +camera to extract more information. We also proposed a +new way to normalize partial views of PCD, solving the +misalignment problem that improves robotic grasp success +rate and reduces PCD completion error. With the experiment, +we show that our network achieves a state-of-the-art result +on PCD completion tasks and improves the average grasp +success rate by a large margin. In future work, we will extend +our work to multi-object shape completion and grasping. +Moreover, we will investigate the possibility of a fusion of +single-view RGB image and 3D PCD within the framework +of 3DSGrasp , to boost the 3D completion accuracy and +grasp success rate. + +REFERENCES +[1] A. Ten Pas and R. 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Savarese, +“Topnet: Structural point cloud decoder,” in Proceedings of the + +IEEE/CVF Conference on Computer Vision and Pattern Recognition, +pp. 383–392, 2019. + diff --git a/WNAyT4oBgHgl3EQf8_py/content/tmp_files/load_file.txt b/WNAyT4oBgHgl3EQf8_py/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2604951d6960329bd07c7b4f6a38a87c4d194e9a --- /dev/null +++ b/WNAyT4oBgHgl3EQf8_py/content/tmp_files/load_file.txt @@ -0,0 +1,761 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf,len=760 +page_content='3DSGrasp: 3D Shape-Completion for Robotic Grasp Seyed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Mohammadi2,3 Nuno F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Duarte1 Dimitris Dimou1 Yiming Wang3,4 Matteo Taiana3 Pietro Morerio3 Atabak Dehban1 Plinio Moreno1 Alexandre Bernardino1 Alessio Del Bue3 Jos´e Santos-Victor1 Abstract— Real-world robotic grasping can be done robustly if a complete 3D Point Cloud Data (PCD) of an object is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, in practice, PCDs are often incomplete when objects are viewed from few and sparse viewpoints before the grasping action, leading to the generation of wrong or inaccurate grasp poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We propose a novel grasping strategy, named 3DSGrasp, that predicts the missing geometry from the partial PCD to produce reliable grasp poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our proposed PCD completion network is a Transformer-based encoder- decoder network with an Offset-Attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our network is inherently invariant to the object pose and point’s permutation, which generates PCDs that are geometrically consistent and completed properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Experiments on a wide range of partial PCD show that 3DSGrasp outperforms the best state-of-the- art method on PCD completion tasks and largely improves the grasping success rate in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The code and dataset will be made available upon acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' INTRODUCTION Robotic grasping has recently gained increasing attention because of its essential role in many real-world applications, such as domestic and collaborative robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The seminal work of Pas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' [1] uses 3D Point Cloud Data (PCD) to generate grasp poses directly on the available 3D object structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, in real practical scenarios we often have to rely on incomplete geometric information acquired from single or few viewpoints, which leads to drastic reduction of grasping success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Researchers bypassed this problem by acquiring complete 3D object scans [2] but this requires a feasible camera path around the object, which is time consuming to obtain and not always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Another strategy is to place additional sensors around the object of interest [3], but this is not cost- effective and it requires careful calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Instead, this paper aims at improving single-view grasping by predicting the missing geometrical structure from a partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3D shape completion is an inherently ambiguous This work has partially received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 964854;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' the FCT funding to the ISR/LARSyS Associated Laboratory UID/EEA/50009/2020 and LA/P/0083/2020 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Duarte is supported by FCT-IST fellowship grant PD/BD/135116/2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 1Vislab, Institute for Systems and Robotics—Lisboa, Instituto Superior T´ecnico, Universidade de Lisboa, Portu- gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Email:{nferreiraduarte, plinio, alex, jasv}@isr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='tecnico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='ulisboa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='pt 2Department of Marine, Electrical, Electronic and Telecommunications Engineering, University of Genoa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3Pattern Analysis & Computer Vision (PAVIS), Istituto Italiano di Tecnologia (IIT), Genoa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Email:{seyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='mohammadi, yiming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='Wang, matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='taiana, pietro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='morerio, alessio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='delbue}@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='it 4Deep Visual Learning (DVL), Fondazione Bruno Kessler, Trento, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Overall pipeline of the proposed 3D robotic grasping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We first capture a partial PCD from a single view of the object using a depth sensor located on the Kinova robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then feed the single-view PCD to the completion network and produce a completed PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, we generate the grasp pose and execute the grasp with feasible trajectory for the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' problem but recent learning-based approaches have provided encouraging results on different classes of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Initial shape completion solutions [4], [5] converted the 3D point cloud to a voxel grid with the rendering of additional data that increases processing time and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' More efficient networks [6], [7] were inspired by the Point- Net [8] architecture that directly processes unordered PCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, most of these methods have been evaluated on synthetic, noise-free datasets, far from real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Differently, this work proposes a new model for 3D point completion that can operate in a realistic scenario for robot grasping with arbitrary object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our method adopts a transformer-based network [9] and it proposes a modification of an Offset-Attention layer [10], [11] with the introduction of skip connections that is able to complete the partial PCD as extracted from just a single depth camera frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' By completing the point cloud, the computation of the grasp poses can leverage the additional information of a full PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our proposed grasping pipeline is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' With the calibrated camera equipped on the robotic arm, we first acquire the PCD and segment the background information using PointNet++ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The segmented partial PCD of the object is then normalised, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' scaled and centered, and fed to the PCD completion network to predict the missing geometry of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then map back the predicted point cloud in the real-world scene reference in order to merge the predicted missing PCD with the observed partial input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we generate the grasp pose on the top of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='00866v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='RO] 2 Jan 2023 Initialisation Single-view PCD completion Completed PCD network PCD Encoder Decoder Grasp Trajectory generationvirtually completed point cloud using the method proposed in Grasp Pose Detection in Point Clouds (GPD) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, we utilise Moveit!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' [14] to plan the arm trajectory that moves the gripper to the pose estimated by GPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We first evaluate our PCD completion method on a PCD completion benchmark dataset [4] that has been generated on the top of YCB dataset [15], by training all the state- of-the-art methods (from scratch) using the same dataset (and split), and outperform the reconstruction error of the best state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, we test the proposed grasping pipeline in a real scenario using a Kinova arm, our completion network, and GPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our method provides accurate completions for successful grasp poses, which en- close the self-occluded parts of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Thus, the set of promising grasp hypothesis is larger, which improves the overall success rate score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' To summarise, these are our main contributions: We propose a novel partial PCD completion network based on the Offset-Attention encoder-decoder Trans- former, that achieves state-of-the-art PCD completion performance when evaluated on the partial version of the YCB dataset proposed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We integrate and test our grasping pipeline with a Ki- nova arm, showing a significant improvement in robotic grasping success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We present extensive ablation studies on the architecture of our proposed completion network to best justify our design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' RELATED WORK We mainly cover related works addressing shape comple- tion with 3D data and robotic grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3D shape completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In environments where objects are not placed on top of others, such as cupboards and shelves, object shape completion can provide additional grasp poses that augment the selection range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Given the incomplete partial 3D data as the input, the aim is to predict an approximation of the complete shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3D shape completion methods can be categorised into geometric and data-driven approaches [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Geometry-based methods [17], [18] assume the presence of shape priors, such as ge- ometric primitives, symmetry and structural repetition [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, the application of these priors may lead to less ac- curate reconstructions for large-scale datasets and real-world 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Data-driven (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' learning-based) approaches rely on deep neural networks that discover the shape completion priors from the data both at local and global point cloud level [20], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In earlier works, the irregular 3D data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', raw point cloud) is converted to a regular data representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=',voxel grid), where 3D CNNs applied on voxelized inputs have been widely adopted for the pure 3D shape completion task [16] and for shape completion for improving grasp estimation [5], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, the cost of memory usage and computational time for such methods is very large [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Instead, PCN [6] directly uses raw PCD for shape comple- tion tasks, and it is based on an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The encoder is a PointNet-based backbone network that provides global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The decoder has two stages, one that estimates a coarse point cloud by applying an MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' After that, FoldingNet [21] is used to generate the detailed and completed point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Following PCN, a range of learning- based methods for pure 3D shape completion tasks from PCD were proposed [7], [22], [23], [24] to improve the resolution and robustness of the reconstructed PCD, while others [25], [26], [27], were proposed for improving the performance of grasp success rate by directly processing 3D PCD for completing the shape of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' PoinTr [24] was the first PCD completion system to adopt the Transformer architecture [9], leading to a signifi- cant improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Later, [27] introduced a transformer-based network for object completion that con- sists of an encoder-decoder architecture, where the encoder is a conventional Multi-Head Self-Attention module, and the decoder is based on the AtlasNet [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Although the authors improve the reconstruction result of the GRnet [29] network that use 3D grids to regularize unordered PCD for point cloud completion, they do not compare their results with PoinTr [24], which consistently outperforms GRnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In addition, the alignment between the partial point cloud and the reconstructed one requires a 6D pose estimation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In contrast, our method accurately aligns the observed point cloud with the reconstructed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Additionally, according to our experiments, we improve the reconstruction results compared to the state-of-the-art and provide more promising grasp poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Vision based Robotic Grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Robotic Grasping aims to find the optimal pose of the robot’s end-effector that leads to a successful grasp of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In one way, model-based methods consider contact points and exerted forces to select the grasps that provide more stability, but the evaluation is usually in simulation, which suffers from a large reality gap [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In the other, data-driven approaches aim to map directly perceptual input such as RGB [31], [32] and RGB-D images [33], [34], [35], to the grasp success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Recent methods take advantage of model-based and data-driven approaches by generating data samples and labels from simulations using domain randomization [33], [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Current approaches are able to map 6DoF pose candi- dates to point clouds [36], [13], [35], [37], [12], address- ing successful grasping in cluttered scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' From the perceptual point of view, segmentation of the objects is very challenging, so these approaches start by sampling grasp poses, followed by grasp pose score computation and finally a refinement pose procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Amongst the 6DoF grasping approaches, we select Grasp Pose Detection in Point Clouds (GPD) [13] to be used in our system, due to the computational efficiency of the grasp sampling and score computation [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The main steps of GPD are: (i) heuristics- based grasp candidate sampling and (ii) binary classification of candidates by a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' A detailed description of GPD is in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Architecture of our point cloud completion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Given the partial PCD as the input, we first apply FPS to the subset of the points representing the center point CR of each local region LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then we use KNN to gather the points around each CR and send them to DGCNN to extract embedding feature FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then send the CR to a FC layer to learn the Positional Embedding PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we concatenate PE and with the corresponding FE to be the input of the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' As the output we predict the shape feature for a missing PCD PM and fed to the FoldingNet to generate high resolution PCD, then we merge the input PCD with the predicted output PCD to shape the completed PCD PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' APPROACH We assembled the setup as a Kinova robotic arm equipped with a RealSense depth camera and Robotiq gripper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' and an object O to be grasped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then utilise the depth camera to capture a depth image from a single viewpoint of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we convert the depth image to PCD using camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The reconstructed PCD contains only the visible part of the object from the camera’s point of view (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e partial 3D scan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Given a partial 3D scan, containing background infor- mation and a colourless partial PCD, we first segment the partial PCD PP, PP = � PPi | PPi ∈ R3,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='N � N=2048, using the PCD segmentation network presented in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, we use our proposed completion network for pro- cessing PP to predict the missing PCD PM, PM = � PMi | PMi ∈ R3,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='M � M=6144, representing the miss- ing point cloud of the complete shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally we map back the predicted missing PCD to the real scene and merge it with the partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we generate a grasp candidate GK on the completed PCD using the Grasp Pose Detection (GPD) network [13], which outputs a set of grasp poses {Gk}, GK = {GK1,GK2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='GKV}V=5 with their corresponding classification scores CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Lastly, the grasp with the best classification score GKBCS that is considered feasible by MoveIt!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' [14], is executed on a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In Section III-A, we introduce a dataset pre-processing step and address the PCD alignment problem for PCD completion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Section III-B describes the proposed point cloud completion network in detail and the defined loss functions used for training the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, Section III-C describes the grasp pose generation and evaluation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' PCD alignment pre-processing Data normalisation is a primary stage for improving the generalisation of deep models on the learning process [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, standard PCD completion approaches use a data normalisation that is not applicable to grasping problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The centroid of each PCD in training is given by the centroid of the completed (full PCD) object, either from CAD model [39] or the GT PCD [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' This is not an issue in general, as PCD completion protocols during testing provide the partial shape aligned with the centroid of GT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Differently, in a real testing scenario, the shape centroid can only be computed from partial PCDs and thus being different from the GT one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' For this reason, the pre-processing of PCD in training has to take into account that the centroid available is only related to the partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Otherwise the completed PCD would be misaligned as shown in the ablation studies in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In this work, we have proposed a simple but an effective technique to solve this problem without using GT informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Given the partial PCD {PP}, we first calculate the translational offset vector {Tp ∈ R3}, where Tp = 1 N ∑N i=1 PPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then calculate the centered PCD {PP}c as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' PPc = PP−Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we normalise the scale {SP ∈ R} as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' SP = maxi ∥PPi −Tp∥2, where ∥·∥2 is norm-2 and the final normalise PCD {PPn} will define as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' PPn = PPc/SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Tp and SP are the normalisation parameters calculated from the partial point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, to avoid the misalignment phenomenon, instead of separately calculating the offset and the scaling for ground-truth point cloud PGT, we simply apply the same parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', Tp , Sp ) achieved by the normalisation of PP on Pgt such that: PGT c = PGT − Tp and PGT n = PGT c/Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In this way, we precisely align the ground-truth PCD with the partial one, and also we consider the partial PCD as a reference PCD which is the application Folding Net Output DGCNN Transformer Decoder Transformer Encoder Input FE PE F1 FC MP Decoder layer Extracted feature Positional embedding Encoder layer Fully connected Max pooling Backbone Sum Concatenation Input feature Queryfor a real-world scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' After normalising the dataset, we apply FPS to sample 2048 points for a partial PCD and 8192 points for the Ground-Truth PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Point cloud completion network This section illustrates in detail how the proposed Trans- former completion network predicts the missing geometry of the 3D data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The architecture is inspired from [24], but using an Offset-Attention [11] instead of the usual Self-Attention encoder-decoder block, which was shown be more suitable to process PCD given its intrinsic invariance to rigid trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Moreover, we propose Skip-Connections among the layer of the encoder and decoder for the better gener- alisation of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The network is composed of three main blocks: The PCD embedding, the Transformer block consisting of the Offset-Attention encoder-decoder layer, and the block that generates the PCD for the missing part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 1) Point cloud embedding: The Transformer architecture requires an ordered sequence of vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' like words in a sentence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' However, PCD is invariant to permutations, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', by changing the point sequence order there should be no difference in the description of the shape of the object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' To address this property of PCD, in this work, we follow the pipeline as in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We divide partial PCD PP into the set of Local Regions LR, LR = {LR1,LR2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='LRR}R=128 by ap- plying FarthestPointSampling (FPS) [12] and then we rep- resent the centroid as CR, � CRi | CRi ∈ R3,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='B � B=128, of each LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then apply KNN [40] to find the points around each CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, we feed the points in each LR into the PCD-backbone network [40] (DGCNN) to compute the embedding feature FE, FE = {FE1,FE2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=',FEB}B=128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We also feed each CR to the fully−connected (FC) layer to extract the positional embedding PE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', describe the location of each subset of the points in each LR [9]) for each FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, we concatenate the PE with the corresponding FE to be the input FI, FI = {FI1,FI2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='FIJ}J=128 of the Offset-attention Transformer encoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 2) Transformer architecture: We propose to use a multi- head Offset-Attention encoder-decoder Transformer layer [11], [10] for PCD completion task, since Offset-Attention layers have been shown to be advantageous over the usual self-attention layer on point cloud segmentation and clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' This is specially important in robotic grasping contexts where the relative pose between the object and end- effector is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The real-world point cloud completion task must be independent from the initial pose of the object as the camera can see the object from different positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' By using the Offset-Attention layer, we take advantage of its invariance to rigid transformations, resulting in a more robust object completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3 shows the architecture of the Offset-Attention layer, where the offset is calculated by measuring the difference between the input features FI and Self-attention features SA, SA = {SA1,SA2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='SAJ} by subtracting one from the other FIJ −SAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Given a sequence of the input features FI, we formulate the encoder as: AE = E(FI), where E is the encoder and AE = {AE1,AE2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='AEw}W=1024 is the output feature vector Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The Offset-Attention layer measure the difference between the Attention and the input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The Offset-Attention in the encoder layer first updates the input features FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, we feed the output of the encoder to the FC layer, followed by a Max − Pooling (MP) operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Moreover, to force the encoder to learn and generalise better about the global complete shape information, we predict the sparse PCD PS, PS = � PSi | PSi ∈ R3,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='S � S=192, where PS is the predicted PCD, representing the complete shape of the object with lower number of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We predict PS by passing the generated feature vector AE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', the output of the encoder) to the queries Q layer that containing FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then we reshape the output of the FC layer to S×3 to create PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' On the other hand, the Offset-Attention decoder layer D shares the exact architecture of the encoder network except having cross-attention mechanisms [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We for- mulate the decoder architecture as AD = D(Q,H), where Q = {Q1,Q2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='QX}X=192 is a set of queries and AD = {AD1,AD2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='ADY}Y=512 are the predicted output features representing the feature vector of the missing point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 3) Point Cloud Generation: The main objective of our proposed PCD completion network is to predict the missing point cloud representing the unseen part of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' To do that, we feed the predicted feature vector AD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', the output of the decoder) to the FC layer, followed by Max- pooling and another FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Furthermore, the output of the last FC layer will be concatenated with the predicted sparse point cloud PS reconstructed by query Q, and pass through another FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, we utilise FoldingNet [21] FN which is able to output a high resolution PCD by applying Fold operation on the output predicted feature vector of the missing PCD from decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We can define the point cloud generation process as: PM = FN(AD) + PS, (symbol ’+’ represents set concatenation) where PM is the prediction of the missing parts of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The predicted missing point cloud PM will be merged with the partial input point cloud PP to shape the final complete point cloud PC where PC = � PCi ∈ R3,i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='Z � Z=8192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We also fed the output feature vector of each encoder layer (by an element-wise summation) to the corresponding decoder layer using Skip- Connections (see Table II for our design choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 4) Network training: Training is achieved by summation of the Chamfer-Distance (CD) loss between the sparse and completed point cloud and ground-truth point cloud: H H FC Input Key Query Soft-Max Map Value Feature Output subtract multiplyTABLE I COMPARISON OF L2 CD LOSS IN DIFFERENT POINT CLOUD COMPLETION MODELS ON YCB DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' WE REPORT THE RESULT OF 14 SEEN CATEGORIES AND 4 UNSEEN CATEGORIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Method Avg Drill box Mini Soccer ball Tomato Soup Cleanser Comet Bleach Box of Sugar Mustard Lemon Morton Salt Pringles Pitcher Sponge Cup Block Cracker Box Banana Stack Blocks TopNet [42] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='24 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='89 TABLE II ABLATION STUDY ON THE NETWORK DESIGNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Model Skip-connection Offset-Attention L2 CD A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='15 B ✓ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='02 C ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='98 D ✓ ✓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='92 TABLE III ABLATION STUDY ON THE NORMALISATION TECHNIQUE EFFECT IN L2 CD LOSS Method Baseline norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Our norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' PCN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='07 PoinTr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='15 Ours 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='92 L = Lcd (PS,PGT)+Lcd (PC,PGT), where Lcd is the Chamfer-Distance loss [6], PGT is the ground-truth PCD, PC is the the completed PCD, and PS is the predicted sparse PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We supervise both PS and PC using ground-truth completed point cloud during training to force both encoder and decoder about the complete shape of the GT PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Grasp pose generation To generate the 6DoF grasp pose candidate for the two- fingered gripper, we use the Grasp Pose Detection (GPD) network introduced in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' GPD uniformly samples points in a Region of Interest (ROI) at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' ROIs are selected from an image-based object detection algorithm, but the algorithm can be tailored to the application’s constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' On the randomly sampled points of the ROIs, a local search heuristic is applied to find suitable orientations in the vicinity of each point, so a grasp candidate GK corresponds to the sampled point and selected orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, the candidates GK are classified as graspable by a four-layer CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The input of the CNN is a multiple view representation of the (clipped by the gripper) point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' To obtain the views, the PCD is voxelized, and the voxels are projected onto orthogonal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, the grasps are ranked according to the output of the last layer of the CNN before the application of the Soft-max function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Thus, the classification score CS provides the ranking of GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In real-world experiments, grasps are executed according to their ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Each grasp candidate corresponds to the goal pose of the end-effector of the robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We use MoveIt to compute a collision- free trajectory for the arm to reach the target pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Qualitative results of generated grasp proposal on the top of partial and completed PCD of 4 objects using our PCD completion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The partial PCDs are acquired by the real sensor on the Kinova robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Each candidate grasp pose generated by GPD is color-coded with green to red representing the score from high to low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' EXPERIMENTS The experiments are divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' In Section IV-A, we first evaluate the performance of our proposed PCD completion network against a range of state-of-the-art methods on the partial YCB dataset [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We also perform extensive ablation studies to justify the design choices of our PCD completion network architecture in terms of the Offset- Attention encoder-decoder and the skip-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We then present real robotic experiments in Section IV-B utilising a Kinova robotic arm equipped with an RGB-D sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We evaluate the grasp success rate of 3DSGrasp (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', GPD with our PCD completion network) in comparison to the one using only partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' A grasp is considered successful if the object is held (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' does not fall) after lifting it up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Evaluation on 3D shape completion Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We use the partial version of the YCB dataset [4], which is a popular choice in PCD completion for robotic grasping [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We randomly sample 50 views from the training set (Training Views), 50 views from the holdout view set (Holdout Views), and 50 views from the holdout models set (Holdout Models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We evaluate the completion network on holdout views and holdout model sets and the training is achieved with only the training set split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' As the exact Grasps Grasps Input bu BLTABLE IV REAL ROBOT EXPERIMENT RESULT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Method Avg Pringles Drill box Mustard Mug Cleanser Clamps (biggest) Drill Jell-o Baseball Pitcher with lid GPD [13] 46% 50% 0% 50% 70% 60% 30% 40% 80% 60% 20% GPD + ours 76% 80% 80% 80% 80% 70% 70% 80% 90% 90% 40% train/test split is unspecified in [4], for fair comparison with the state-of-the-art methods on PCD shape completion, we train all compared methods using our own split dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' All PCDs are pre-processed as described in Section III-A for normalisation and sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We compare our proposed PCD completion network against a range of state-of-the-art methods for shape completion in terms of the L2 Chamfer-Distance loss [6] (multiplied by 1000) between the reconstructed, and the ground truth PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' For a fair comparison, we train from scratch (all the mentioned methods using the same dataset and split) and test against the existing PCD completion networks such as FoldingNet [21], PCN [6], MSN [7], and PoinTr [24] on the partial YCB dataset using their open- source code with their best hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We are unable to fairly compare against [27] as the code is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' As shown in Table I, on average, our completion network achieves the lowest reconstruction loss among the competi- tors, outperforming the state-of-the-art method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=', PoinTr) by +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We perform extensive experiments to justify our network design choices in terms of the Offset-attention and skip-connection using the partial YCB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Moreover, we evaluate the effect of our proposed PCD alignment processing technique on PCD completion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Do the Offset-attention and skip-connection improve the PCD completion accuracy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We evaluate the impact of our proposed Offset-Attention encoder-decoder layer and skip connection on the PCD reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' A set of variant models are studied: model A is the baseline Transformer with Self-Attention encoder-decoder layer, model B adds Skip-connection between the encoder and decoder to the baseline model, model C replaces Self-Attention with Offset- Attention layer and model D adds both skip-connection and Offset-Attention layer to the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' As shown in Table II, we observed that using skip-connection can improve the performance of the baseline model by +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' When using Offset-Attention layer (model C) instead of Offset-Attention layer (model A), we observe an improvement to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The best result is achieved by model D when adding both skip- connection and Offset-Attention layer to the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Does the PCD pre-processing help with PCD completion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We evaluate the effect of our proposed PCD pre-processing technique on point cloud completion using our completion network, PCN [6] and PoinTr [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The Baseline norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' stands for normalising the GT and partial PCD with same formula but different parameters and Our norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' use the parameters of partial PCD to normalise GT PCD (See Section III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' As shown in Table III, our network achieved the low- est reconstruction error compared to the other two methods using both PCD processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Moreover, there is a large reduction in PCD reconstruction error when applying our proposed pre-processing technique to all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Evaluation on robotic grasping We perform the real-world experiments utilising a Kinova Gen3 robot equipped with a Robotiq 2F-85 gripper for grasping and an Intel RealSense D430 depth camera to capture the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' During the experiments, an object is placed on the table and the robot starts at a predefined initial pose facing the object as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' For each test, the partial PCD of the object is extracted by removing the background information using a PCD segmentation network [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Then, the segmented point cloud is fed to our comple- tion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Finally, the GPD [13] network generates and ranks grasp candidates for both the completed and the partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The final grasp is chosen as the best ranked and with a feasible solution when sent to the MoveIt!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' motion planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Each object is grasped 10 times at different poses w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' the arm in its workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We compare GPD vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' GPD with 3DSGrasp in Table IV, where GPD with 3DSGrasp con- sistently outperform GPD without object PCD completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' For example, the drill box dimensions (specially width, See figure 4) are not captured by the partial PCD, which results in a low success rate due to collisions with the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' With our method, the completed PCD can better address this issue after correctly reconstructing its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Another hard case is the Pitcher with a lid given the size of the object and the gripper’s maximum aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Since the lid reduces the available grasp poses from the top, only grasps from the handle are feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' Nevertheless, our 3DSGrasp doubles the successful grasps compared to the baseline with partial PCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' CONCLUSIONS In this work, we propose a new system called 3DS- Grasp for improving the robotic grasp success rate in a real-world experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' The central core of the proposed system is the PCD completion head with the ability to complete accurately the missing geometry of the 3D objects that have not been observed before and without moving the camera to extract more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' We also proposed a new way to normalize partial views of PCD, solving the misalignment problem that improves robotic grasp success rate and reduces PCD completion error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAyT4oBgHgl3EQf8_py/content/2301.00866v1.pdf'} +page_content=' With the experiment, we show that our network achieves a state-of-the-art 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Rossi,2 Nicola Tamanini,1 and Giulia Cusin3,4 +1 Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3), Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France +2 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, the Netherlands +3 Institut d’Astrophysique de Paris, Sorbonne Université, CNRS, UMR 7095, 98 bis bd Arago, 75014 Paris, France +4 Université de Genéve, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genéve 4, Switzerland +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +In this work, we investigate the effect of gravitational lensing on the gravitational wave (GW) signals of a population of tidal +disruption events (TDEs). We estimate the number of lensed-magnified signals that we expect to detect with future space-based +GW observatories, in particular LISA and DECIGO. We model the lens distribution using an hybrid approach that combines +semi-analytic methods with numerical results from ray tracing simulations. We divide the TDE population in two classes, nuclear +TDEs (main sequence stars tidally disrupted by massive black holes in the cores of galaxies) and globular TDEs (white dwarfs +tidally disrupted by intermediate mass black holes in globular clusters). We find that, even considering the effect of lensing, +LISA will not be able to observe any TDEs, while DECIGO could detect ∼10 strongly lensed (𝜇 > 2) globular TDEs and ∼130 +strongly lensed nuclear TDEs, over an observational period of 10 years. Our results reveal the role that lensing will play in future +deci-Hertz GW observatories, indicating exciting multi-messenger opportunities with TDEs but at same time signalling the need +to develop adequate data analysis techniques to correctly reconstruct the astrophysical properties of the source. +Key words: gravitational lensing: strong – gravitational waves – transients: tidal disruption events +1 INTRODUCTION +Stars orbiting around a massive black hole (BH) can be shred into +pieces due to tides induced by the BH’s gravitational field. We call +these extreme phenomena tidal disruption events (TDEs; Luminet +& Carter 1986; Rees 1988; Phinney 1989; for a recent review see +Rossi et al. 2020). Thanks to the bright electromagnetic (EM) flares +produced by the stellar debris during the following accretion, TDEs +have been one of the most powerful ways to reveal the presence of +otherwise quiescent massive BHs. To date, we have around 100 ob- +servations of these events, in different bands of the EM spectrum +(see for recent reviews: Saxton et al. 2020 for X-ray; van Velzen et al. +2020 for optical; Alexander et al. 2020 for radio and all the references +therein). +Besides being multiband emitters in the EM spectrum, TDEs are +potential multimessenger sources. Recently, a few TDEs have been +claimed to be associated to observed astrophysical neutrinos (Stein +2022; Reusch et al. 2022) as theoretically expected in the presence +of jets (Hayasaki 2021). Moreover, TDEs are also predicted to emit +gravitational waves (GWs). In particular, we can distinguish between +three main GW contributions: GWs due to the internal stellar mass +quadrupole, generated by the stretching and compressing action of +the BH tidal forces (see, e.g, Stone et al. 2013); GWs associated +to the BH-star system mass quadrupole, emitted during the disrup- +tion phase (see, e.g., Kobayashi et al. 2004; Toscani et al. 2020, +2022); GWs produced at later stages, along the circularization pro- +cess (Toscani et al. 2022) and in presence of an accretion disc (see, +∗E-mail: martina.toscani@l2it.in2p3.fr +e.g., Toscani et al. 2019). For standard values of the parameters in- +volved, the strongest gravitational contribution is the burst emitted +during the disruption phase, that has typical frequencies in the range +10−3 − 10−2 Hz. +Being low frequency GW sources, TDEs could be detected by +future space-based interferometers such as the Laser Interferometer +Space Antenna (LISA - Amaro-Seoane et al. 2017, 2022), currently +scheduled for launch in the mid-2030s, and the proposed deci-Hertz +Gravitational Observatory (DECIGO - Sato et al. 2017). A detailed +study about TDEs detectability with these next-generation detectors +has been carried out in Pfister et al. 2022. Their work shows that while +detection of individual TDEs by LISA seems unlikely, these events +are promising sources for deci-Hz observatories. Future instruments +with a DECIGO-like sensitivity (e.g., DECIGO, BBO - Harry et al. +2006, ALIA - Baker et al. 2019, DO - Sedda et al. 2020, 2021) could +observe hundreds of thousands TDEs per year. +Given these expectations, it becomes relevant to assess the effects +that a distribution of lenses produces on the GW emission from a +TDE population. We refer to gravitational lensing (see, e.g., Schnei- +der et al. 1992) when a massive object (i.e. the lens), which lies +along the line of sight between the observer and the source, curves +the surrounding space-time, causing the signal to deviate from its +original path. This effect has interesting consequences: for example +it may (de-)magnify the signal or produce multiple images of the +source. Moreover, different images typically arrive at the detector at +different times (time-delay effect) and they interfere if the duration +of the signal is larger than the typical delay in the time of arrival. +In this work, we study for the first time the effect of gravitational +lensing on a TDE population and provide estimates on the expected +© 2022 The Authors +arXiv:2301.01804v1 [astro-ph.HE] 4 Jan 2023 + +2 +M.Toscani et al. +number of observed lensed-magnified TDEs. We consider both LISA, +for which a magnified TDE could be the only way to have a signal +above the detectability threshold, but also DECIGO, for which the +ability to distinguish lensed TDEs would avoid errors in the recon- +struction of the parameters describing the source (e.g., distance and +mass), as well as provide additional information on the astrophysical +properties of the source and lens populations. We perform this inves- +tigation dividing TDEs in two different classes: nuclear TDEs, where +we consider main sequence (MS) stars disrupted by massive BHs in +the cores of galaxies, and globular TDEs, where we consider white +dwarfs (WDs) tidally disrupted by intermediate mass BHs (IMBHs) +located in globular cluster (GCs). The structure of the paper is the +following: in Section 2 we illustrate the basis of gravitational lens- +ing, in Section 3 we describe the distribution of lenses and the TDE +populations in details, in Section 4 we show and discuss the results +and finally in Section 5 we draw our conclusions. +Throughout this work, we adopt a ΛCDM cosmological model, +with matter density parameter Ωm = 0.274, dark energy density +parameter ΩΛ = 0.726 and Hubble constant 𝐻0 = 70 Km/sMpc. +2 GRAVITATIONAL LENSING IN A NUTSHELL +We want to determine the number of TDEs that, having a given +strain amplitude, or rather a given signal to noise ratio (SNR) 𝜌 +for a specified interferometer, are significantly magnified by lensing. +Following Cusin et al. (2021), we present semi-analytic formulae +which can be applied to an arbitrary lens and source distribution, +keeping full control of modeling and transparency of all physical +effects. We mainly focus our analysis on strong lensing, working out +the distribution of magnification for 𝜇 > 1, considering LISA and +DECIGO. It is indeed true that the lensing induced by the Cosmolog- +ical Large Scale Structure can also lead to de-magnification (𝜇 < 1) +of a signal. However de-magnification from the Large Scale Structure +usually does not reach values 𝜇 ≪ 1, meaning that its contribution +should not significantly affect the detection rates of the observed pop- +ulation. Furthermore the TDE population is well below the detection +threshold of LISA, hence de-magnified events will anyway remain +undetected. For these reasons in what follows we ignore the effect of +de-magnification. +Our description of strong lensing relies on the geometric optics +approximation (we do not describe wave effects such as diffraction +and interference). This is a well-justified approximation when look- +ing at TDEs lensed by a population of galaxies. Indeed, given a lens +of mass 𝑀l, diffraction effects are relevant when (see, e.g, Takahashi +& Nakamura 2003b) +𝑀l ≲ 108M⊙ +� +𝑓 +mHz +�−1 +, +(1) +where 𝑓 is the frequency of the lensed signal. Since GWs by TDEs +have typical frequencies in the range 10−3 − 10−2 Hz, wave effects +can be neglected in our work, as we consider galaxy stellar masses +between 108 − 1012M⊙.. +To predict the number of magnified TDEs observable with a given +instrument, we need to take the following steps: +(i) choose a lens model, and a model to describe the lens distri- +bution and the population of sources; +(ii) derive the probability density function (PDF) for a generic +source at redshift 𝑧s to be amplified more than 𝜇 by the population +of foreground lenses; +Figure 1. Sketch of the geometry for a SIS lens. +(iii) convolve the magnification PDF with the population of ob- +servable events for a given instrument, in presence of magnification +𝜇. +In the following, we model the lenses as singular isothermal +spheres (SIS), that we describe in detail in Section 2.1. While SIS +are not very realistic when considering lensing by individual clusters, +they are sufficient for statistical purposes. In particular, the main ad- +vantage of the SIS model is that it can be studied analytically, which +allows us to have a better (even tough idealised) comprehension of +the physics behind lensing. To follow a more realistic approach in +addition to strong lensing, described by our semi-analytic approach, +we also consider the contribution from weak lensing due to the grav- +itational potential of the Large Scale Structure. To derive numerical +results, we use ray-tracing simulations by Takahashi et al. (2011), +which include both weak lensing contribution, and strong lensing +tails in the magnification PDF. +2.1 SIS and gravitational lensing statistics +In the SIS model all the mass components of the galaxy behave like +particles of an ideal gas, confined by their combined, spherically +symmetric gravitational potential, in thermal equilibrium (see, e.g., +Narayan & Bartelmann 1996). The mass density of a SIS is described +by +𝜚(𝑟) = +𝜎2v +2𝜋𝐺𝑟2 , +(2) +where 𝜎v is the velocity dispersion in the galaxy, 𝐺 is the gravitational +constant and 𝑟 is the distance from the center. Integrating Equation +(2) along the line of sight, we get the surface mass density +Σ(𝜉) = 𝜎2v +2𝐺𝜉 , +(3) +where 𝜉 is the bi-dimensional vector in the lens plane, called lens +impact parameter. The geometry of such system is illustrated in +Figure 1, where the following elements are shown: +(i) the source, its lensed image, the observer (e.g., LISA) and the +lens, +(ii) 𝛾, angle between the line of sight and the unlensed source, +MNRAS 000, 1–11 (2022) + +source +image +n +dis +lens +Sm +ds +α +di +observer +(e.g.,LISA)Lensed TDEs +3 +(iii) 𝜃, angle between the line of sight and the lensed image, +(iv) ˆ𝛼 deflection angle induced by the presence of the lens, +(v) 𝑑𝑥, angular diameter distances between the lens and the source +(x=ls), the lens and the observer (x=l), the source and the observer +(x=s), +(vi) 𝜉 = 𝜃𝑑l, lens impact parameter, in the lens plane, +(vii) 𝜂 = 𝛾𝑑s, source impact parameter, in the source plane. +We recall (see, e.g, Hogg 1999) that the angular diameter distance of +an object is related to its luminosity distance, 𝐷, by +𝑑 = +𝐷 +(1 + 𝑧)2 , +(4) +and the angular diameter distance between two objects at redshifts +𝑧1 and 𝑧2 is +𝑑12 = +1 +1 + 𝑧2 +[𝜒(𝑧2) − 𝜒(𝑧1)] = +1 +1 + 𝑧2 +𝜒(𝑧1, 𝑧2), +(5) +where 𝜒 is the comoving distance +𝜒(𝑧) = 𝑐 +𝐻0 +∫ 𝑧 +0 +𝑑𝑧′ +𝐸(𝑧′) . +(6) +In +the +above +Equation, +we +have +introduced +𝐸(𝑧′) += +(Ωm(1 + 𝑧′)3 + Ω𝜆)1/2, where Ωm and Ω𝜆 are the present values +of the matter and cosmological constant density contrasts and 𝑐 is +the speed of light in vacuum. +The basic quantity for statistical analysis is the cross section of +the lens for producing the desired lensing effect (e.g. magnification +larger than 𝜇). The corresponding optical depth is the fraction of +the sky where, given the lenses, one can place a source and observe +this magnification (see, e.g, Kochanek 2006). In the case of the SIS, +the area on the source plane in which a source at redshift 𝑧s will +be magnified ≥ 𝜇 is given by 𝜍sis(𝜇, 𝑧s, 𝑧l, 𝜎v) = 𝜋𝜂2(𝜇, 𝑧s, 𝑧l, 𝜎v). +The corresponding optical depth is +𝜏(𝜇, 𝑧s) = +∫ 𝑧s +0 +𝑑𝑧l +𝑑𝑟 +𝑑𝑧l +� 𝑑l +𝑑𝑠 +�2 ∫ +𝑑𝜎v𝑛(𝜎v, 𝑧l)𝜍sis(𝜇, 𝑧l, 𝑧s, 𝜎v) , +(7) +where 𝑑𝑟 is the physical length element at redshift 𝑧l, while 𝑛(𝜎v, 𝑧l) +is the physical number density of lenses per bin of 𝜎v (Kochanek +2006). +In the SIS model, there are two lensed images when the source +satisfy the following criterion (Schneider et al. 1992) +𝛾 < 𝛼0 = 4𝜋 𝜎2v +𝑐2 +𝑑ls +𝑑s +, +(8) +where 𝛼0 is usually called Einstein angle. In our study we consider +this scenario, but we restrict our attention to the primary, i.e. more +magnified, image1, for which we provide the cross section and the +explicit final formula for 𝜏 in Appendix A. This choice is justified +since we expect to see a short burst of GWs from a TDE which +comes only from one image. The second image is in fact delayed +in time, with typical time delay of the order of a few months (see +e.g. Oguri 2018), much longer than the GW burst itself. The problem +of correctly identifying secondary images, and associating them to +their primaries, requires a dedicated data analysis investigation which +goes beyond the scope of our study. +1 In lensing analyses it is common jargon to refer to the observed signals as +“images”, even though they are not EM signals. In this paper we follow this +convention, calling images the lensed GW signals. +The probability that a source at redshift 𝑧s is magnified more than +𝜇 is +𝑃(> 𝜇, 𝑧s) = 1 − exp(−𝜏(𝜇, 𝑧s)) = +∫ +∞ +𝜇 +𝑝(𝜇, 𝑧s)𝑑𝜇, +(9) +where +𝑝(𝜇, 𝑧s) = − 𝑑𝜏 +𝑑𝜇 exp(−𝜏(𝜇, 𝑧s)) +(10) +is the magnification PDF. +To understand Equation (9) note that 𝑑𝜏/𝑑𝑧 can be interpreted as a +sort of GW scattering rate leading to magnification larger than 𝜇 (per +bin of redshift). Hence the probability for magnification larger than 𝜇 +satisfies the differential equation 𝑑𝑃(> 𝜇, 𝑧)/𝑑𝑧 = (1−𝑃)𝑑𝜏/𝑑𝑧 with +solution (9). The factor (1 − 𝑃) is essential to keep the probability +normalized also when 𝜏 becomes large. In the limit of small optical +depth, 𝑃(> 𝜇, 𝑧𝑠) ≈ 𝜏(𝜇, 𝑧𝑠). +As mentioned above, note that our approach does not describe +de-magnification which happens when a signal crosses a cosmic +under-density. +2.2 Gravitational lensing applied to a source population +We consider a population of sources, that we describe as a function of +source redshift 𝑧s and SNR 𝜌 and we denote the number of sources +per bin of redshift and SNR as 𝑑N/(𝑑𝜌𝑑𝑧s). If the magnification +is 𝜇, the number of observable sources per bin of 𝑧s, for a given +interferometer, reads +𝑑N (𝜇, 𝑧s) +𝑑𝑧s += +∫ ∞ +𝜌lim/√𝜇 +𝑑N +𝑑𝜌𝑑𝑧s +𝑑𝜌 , +(11) +where 𝜌lim is the threshold above which we have a GW detection. In +the rest of the paper, we take 𝜌lim = 8. +Convolving this quantity with the magnification PDF (Equation +10), we get the total number of observed objects in presence of +magnification +Nobs = +∫ 𝑧max +0 +𝑑𝑧s +𝑑N +𝑑𝑧s +(𝑧s) = +∫ 𝑧max +0 +𝑑𝑧s +∫ +∞ +𝜇min +𝑑𝜇𝑝(𝜇, 𝑧s) 𝑑N (𝜇, 𝑧s) +𝑑𝑧s +, +(12) +where 𝑧max corresponds to the maximum redshift at which we expect +to find sources and 𝜇min is the minimum value of magnification +considered. The probability that if an instrument sees a source from +redshift 𝑧s this is magnified more than 𝜇 is given by +P(𝑧s, 𝜇) = C +∫ ∞ +𝜇 +𝑝(𝜇′, 𝑧s) 𝑑N (𝜇′, 𝑧s) +𝑑𝑧s +𝑑𝜇′ , +(13) +where C is a normalization constant. +3 METHODS +In this section we illustrate how we build our model for lens and +source distributions. +3.1 Lens distribution +We model the number density of galaxies (lenses) as a function of +redshift and of the velocity dispersion 𝜎v. One can show that if the +MNRAS 000, 1–11 (2022) + +4 +M.Toscani et al. +101 +102 +10 +11 +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +p( , zs) +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +Figure 2. Magnification PDF for different values of source redshift: 𝑧s = 0.28 +(light blue), 𝑧s = 0.49 (orange), 𝑧s = 1.10 (green), 𝑧s = 01.72 (red). +evolution of sources is neglected, 𝜏 reduces to (see Cusin et al. 2019a) +𝜏(𝜇, 𝑧𝑠) ≃ +0.001 +(𝜇 − 1)2 +� 𝐻0𝜒(𝑧𝑠) +𝑐 +�3 � +𝑁𝑐3 +109𝐻3 +0 +⟨𝜎4𝑣⟩ +𝑐4 × 5 × 10−14 +� +, +(14) +where 𝑁 is the present galaxy density, +𝑁 = +∫ ∞ +0 +𝑑𝜎𝑣𝑛(𝜎𝑣, 𝑧 = 0) , +(15) +and ⟨𝜎4v ⟩ a mean of the velocity dispersion to power 4. Crude esti- +mates for 𝑁 and 𝜎v are +𝑁 = 109 𝐻3 +0 +𝑐3 , +⟨𝜎4 +v ⟩ = (150 km/s)4. +(16) +Then the PDF magnification reads +𝑝(𝜇, 𝑧s) = 2𝑝1(𝑧s) +(𝜇 − 1)3 exp +� 𝑝1(𝑧s) +(𝜇 − 1)2 +� +, +𝑝1(𝑧s) = 0.001 +� 𝐻0𝜒(𝑧s) +𝑐 +� +. +(17) +As shown in Cusin et al. (2021), this simplified analytic model gives +a result for optical depth in good agreement with the one obtained +considering a more realistic distribution of lenses, which evolve with +redshift (fractional deviations of a few percent). +We add to this strong lensing PDF the contribution of weak lens- +ing due to the gravitational potential of the Large Scale Structure, +which allows us to build a more realistic lens distribution. To do so, +we use results for the magnification probability density of Takahashi +et al. 2011 (for 𝑧 > 1), which reconstruct the path of light through +inhomogeneous clumps of matter in the Universe via high-resolution +ray-tracing approximation. We interpolate their results2 for the red- +2 The probability densities from Takahashi et al. 2011 are available +on this website http://cosmo.phys.hirosaki-u.ac.jp/takahasi/ +raytracing/open_data/ +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +f(Hz) +10 +24 +10 +23 +10 +22 +10 +21 +10 +20 +10 +19 +hc +LISA +DECIGO +Figure 3. Sensitivity curves for LISA (blue) and DECIGO (magenta). +shift values we want to study. In their simulations they used the box +size of 50ℎ−1 Mpc with 10243 particles, the mean particle separation +of 50ℎ−1 kpc, and the softening length of 2ℎ−1 kpc. +The overall magnification PDF that we obtain is shown in Figure 2, +for some selected values of 𝑧s. From this plot, we see that this function +in general increases with the source redshift, which is reasonable +since for bigger 𝑧s we expect more foreground lenses between the +source and the observer. Furthermore, over the magnification interval +1 ≤ 𝜇 ≤ 500, the function decreases very steeply, showing a 10 +order-of-magnitude lowering, which shows how higher values of the +magnification are generally suppressed in favor of lower values. +3.2 Source population +Before illustrating how we built the source population, we briefly +recall the main formulas for the TDE gravitational emission. +The GWs associated with the disruption of a star can be approxi- +mated as a monochromatic burst, with strain and frequency given by +(see, e.g., Toscani et al. 2022) +ℎgw ≈ 2 × 10−22𝛽 × +� M∗ +M⊙ +�4/3 +× +� +𝑀bh +106M⊙ +�2/3 +× +� 𝑅∗ +R⊙ +�−1 +× +� +𝜒 +16 Mpc +�−1 +, +(18) +𝑓gw ≈ 𝛽3/2 × 10−4 Hz × +� 𝑀∗ +M⊙ +�1/2 +× +� 𝑅∗ +R⊙ +�−3/2 +, +(19) +where 𝑀∗ and 𝑅∗ are the stellar mass and radius, 𝑀bh the BH mass, +M⊙ and R⊙ the solar mass and radius, and 𝛽 = 𝑅t/𝑅p is the pene- +tration factor, i.e. the ratio between the maximum distance from the +BH to have a full disruption, a.k.a. the tidal radius +𝑅t ≈ 7 × 1012 cm × +� 𝑅∗ +𝑅⊙ +� +× +� +𝑀bh +106M⊙ +�1/3 +× +� 𝑀∗ +M⊙ +�−1/3 +, +(20) +MNRAS 000, 1–11 (2022) + +Lensed TDEs +5 +and the stellar pericenter 𝑅p. Requiring 𝑅p bigger than BH +Schwarschild radius, we get the following limits for 𝛽 +1 ≲ 𝛽 ≲ 𝛽max ≈ 20 × +� 𝑅∗ +R⊙ +� +× +� 𝑀∗ +M⊙ +�−1/3 +× +� +𝑀bh +106M⊙ +�−2/3 +. +(21) +The SNR 𝜌 for such a signal can be written as (Pfister et al. 2022, +see also Appendix B) +𝜌 = +ℎgw +ℎc( 𝑓gw/(1 + 𝑧)) = += 𝛽 × 2 × 10−22 × +� 𝑀∗ +𝑀⊙ +�4/3 +× +� +𝑀bh +106M⊙ +�2/3 +× +� 𝑅∗ +R⊙ +�−1 +× +� +𝜒 +16 Mpc +�−1 +× +1 +ℎc( 𝑓gw/(1 + 𝑧)) , +(22) +where ℎc is the characteristic noise of the instrument (Moore et al. +2015). Throughout this work, we consider the sensitivity curves of +LISA (LISA Science Study Team 2018) and DECIGO (Sato et al. +2017), illustrated in Figure 3. +In this study, when considering MS stars, we adopt the following +approximated scaling relation (Kippenhahn & Weigert 1990) +𝑀ms +𝑀⊙ +≈ 𝑅ms +R⊙ +, +(23) +while for the WD case we assume fixed values for the mass and +radius, 𝑀wd = 0.5M⊙, 𝑅wd = 0.01R⊙. +Note that, from this point forward, we use 𝑀bh ≡ 𝑀•, when +referring to massive BHs residing in galaxy cores, while we write +𝑀bh ≡ 𝑀h when referring to IMBHs located in GCs. +3.2.1 Nuclear TDEs +We build the population of MS stars tidally disrupted by massive +BHs residing in galaxy cores following the same steps as in Toscani +et al. 2020, +𝑑Nms +𝑑𝑧𝑑𝑀•𝑑𝑀★𝑑𝛽 = 4𝜋𝑐𝜒(𝑧)2 +𝐻0𝐸(𝑧) Φ(𝑀•)𝜓(𝛽)𝜙(𝑀∗) Γ(𝑀•) +(1 + 𝑧)𝑇, +(24) +where we have the following terms: +(i) the comoving volume term 4𝜋𝑐𝜒(𝑧)2/𝐻0𝐸(𝑧); +(ii) the distribution of nuclear massive BHs that we build from +a Schechter mass function with z-dependence (see Gabasch et al. +2006), expressed in terms of 𝑀• using the Faber-Jackson relation +(Faber & Jackson 1976) and the 𝑀• − 𝜎 relation (McConnell & Ma +2013), as done in Toscani et al. 2020, +𝜙(𝑀•) = +0.003Mpc−3 +(1 + 𝑧)0.48108M⊙ +× +� +𝑀• +108M⊙ +�−1.24 +(25) +× exp +� +− +0.59 +(1 + 𝑧)0.7 +� +𝑀• +108M⊙ +�0.7� +; +(iii) the normalized distribution for 𝛽 (Stone & Metzger 2016) +𝜓(𝛽) = +𝛽max(𝑀•, 𝑀∗) +𝛽2(𝛽(𝑀•, 𝑀∗max) − 1) ; +(26) +(iv) the normalized Salpeter initial stellar mass function (Salpeter +1955) +𝜙(𝑀∗) = +1.35 +𝑀−1.35 +∗min − 𝑀−1.35 +∗max +𝑀−2.35 +∗ +; +(27) +(v) the galaxy rate for nuclear TDEs (see, e.g, Stone & Metzger +2016) +Γ(𝑀•) = 2.9 × 10−5 /(yr gal) +� +𝑀• +108M⊙ +�−0.404 +; +(28) +(vi) the observation time𝑇, that we take equal to the lifetime of the +mission. Here we assume, both for LISA and DECIGO, 𝑇 = 10 yr. +3.2.2 Globular TDEs +We build the population of WDs tidally disrupted by IMBHs located +in GCs in a similar way as done in Toscani et al. 2020, +𝑑Nwd +𝑑𝑧𝑑𝑀•𝑑𝛽 = 4𝜋𝑐𝜒(𝑧)2 +𝐻0𝐸(𝑧) Φ(𝑀•)𝑁gc +gal(𝑀•)𝜓(𝛽) Π(𝑀h, 𝑀wd) +(1 + 𝑧) +𝑇, (29) +where we have the following terms: +(i) the comoving volume term 4𝜋𝑐𝜒(𝑧)2/𝐻0𝐸(𝑧); +(ii) the distribution of nuclear massive BHs, Φ(𝑀•); +(iii) a scaling relation between the number of GCs per galaxy and +the mass of the BH in the core (Harris & Harris 2011; Burkert & +Tremaine 2010) +𝑁gc +gal = +𝑀• +4.07 × 105𝑀⊙ +; +(30) +(iv) the rate of globular TDEs per GCs (Baumgardt et al. 2004) +Π ∼ 60Myr−1 × +� 𝑅wd +R⊙ +�4/9 +× +� 𝑀wd +M⊙ +�−95/54 +× +× +� +𝑀h +103M⊙ +�61/27 +× +� 𝑛c +pc−3 +�−7/6 +× +� 𝑟c +1pc +�−49/9 +, +(31) +where we take the GC core density equal to 𝑛c = 105 pc−3 and the +GC core radius equal to 𝑟c = 0.5 pc. We remind that we assume the +WD mass and radius to be fixed, 𝑀wd = 0.5M⊙, 𝑅wd = 0.01R⊙. +We assume that the mass distribution of IMBHs is a 𝛿 function at a +fixed value of 𝑀h. In particular, we will build two populations, one +with 𝑀h = 103M⊙, the other with 𝑀h = 104M⊙. +3.2.3 Range of parameters for the TDE populations +To derive the total number of observed TDEs in presence of mag- +nification, Nobs, we start by building the nuclear and globular TDE +populations according to the aforementioned description. In particu- +lar we choose the following ranges: +• for the source redshift we take 𝑧s ∈ [0.001, 2], where the min- +imum value of redshift corresponds to ≈ 20Mpc (average distance +of the Virgo Cluster), while the maximum value corresponds to the +redshift after which the GW emission from a TDE population is +negligible (Toscani et al. 2020). +• for the central BH mass we take 𝑀• ∈ [104M⊙, 109M⊙], thus +considering both dwarf and large galaxies; +• for the stellar mass we take 𝑀∗ ∈ [1M⊙, 100M⊙] for the MS +star case, hence young stellar population, while for the WD case we +assume fixed mass and radius 𝑀wd = 0.5M⊙, 𝑅wd = 0.01R⊙. +• for the penetration factor we take 𝛽 ∈ [1, 𝛽max], where the +formula for 𝛽max, which in general will depend on the BH and star +mass, is illustrated in Equation (21). +MNRAS 000, 1–11 (2022) + +6 +M.Toscani et al. +0 +2 +4 +6 +8 +10 +10 +1 +101 +103 +105 +107 +d +ms/d dzs +LISA +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +0 +1000 +2000 +3000 +4000 +5000 +6000 +10 +1 +101 +103 +105 +107 +d +ms/d dzs +DECIGO +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +Figure 4. Number of nuclear TDEs per bin of SNR and bin of source redshift as a function of SNR 𝜌. On the left panel we consider LISA, on the right panel +we consider DECIGO. The pink horizontal dashed line represents 1 TDE per SNR and redshift bin in 10 years. +0 +100 +200 +300 +400 +500 +101 +102 +103 +104 +d +ms/dzs +LISA +z=0.28 +z=0.49 +z=1.10 +z=1.72 +100 +200 +300 +400 +500 +105 +106 +d +ms/dzs +DECIGO +z=0.28 +z=0.49 +z=1.10 +z=1.72 +Figure 5. Number of visible (i.e. above the threshold 𝜌lim/√𝜇) nuclear TDEs if the magnification is 𝜇. On the left panel we consider LISA, on the right panel +we consider DECIGO. +4 RESULTS +Once we have built the two TDE populations, we can study which +is the probability that they are lensed by a foreground population of +lenses, using the framework presented in Section 3.1. We illustrate +the main results of our lensing study in the following, distinguishing +between the case of nuclear and globular TDEs. +4.1 Strong lensing of GWs from nuclear TDEs +In Figure 4 we show the number of MS stars tidally disrupted by +massive BH per bin of SNR and 𝑧s, 𝑑Nms/𝑑𝜌𝑑𝑧s (cf. Equation 11). +On the left side we calculate the SNR considering LISA, on the +right side DECIGO. Each colour represents the number of TDEs in a +redshift bin 𝑧s ± 0.068, and in particular we show selected values of +𝑧s: 0.28 (blue), 0.49 (orange), 1.10 (green), 1.72 (red). In general, we +MNRAS 000, 1–11 (2022) + +Lensed TDEs +7 +0 +1 +2 +3 +4 +5 +6 +7 +8 +10 +1 +100 +101 +102 +103 +104 +d +wd/d dzs +DECIGO, Mh = 103M +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +10 +1 +100 +101 +102 +103 +104 +105 +106 +d +wd/d dzs +DECIGO, Mh = 104M +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +Figure 6. Number of globular TDEs per bin of SNR and bin of source redshift. On the left panel we consider 𝑀h = 103M⊙, on the right 𝑀h = 104M⊙. We +assume that the interferometer is DECIGO. The pink horizontal dashed line represents 1 TDE per SNR and redshift bin in 10 years. +0 +100 +200 +300 +400 +500 +102 +103 +104 +d +wd/dzs +DECIGO, Mh = 103M +z=0.28 +z=0.49 +z=1.10 +z=1.72 +0 +100 +200 +300 +400 +500 +105 +106 +d +wd/dzs +DECIGO, Mh = 104M +z=0.28 +z=0.49 +z=1.10 +z=1.72 +Figure 7. Number of visible (i.e. above the threshold 𝜌lim/√𝜇) globular TDEs if the magnification is 𝜇. On the left panel we consider 𝑀h = 103M⊙, on the +right panel we consider 𝑀h = 103M⊙. +see that 𝑑Nms/𝑑𝜌𝑑𝑧s diminishes for higher values of 𝜌 as expected. +In addition to this, the maximum SNR decreases for higher redshift. +As for the minimum SNR, this is always ≈ 0, which is a reasonable +result since for each redshift bin we have TDEs below the instruments +sensitivity curves. +In Figure 5, we show the number of visible (i.e. above the threshold +𝜌lim/√𝜇) nuclear TDEs if the magnification is 𝜇, 𝑑Nms(𝜇, 𝑧s)/𝑑𝑧s, +calculated through Equation (11). The layout and colour are the +same as in Figure 4. From these plots, we see that while for LISA +𝑑Nms(𝜇, 𝑧s)/𝑑𝑧s decreases for higher values of redshift, the same +quantity for DECIGO first increases for higher values of 𝑧s, than it +starts to lower again. This behavior can be explained in the following +way. Since DECIGO has a better sensitivity than LISA, there are two +opposite trends that interplay between each other: i) the total number +of TDEs increases for higher values of redshift (volume effect), ii) +the number of visible TDEs decreases for high values of redshift +MNRAS 000, 1–11 (2022) + +8 +M.Toscani et al. +(SNR limitation). In the case of LISA however, which presents a +worse sensitivity to TDEs, the ii) effect always prevails. In other +words, LISA is always SNR limited and the very few detectable +events decrease rapidly with redshift. +Finally, we have all the ingredients to calculate the total number of +observed TDEs in presence of magnification, Nobs, through Equation +(12). Restricting to magnification 𝜇 > 2 (i.e. focusing on the stronger +lensed image, cf. Section 2 and Appendix A) , we find that for LISA +the number of lensed-magnified TDEs is 0, while for DECIGO we +expect the detection of ∼130 magnified TDEs (𝜇 > 2). Yet, this +number decreases quite rapidly as we increase the magnification +threshold: it reduces to ∼13 for 𝜇 > 3, ∼3 for 𝜇 > 4 and goes to ∼ 0 +for 𝜇 > 5. This fast drop is in agreement with the steep decreasing +presented by the magnification PDF illustrated in Figure 2. +4.2 Strong lensing of GWs from globular TDEs +As for the case of globular TDEs, we consider two sub-populations: +one where WDs are disrupted by IMBHs of mass 103M⊙, the other +with an IMBH mass of 104M⊙. We assume that in each GC there is +an IMBH. +If we consider LISA, both these sub-populations of globular TDEs +are below threshold and not even lensing can make part of these +sources detectable. +The situation is instead more interesting if we consider DECIGO. +In Figure 6 we show the number of WDs tidally disrupted by IMBHs +per bin of SNR and 𝑧s, 𝑑Nwd/𝑑𝜌𝑑𝑧s. On the left side we consider +𝑀h = 103M⊙, while on the right 𝑀h = 104M⊙. The colour scheme +is the same as previously described. In a similar way as for the nu- +clear TDE scenario, we note that: i) 𝑑Nwd/𝑑𝜌𝑑𝑧s shows a general +decreasing trend while 𝜌 increases; ii) the maximum SNR decreases +for for higher 𝑧s; iii) the minimum SNR is ≈ 0. +In Figure 7, we show the number of observable globular TDEs if +the magnification is 𝜇, 𝑑Nwd(𝜇, 𝑧s)/𝑑𝑧s, calculated through Equa- +tion 11. The layout and colour are the same as in the previous plot. +Also in this case, we note the interplay between the volume effect +against the SNR effect already presented in Figure 5 for DECIGO. +Finally we have all the ingredients to calculate the number of ob- +served magnified TDEs. For the case 𝑀h = 103M⊙, DECIGO will +not observe any TDEs with 𝜇 > 2. Thus, DECIGO will not detect +any TDEs from this population. As for the scenario 𝑀h = 104M⊙, +the number of TDEs with 𝜇 > 2 is ∼10, with 𝜇 > 3 is ∼3, and it +drops to 0 for higher magnification. +5 DISCUSSION AND CONCLUSIONS +In this paper, we have investigated the effects of gravitational lensing +of GW signals from TDEs of MS stars disrupted by massive BHs in +galaxy cores (nuclear TDEs) and from TDEs of WDs disrupted by +IMBHs in GCs (globular TDEs). In order to follow a most realistic +procedure as possible, we built the distribution of lenses following +an hybrid approach. The lenses (galaxies) are modeled as SIS. To +derive numerical results, we add the contribution of weak lensing +from the Large Scale Structure using the results from ray-tracing +simulations of Takahashi et al. (2011), which include both weak +lensing contribution, and strong lensing tails in the magnification +PDF. +For the TDE population, we follow similar steps as in Toscani +et al. (2020). Our work shows that, while LISA shall not be able to +observe lensed-magnified TDEs, the situation will be different for +101 +102 +10 +10 +10 +8 +10 +6 +10 +4 +10 +2 +100 +(zs, +) +zs = 0.28 +zs = 0.49 +zs = 1.10 +zs = 1.72 +Figure 8. Probability that, if DECIGO detect a nuclear TDE from redshift 𝑧s, +this is magnified more than 𝜇. +an interferometer with a DECIGO-like sensitivity. While this inter- +ferometer will observe ∼10 magnified globular TDEs (𝜇 > 2), we +expect to observe ∼130 lensed-magnified nuclear TDEs for 𝜇 > 2. In +Fig. 8 we show a summary plot illustrating, for the most promising +scenario of nuclear TDEs observed by DECIGO, the probability that +a TDE from redshift 𝑧s is magnified more than 𝜇 (see Equation 13). +The probability that a TDE observed at redshift 𝑧s = 0.28 would be +magnified more than 𝜇 > 2 is ∼10−4, and increases up to one order +of magnitude if we go to higher redshift (𝑧s = 1.72). At fixed redshift +bin, the probability to have higher magnification decreases steeply +(roughly 9 to 10 orders of magnitudes in the interval 1 ≤ 𝜇 ≤ 100), +which justifies why the number of magnified TDEs drops rapidly as +explained above. +Our results point out that DECIGO will observe a non-negligible +fraction (∼0.1%) of strongly lensed TDEs. Hence data analysis tech- +niques need to be developed to be able to distinguish lensed TDEs +from unlensed ones. This will be important to prevent a biased recon- +struction of the parameters of the source. Lensed events will in fact +have a (de-)magnified GW amplitude at the detector, which could +bias the measurement of source parameters such as its distance. Fur- +thermore, we expect lensing to induce lensing selection effects on the +study of the TDE population, in analogy with what found for example +in Cusin & Tamanini (2021) for a population of supermassive black +hole binaries visible by LISA. Indeed, a realistic GW detector has a +finite sensitivity: magnified sources are on average easier to detect +than de-magnified ones and this affects the distribution of lensing +magnification of an observed source sample. These lensing selection +effects, which should disappear in the limit of a perfect instrument, +are then expected to introduce a bias on the reconstruction of the +source parameters (e.g. the luminosity distance), independent of the +sample size. Hence, the characterisation of all the implications due +to lensing, including selection effects due to the specifics of a given +instrument, is necessary to accurately infer the source population as- +MNRAS 000, 1–11 (2022) + +Lensed TDEs +9 +trophysical properties across cosmic time, but also to be able to use +high-redshift GW sources as a new cosmological probe. +We would like to remind that our study of lensing relies on the geo- +metric optics approximation. We expect wave effects to become non- +negligible in the mHz waveband only when dealing with diffusion +off sub-galactic structures, see e.g. Takahashi & Nakamura (2003a); +Nakamura (1998); Takahashi (2017); Dolan (2008); Cusin et al. +(2019b); Cusin & Lagos (2020); Dalang et al. (2022). Diffraction +on sub-galactic scales makes lenses on those scales effectively trans- +parent to GW in the LISA band (Takahashi & Nakamura 2003a), in +contrast with what happens for lensing of EM sources (see e.g. Fleury +et al. 2015) as the EM spectrum is at much lower wavelengths than +any relevant astrophysical structure at cosmological scales. +We conclude by pointing out that TDEs may indeed constitute +highly interesting multi-messenger sources, as they emit not only +EM radiation, but also GWs and high-energy neutrinos. The obser- +vation of both EM and GW signals from the same TDE, could in +fact enable spectacular multi-messenger analyses which may well +unveil new secrets on the intrinsic mechanisms behind these events. +Concretely for example, the GW signal would mark the moment +of stellar disruption at the first pericenter passage, otherwise unde- +tectable (Rossi et al. 2020). A measurement of the time delay between +the GW signal and the subsequent EM signals would decisively help +discriminating between EM emission mechanisms, currently highly +debated (Bonnerot & Stone 2021). We may compare this scenario +with GW170817 which revolutionised our understanding of binary +neutron star mergers (Abbott et al. 2017a,c, 2019a), and triggered +new tests of general relativity (Abbott et al. 2019b) and new cosmo- +logical measurements (Abbott et al. 2017b). TDEs may well be used +for similar measurements in the future: for example they could allow +us to probe the expansion of the universe if the luminosity distance +is extracted from the GW signals and the host galaxy is identified +from the EM emission, in analogy to massive BH binary mergers +with LISA (Tamanini et al. 2016; Belgacem et al. 2019) and double +WD mergers with DECIGO (Maselli et al. 2020). +The observation of a multi-messenger lensed TDE would not only +provide the data for the analyses outlined above, but the differences in +the observed EM and GW lensed signals would yield unprecedented +opportunities to study properties of both the source and the lens. +For example the EM radiation will always fall within the geometric +optics approximation, while as mentioned above the GW signal could +show signs of wave optics effects, which may then be used to infer +additional information on the lens. A detailed analysis of lensed EM +emission from TDEs is currently missing in the literature and this +will be the subject for a future work. +ACKNOWLEDGEMENTS +MT and NT acknowledge support form the French space agency +CNES in the framework of LISA. The work of GC is supported by +CNRS and by the Swiss National Science Foundation (Ambizione +grant–Gravitational wave propagation in the clustered Universe). +EMR acknowledges that this project has received funding from the +European Research Council (ERC) under the European Union’s Hori- +zon 2020 research and innovation programme (Grant agreement No. +101002511 - VEGA P). +DATA AVAILABILITY +The data underlying this article will be shared on reasonable request +to the corresponding author. +REFERENCES +Abbott B. P., et al., 2017a, Phys. Rev. Lett., 119, 161101 +Abbott B. P., et al., 2017b, Nature, 551, 85 +Abbott B. P., et al., 2017c, Astrophys. J. Lett., 848, L13 +Abbott B. P., et al., 2019a, Phys. Rev. X, 9, 011001 +Abbott B. 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Rev., 216, 124 +APPENDIX A: STRONG LENSING IN A NUTSHELL +In this Appendix we present details of the derivation of cross-section +and optical depth for a SIS lens. +A1 Basic strong lensing quantities +A typical situation considered in gravitational lensing is the one +illustrated in Figure 1, where a lens of mass 𝑀l at redshift 𝑧l deflects +the signal from a source at redshift 𝑧s. The actual path followed by the +signal3, which is smoothly curved in the space-time surrounding the +lens, can be - as a first approximation - replaced by two straight rays +with a kink near the deflector. The difference between the angular +position of the image and the angular position of the source is called +deflection angle, and we denote it as ˆ𝛼. +The true position of the source is related to its lensed image on the +sky through the lens equation, which reads (Schneider et al. 1992) +𝜂𝜂𝜂 = 𝑑s +𝑑l +𝜉𝜉𝜉 − 𝑑ls ˆ𝛼ˆ𝛼ˆ𝛼(𝜉𝜉𝜉). +(A1) +Taking into account the geometry illustrated in Figure 1, the source +and lens impact parameters can be written as +𝜂𝜂𝜂 = 𝛾𝛾𝛾𝑑s, +𝜉𝜉𝜉 = 𝜃𝜃𝜃𝑑l, +(A2) +and thus the Equation (A1) becomes +𝛾𝛾𝛾 = 𝜃𝜃𝜃 − 𝑑ls +𝑑s +ˆ𝛼ˆ𝛼ˆ𝛼(𝜃𝜃𝜃𝑑d) ≡ 𝜃𝜃𝜃 − 𝛼𝛼𝛼(𝜃𝜃𝜃), +(A3) +where we have introduced the scaled deflection angle 𝛼𝛼𝛼 = (𝑑ls/𝑑s) ˆ𝛼ˆ𝛼ˆ𝛼. +The scaled deflection angle can be expressed in terms of the conver- +gence 𝜅, as +𝛼𝛼𝛼(𝜃𝜃𝜃) = 1 +𝜋 +∫ +𝑑2𝜃′𝜅(𝜃′𝜃′𝜃′) 𝜃𝜃𝜃 − 𝜃′𝜃′𝜃′ +|𝜃𝜃𝜃 − 𝜃′𝜃′𝜃′| , +(A4) +where 𝜅 is defined as +𝜅(𝜃𝜃𝜃) = Σ(𝑑l𝜃𝜃𝜃) +Σcr +, +with Σcr = +𝑐2𝑑s +4𝜋𝐺𝑑l𝑑ls +. +(A5) +3 We recall that we are here working in the geometric optics limit. +Note that the surface mass density Σ is obtained by integrating the +mean mass density of the lens along the line of sight. We are in the +strong lensing limit when Σ > Σcr, i.e. when the mass distribution +of the lens allows the production of multiple images of the source +(Kochanek 2006). +Let us now focus on the SIS model. The mass density of a SIS is +given by (Narayan & Bartelmann 1996) +𝜚(𝑟) = +𝜎2v +2𝜋𝐺𝑟2 , +(A6) +where 𝜎v denotes the velocity dispersion of the lens. Despite the +singularity in the center and the infinite total mass, this can be con- +sidered as a rather realistic mass distribution for lensing by a galaxy +(see, e.g., Schneider et al. 1992), with 𝜎v velocity dispersion within +the galaxy. Integrating along the line of sight we obtain the surface +density +Σ(𝜃) = 2 × 𝜎2v +2𝜋𝐺 +∫ +∞ +0 +1 +𝜉2 + 𝑧2 += 𝜎2𝑣 +2𝐺 +1 +𝜉 += 𝜎2𝑣 +2𝐺 +1 +𝑑l𝜃 . +(A7) +Thus, for a SIS the convergence reads +𝜅(𝜃) = Σ(𝜃) +Σc += 2𝜋 𝜎2v +𝑐2 +𝑑ls +𝑑s +1 +𝜃 , +(A8) +with a constant deflection angle +𝛼(𝜃) = 4𝜋 𝜎2v +𝑐2 +𝑑ls +𝑑s += 2𝜃𝜅(𝜃) ≡ 𝛼0 . +(A9) +This is usually called Einstein angle (see, e.g., Schneider et al. 1992), +and in order to have multiple images of the source the following +condition needs to be fulfilled 𝛾 < 𝛼0. +If we rescale our variables by 𝛼0, we can define the rescaled image +and source positions as x = 𝜃𝜃𝜃/𝛼0 and y = 𝛾𝛾𝛾/𝛼0, hence Equation +(A3) becomes +y = x − x +|x| . +(A10) +We can distinguish three cases: i) for y = 0 the solution is the Einstein +ring |x| = 1, ii) for 𝑦 = |y| < 1 one solution is 𝑥1 = |x1| = 1 + 𝑦 +(on the same side of the line of sight as the source), the other one is +𝑥2 = |x2| = 1 − 𝑦 (on the opposite side), iii) for 𝑦 > 1 the second +solution no longer exists. +The Jacobian of the lens map is +𝐴𝑖 𝑗 = 𝛿𝑖 𝑗 +� +1 − 1 +|x| +� ++ 𝑥𝑖𝑥 𝑗 +|x|3 , +det𝐴 = 1 − 1 +|x| , +𝜇 = +1 +|det𝐴| = +|x| +|1 − |x|| , +(A11) +where we have formally introduced the magnification 𝜇. Expressing +the total magnification of a point source at position 𝑦 in terms of 𝑦 +we find +𝜇(𝑦) = +� +𝜇(x1) + 𝜇(x2) = 𝑦+1 +𝑦 ++ 1−𝑦 +𝑦 += 𝜇1 + 𝜇2 = 2 +𝑦 , +𝑦 ≤ 1 , +𝑦+1 +𝑦 += 1 + 1 +𝑦 , +𝑦 ≥ 1 . +(A12) +We observe that the magnification is always positive. This is a con- +sequence of the fact that SIS is an overdensity, hence it cannot de- +magnify the signal of a background source. +MNRAS 000, 1–11 (2022) + +Lensed TDEs +11 +A2 Cross-section and optical depth +The impact parameter of the source (in the source plane) is given by +|𝜂| = 𝜂 = 𝛾𝑑𝑠 = 𝑦𝛼0𝑑𝑠. A source with impact parameter smaller or +equal to 𝜂 is amplified by at least a factor 𝜇(𝑦). Hence, considering +a SIS with velocity dispersion 𝜎𝑣, the cross section for amplification +≥ 𝜇1 of the stronger image is +𝜍(𝜇1, 𝑧ℓ, 𝑧𝑠, 𝜎𝑣) = 𝜋𝜂2 = 𝜋(𝑦𝛼0𝑑𝑠)2 = +𝜋(4𝜋)2𝜎4𝑣𝑑2 +ls +𝑐4(𝜇1 − 1)2 . +(A13) +Note that this cross section gives the area, centered along the line of +sight of the lens, within which a source at 𝑧𝑠 must lie so that it is +amplified by a factor 𝜇1 or larger by the lens at 𝑧𝑙. The expression +(A13) remains valid also for 𝑦 ≥ 1, where we have only one image +with magnification 𝜇1 which tends to 1 when 𝑦 → ∞. +In our study of strong lensing of gravitational waves we consider +only one image and not the sum of both, since we expect to see a short +burst of GWs which comes only from one image. The second image is +delayed in time, with typical time delay of the order of a few months +(see e.g. Oguri 2018), much longer than the GW burst. Since we are +interested in magnification, we shall compute the cross section for the +stronger image. This point has been raised in Cusin et al. (2021) but +it was neglected in the previous literature: usually in Equation (A13) +𝑦−1 = 𝜇1 − 1 is replaced by 𝑦−1 = 𝜇/2 = (𝜇1 + 𝜇2)/2 which is +the correct expression for a static situation where both images are +seen together. For strong amplification, 𝜇1 ∼ 𝜇2 ≫ 1 this difference +reduces the cross section by a factor 4. +To compute the corresponding optical depth, denoted 𝜏(𝜇, 𝑧𝑠), we +need to know the physical density 𝑛(𝜎𝑣, 𝑧l) of lenses (galaxies) with +a given velocity dispersion 𝜎v at redshift 𝑧l. We define the density +function 𝑛(𝜎v, 𝑧l) such that its integral +∫ +𝑛(𝜎v, 𝑧l)𝑑𝜎v simply gives +the total density of lenses at redshift 𝑧l. +The optical depth for lensing with magnification ≥ 𝜇 (of the most +strongly magnified image in the case of two images) for a source at +redshift 𝑧s is +𝜏(𝜇, 𝑧s) = +∫ 𝑧s +0 +𝑑𝑧l +𝑑𝑟 +𝑑𝑧l +� 𝑑l +𝑑s +�2 ∫ +𝜍(𝜇, 𝑧l, 𝑧𝑠, 𝜎v)𝑛(𝜎v, 𝑧l)𝑑𝜎v , +(A14) +where 𝑑𝑟 is the physical length element at redshift 𝑧l, while 𝑛(𝜎v, 𝑧l) +is the physical number density of lenses, which is related to the +comoving number density by 𝑛(𝜎v, 𝑧l) = (1 + 𝑧)3𝑛com(𝜎v, 𝑧l). Note +also that we have rescaled the cross section to the lens redshift, +𝜍 → 𝜍(𝑑l/𝑑s)2 since in eq. (A14) we multiply by the lens density at +𝑧l. Inserting this and (A13) for the cross section in Equation (A14) +we obtain +𝜏(𝜇, 𝑧𝑠) = += 𝜋(4𝜋)2 +(𝜇 − 1)2 +∫ 𝑧𝑠 +0 +𝑑𝑧 +𝜒2(𝑧, 𝑧𝑠)𝜒2(𝑧) +𝜒(𝑧𝑠)2(1 + 𝑧)3𝐻(𝑧) +∫ ∞ +0 +𝑑𝜎𝑣𝜎4 +𝑣𝑛(𝜎v, 𝑧l) , +(A15) +where 𝜒(𝑧1, 𝑧2) denotes the comoving distance from redshift 𝑧1 to +𝑧2. Eq. (A15) is the optical depth for magnification larger than 𝜇. +Often only the strong lensing case is considered and the magnifica- +tion from the two images is added to give the total magnification. To +do this one has to replace 1/(𝜇 − 1)2 by 4/𝜇2. As already mentioned +above, here we cannot do this since in the case of strong magnifica- +tion and double images we expect a considerable time delay, so that +typically we observe only one image at one given time. Here we as- +sume this to be the stronger image. In the strong magnification case, +𝜇 ≫ 2, this difference is roughly a factor 4, while in the limit 𝑦 → 1 +where 𝜇 → 2 and the second image disappears, the two expressions +agree. +APPENDIX B: SNR CALCULATION +To calculate the SNR 𝜌 of a TDE for a given interferometer, we need +to express the signal in the detector frame of reference +𝜌 = +ℎdgw +ℎc( 𝑓 dgw) += +(B1) += 𝛽𝜈 × 2 × 10−22 × +� +𝑀d∗ +𝑀⊙ +�1/3 +× +� +𝑀d• +106M⊙ +�2/3 +× +� +𝐷 +16Mpc +�−1 +× +1 +ℎc( 𝑓gw/(1 + 𝑧)) . +For sake of simplicity, in the above formula we have considered the +MS star case and already applied the mass-radius scaling relation of +Equation (23). Considering how to convert frequency and mass from +detector to source-frame reference, +𝑀d = 𝑀(1 + 𝑧), +(B2) +𝑓 d +gw = 𝑓gw/(1 + 𝑧), +(B3) +and the relation between comoving and luminosity distance (Hogg +1999) +𝜒 = +𝐷 +1 + 𝑧 , +(B4) +we can re-write 𝜌 as presented in Equation (22). +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–11 (2022) + diff --git a/XNAzT4oBgHgl3EQf1v6R/content/tmp_files/load_file.txt b/XNAzT4oBgHgl3EQf1v6R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12c4587d134d8708372cdd1395c0cd44ca121b12 --- /dev/null +++ b/XNAzT4oBgHgl3EQf1v6R/content/tmp_files/load_file.txt @@ -0,0 +1,828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf,len=827 +page_content='MNRAS 000, 1–11 (2022) Preprint 6 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 Lensing of gravitational waves from tidal disruption events Martina Toscani1∗, Elena M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Rossi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2 Nicola Tamanini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1 and Giulia Cusin3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='4 1 Laboratoire des 2 Infinis - Toulouse (L2IT-IN2P3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Université de Toulouse,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2300 RA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' the Netherlands 3 Institut d’Astrophysique de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' UMR 7095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 98 bis bd Arago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 75014 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' France 4 Université de Genéve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Département de Physique Théorique and Centre for Astroparticle Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 24 quai Ernest-Ansermet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' CH-1211 Genéve 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Switzerland Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' in original form ZZZ ABSTRACT In this work, we investigate the effect of gravitational lensing on the gravitational wave (GW) signals of a population of tidal disruption events (TDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We estimate the number of lensed-magnified signals that we expect to detect with future space-based GW observatories, in particular LISA and DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We model the lens distribution using an hybrid approach that combines semi-analytic methods with numerical results from ray tracing simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We divide the TDE population in two classes, nuclear TDEs (main sequence stars tidally disrupted by massive black holes in the cores of galaxies) and globular TDEs (white dwarfs tidally disrupted by intermediate mass black holes in globular clusters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We find that, even considering the effect of lensing, LISA will not be able to observe any TDEs, while DECIGO could detect ∼10 strongly lensed (𝜇 > 2) globular TDEs and ∼130 strongly lensed nuclear TDEs, over an observational period of 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Our results reveal the role that lensing will play in future deci-Hertz GW observatories, indicating exciting multi-messenger opportunities with TDEs but at same time signalling the need to develop adequate data analysis techniques to correctly reconstruct the astrophysical properties of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Key words: gravitational lensing: strong – gravitational waves – transients: tidal disruption events 1 INTRODUCTION Stars orbiting around a massive black hole (BH) can be shred into pieces due to tides induced by the BH’s gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We call these extreme phenomena tidal disruption events (TDEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Luminet & Carter 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Rees 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Phinney 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' for a recent review see Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Thanks to the bright electromagnetic (EM) flares produced by the stellar debris during the following accretion, TDEs have been one of the most powerful ways to reveal the presence of otherwise quiescent massive BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To date, we have around 100 ob- servations of these events, in different bands of the EM spectrum (see for recent reviews: Saxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020 for X-ray;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' van Velzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020 for optical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020 for radio and all the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Besides being multiband emitters in the EM spectrum, TDEs are potential multimessenger sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Recently, a few TDEs have been claimed to be associated to observed astrophysical neutrinos (Stein 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Reusch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2022) as theoretically expected in the presence of jets (Hayasaki 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Moreover, TDEs are also predicted to emit gravitational waves (GWs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In particular, we can distinguish between three main GW contributions: GWs due to the internal stellar mass quadrupole, generated by the stretching and compressing action of the BH tidal forces (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g, Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' GWs associated to the BH-star system mass quadrupole, emitted during the disrup- tion phase (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' GWs produced at later stages, along the circularization pro- cess (Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2022) and in presence of an accretion disc (see, ∗E-mail: martina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='toscani@l2it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='fr e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For standard values of the parameters in- volved, the strongest gravitational contribution is the burst emitted during the disruption phase, that has typical frequencies in the range 10−3 − 10−2 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Being low frequency GW sources, TDEs could be detected by future space-based interferometers such as the Laser Interferometer Space Antenna (LISA - Amaro-Seoane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2017, 2022), currently scheduled for launch in the mid-2030s, and the proposed deci-Hertz Gravitational Observatory (DECIGO - Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' A detailed study about TDEs detectability with these next-generation detectors has been carried out in Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Their work shows that while detection of individual TDEs by LISA seems unlikely, these events are promising sources for deci-Hz observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Future instruments with a DECIGO-like sensitivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', DECIGO, BBO - Harry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2006, ALIA - Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2019, DO - Sedda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020, 2021) could observe hundreds of thousands TDEs per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Given these expectations, it becomes relevant to assess the effects that a distribution of lenses produces on the GW emission from a TDE population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We refer to gravitational lensing (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Schnei- der et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1992) when a massive object (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' the lens), which lies along the line of sight between the observer and the source, curves the surrounding space-time, causing the signal to deviate from its original path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This effect has interesting consequences: for example it may (de-)magnify the signal or produce multiple images of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Moreover, different images typically arrive at the detector at different times (time-delay effect) and they interfere if the duration of the signal is larger than the typical delay in the time of arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In this work, we study for the first time the effect of gravitational lensing on a TDE population and provide estimates on the expected © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='01804v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='HE] 4 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' number of observed lensed-magnified TDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We consider both LISA, for which a magnified TDE could be the only way to have a signal above the detectability threshold, but also DECIGO, for which the ability to distinguish lensed TDEs would avoid errors in the recon- struction of the parameters describing the source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', distance and mass), as well as provide additional information on the astrophysical properties of the source and lens populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We perform this inves- tigation dividing TDEs in two different classes: nuclear TDEs, where we consider main sequence (MS) stars disrupted by massive BHs in the cores of galaxies, and globular TDEs, where we consider white dwarfs (WDs) tidally disrupted by intermediate mass BHs (IMBHs) located in globular cluster (GCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The structure of the paper is the following: in Section 2 we illustrate the basis of gravitational lens- ing, in Section 3 we describe the distribution of lenses and the TDE populations in details, in Section 4 we show and discuss the results and finally in Section 5 we draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Throughout this work, we adopt a ΛCDM cosmological model, with matter density parameter Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='274, dark energy density parameter ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='726 and Hubble constant 𝐻0 = 70 Km/sMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2 GRAVITATIONAL LENSING IN A NUTSHELL We want to determine the number of TDEs that, having a given strain amplitude, or rather a given signal to noise ratio (SNR) 𝜌 for a specified interferometer, are significantly magnified by lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Following Cusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2021), we present semi-analytic formulae which can be applied to an arbitrary lens and source distribution, keeping full control of modeling and transparency of all physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We mainly focus our analysis on strong lensing, working out the distribution of magnification for 𝜇 > 1, considering LISA and DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' It is indeed true that the lensing induced by the Cosmolog- ical Large Scale Structure can also lead to de-magnification (𝜇 < 1) of a signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' However de-magnification from the Large Scale Structure usually does not reach values 𝜇 ≪ 1, meaning that its contribution should not significantly affect the detection rates of the observed pop- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Furthermore the TDE population is well below the detection threshold of LISA, hence de-magnified events will anyway remain undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For these reasons in what follows we ignore the effect of de-magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Our description of strong lensing relies on the geometric optics approximation (we do not describe wave effects such as diffraction and interference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This is a well-justified approximation when look- ing at TDEs lensed by a population of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Indeed, given a lens of mass 𝑀l, diffraction effects are relevant when (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g, Takahashi & Nakamura 2003b) 𝑀l ≲ 108M⊙ � 𝑓 mHz �−1 , (1) where 𝑓 is the frequency of the lensed signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Since GWs by TDEs have typical frequencies in the range 10−3 − 10−2 Hz, wave effects can be neglected in our work, as we consider galaxy stellar masses between 108 − 1012M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='. To predict the number of magnified TDEs observable with a given instrument, we need to take the following steps: (i) choose a lens model, and a model to describe the lens distri- bution and the population of sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (ii) derive the probability density function (PDF) for a generic source at redshift 𝑧s to be amplified more than 𝜇 by the population of foreground lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Sketch of the geometry for a SIS lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (iii) convolve the magnification PDF with the population of ob- servable events for a given instrument, in presence of magnification 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the following, we model the lenses as singular isothermal spheres (SIS), that we describe in detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' While SIS are not very realistic when considering lensing by individual clusters, they are sufficient for statistical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In particular, the main ad- vantage of the SIS model is that it can be studied analytically, which allows us to have a better (even tough idealised) comprehension of the physics behind lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To follow a more realistic approach in addition to strong lensing, described by our semi-analytic approach, we also consider the contribution from weak lensing due to the grav- itational potential of the Large Scale Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To derive numerical results, we use ray-tracing simulations by Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2011), which include both weak lensing contribution, and strong lensing tails in the magnification PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1 SIS and gravitational lensing statistics In the SIS model all the mass components of the galaxy behave like particles of an ideal gas, confined by their combined, spherically symmetric gravitational potential, in thermal equilibrium (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Narayan & Bartelmann 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The mass density of a SIS is described by 𝜚(𝑟) = 𝜎2v 2𝜋𝐺𝑟2 , (2) where 𝜎v is the velocity dispersion in the galaxy, 𝐺 is the gravitational constant and 𝑟 is the distance from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Integrating Equation (2) along the line of sight, we get the surface mass density Σ(𝜉) = 𝜎2v 2𝐺𝜉 , (3) where 𝜉 is the bi-dimensional vector in the lens plane, called lens impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The geometry of such system is illustrated in Figure 1, where the following elements are shown: (i) the source, its lensed image, the observer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', LISA) and the lens, (ii) 𝛾, angle between the line of sight and the unlensed source, MNRAS 000, 1–11 (2022) source image n dis lens Sm ds α di observer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=',LISA)Lensed TDEs 3 (iii) 𝜃, angle between the line of sight and the lensed image, (iv) ˆ𝛼 deflection angle induced by the presence of the lens, (v) 𝑑𝑥, angular diameter distances between the lens and the source (x=ls), the lens and the observer (x=l), the source and the observer (x=s), (vi) 𝜉 = 𝜃𝑑l, lens impact parameter, in the lens plane, (vii) 𝜂 = 𝛾𝑑s, source impact parameter, in the source plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We recall (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g, Hogg 1999) that the angular diameter distance of an object is related to its luminosity distance, 𝐷, by 𝑑 = 𝐷 (1 + 𝑧)2 , (4) and the angular diameter distance between two objects at redshifts 𝑧1 and 𝑧2 is 𝑑12 = 1 1 + 𝑧2 [𝜒(𝑧2) − 𝜒(𝑧1)] = 1 1 + 𝑧2 𝜒(𝑧1, 𝑧2), (5) where 𝜒 is the comoving distance 𝜒(𝑧) = 𝑐 𝐻0 ∫ 𝑧 0 𝑑𝑧′ 𝐸(𝑧′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (6) In the above Equation, we have introduced 𝐸(𝑧′) = (Ωm(1 + 𝑧′)3 + Ω𝜆)1/2, where Ωm and Ω𝜆 are the present values of the matter and cosmological constant density contrasts and 𝑐 is the speed of light in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The basic quantity for statistical analysis is the cross section of the lens for producing the desired lensing effect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' magnification larger than 𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The corresponding optical depth is the fraction of the sky where, given the lenses, one can place a source and observe this magnification (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g, Kochanek 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the case of the SIS, the area on the source plane in which a source at redshift 𝑧s will be magnified ≥ 𝜇 is given by 𝜍sis(𝜇, 𝑧s, 𝑧l, 𝜎v) = 𝜋𝜂2(𝜇, 𝑧s, 𝑧l, 𝜎v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The corresponding optical depth is 𝜏(𝜇, 𝑧s) = ∫ 𝑧s 0 𝑑𝑧l 𝑑𝑟 𝑑𝑧l � 𝑑l 𝑑𝑠 �2 ∫ 𝑑𝜎v𝑛(𝜎v, 𝑧l)𝜍sis(𝜇, 𝑧l, 𝑧s, 𝜎v) , (7) where 𝑑𝑟 is the physical length element at redshift 𝑧l, while 𝑛(𝜎v, 𝑧l) is the physical number density of lenses per bin of 𝜎v (Kochanek 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the SIS model, there are two lensed images when the source satisfy the following criterion (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1992) 𝛾 < 𝛼0 = 4𝜋 𝜎2v 𝑐2 𝑑ls 𝑑s , (8) where 𝛼0 is usually called Einstein angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In our study we consider this scenario, but we restrict our attention to the primary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' more magnified, image1, for which we provide the cross section and the explicit final formula for 𝜏 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This choice is justified since we expect to see a short burst of GWs from a TDE which comes only from one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The second image is in fact delayed in time, with typical time delay of the order of a few months (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Oguri 2018), much longer than the GW burst itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The problem of correctly identifying secondary images, and associating them to their primaries, requires a dedicated data analysis investigation which goes beyond the scope of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1 In lensing analyses it is common jargon to refer to the observed signals as “images”, even though they are not EM signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In this paper we follow this convention, calling images the lensed GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The probability that a source at redshift 𝑧s is magnified more than 𝜇 is 𝑃(> 𝜇, 𝑧s) = 1 − exp(−𝜏(𝜇, 𝑧s)) = ∫ +∞ 𝜇 𝑝(𝜇, 𝑧s)𝑑𝜇, (9) where 𝑝(𝜇, 𝑧s) = − 𝑑𝜏 𝑑𝜇 exp(−𝜏(𝜇, 𝑧s)) (10) is the magnification PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To understand Equation (9) note that 𝑑𝜏/𝑑𝑧 can be interpreted as a sort of GW scattering rate leading to magnification larger than 𝜇 (per bin of redshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Hence the probability for magnification larger than 𝜇 satisfies the differential equation 𝑑𝑃(> 𝜇, 𝑧)/𝑑𝑧 = (1−𝑃)𝑑𝜏/𝑑𝑧 with solution (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The factor (1 − 𝑃) is essential to keep the probability normalized also when 𝜏 becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the limit of small optical depth, 𝑃(> 𝜇, 𝑧𝑠) ≈ 𝜏(𝜇, 𝑧𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' As mentioned above, note that our approach does not describe de-magnification which happens when a signal crosses a cosmic under-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2 Gravitational lensing applied to a source population We consider a population of sources, that we describe as a function of source redshift 𝑧s and SNR 𝜌 and we denote the number of sources per bin of redshift and SNR as 𝑑N/(𝑑𝜌𝑑𝑧s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' If the magnification is 𝜇, the number of observable sources per bin of 𝑧s, for a given interferometer, reads 𝑑N (𝜇, 𝑧s) 𝑑𝑧s = ∫ ∞ 𝜌lim/√𝜇 𝑑N 𝑑𝜌𝑑𝑧s 𝑑𝜌 , (11) where 𝜌lim is the threshold above which we have a GW detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the rest of the paper, we take 𝜌lim = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Convolving this quantity with the magnification PDF (Equation 10), we get the total number of observed objects in presence of magnification Nobs = ∫ 𝑧max 0 𝑑𝑧s 𝑑N 𝑑𝑧s (𝑧s) = ∫ 𝑧max 0 𝑑𝑧s ∫ +∞ 𝜇min 𝑑𝜇𝑝(𝜇, 𝑧s) 𝑑N (𝜇, 𝑧s) 𝑑𝑧s , (12) where 𝑧max corresponds to the maximum redshift at which we expect to find sources and 𝜇min is the minimum value of magnification considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The probability that if an instrument sees a source from redshift 𝑧s this is magnified more than 𝜇 is given by P(𝑧s, 𝜇) = C ∫ ∞ 𝜇 𝑝(𝜇′, 𝑧s) 𝑑N (𝜇′, 𝑧s) 𝑑𝑧s 𝑑𝜇′ , (13) where C is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3 METHODS In this section we illustrate how we build our model for lens and source distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1 Lens distribution We model the number density of galaxies (lenses) as a function of redshift and of the velocity dispersion 𝜎v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' One can show that if the MNRAS 000, 1–11 (2022) 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 101 102 10 11 10 9 10 7 10 5 10 3 10 1 p( , zs) zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Magnification PDF for different values of source redshift: 𝑧s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 (light blue), 𝑧s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 (orange), 𝑧s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 (green), 𝑧s = 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' evolution of sources is neglected, 𝜏 reduces to (see Cusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2019a) 𝜏(𝜇, 𝑧𝑠) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='001 (𝜇 − 1)2 � 𝐻0𝜒(𝑧𝑠) 𝑐 �3 � 𝑁𝑐3 109𝐻3 0 ⟨𝜎4𝑣⟩ 𝑐4 × 5 × 10−14 � , (14) where 𝑁 is the present galaxy density, 𝑁 = ∫ ∞ 0 𝑑𝜎𝑣𝑛(𝜎𝑣, 𝑧 = 0) , (15) and ⟨𝜎4v ⟩ a mean of the velocity dispersion to power 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Crude esti- mates for 𝑁 and 𝜎v are 𝑁 = 109 𝐻3 0 𝑐3 , ⟨𝜎4 v ⟩ = (150 km/s)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (16) Then the PDF magnification reads 𝑝(𝜇, 𝑧s) = 2𝑝1(𝑧s) (𝜇 − 1)3 exp � 𝑝1(𝑧s) (𝜇 − 1)2 � , 𝑝1(𝑧s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='001 � 𝐻0𝜒(𝑧s) 𝑐 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (17) As shown in Cusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2021), this simplified analytic model gives a result for optical depth in good agreement with the one obtained considering a more realistic distribution of lenses, which evolve with redshift (fractional deviations of a few percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We add to this strong lensing PDF the contribution of weak lens- ing due to the gravitational potential of the Large Scale Structure, which allows us to build a more realistic lens distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To do so, we use results for the magnification probability density of Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2011 (for 𝑧 > 1), which reconstruct the path of light through inhomogeneous clumps of matter in the Universe via high-resolution ray-tracing approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We interpolate their results2 for the red- 2 The probability densities from Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2011 are available on this website http://cosmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='hirosaki-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='jp/takahasi/ raytracing/open_data/ 10 4 10 3 10 2 10 1 100 101 f(Hz) 10 24 10 23 10 22 10 21 10 20 10 19 hc LISA DECIGO Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Sensitivity curves for LISA (blue) and DECIGO (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' shift values we want to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In their simulations they used the box size of 50ℎ−1 Mpc with 10243 particles, the mean particle separation of 50ℎ−1 kpc, and the softening length of 2ℎ−1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The overall magnification PDF that we obtain is shown in Figure 2, for some selected values of 𝑧s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' From this plot, we see that this function in general increases with the source redshift, which is reasonable since for bigger 𝑧s we expect more foreground lenses between the source and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Furthermore, over the magnification interval 1 ≤ 𝜇 ≤ 500, the function decreases very steeply, showing a 10 order-of-magnitude lowering, which shows how higher values of the magnification are generally suppressed in favor of lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2 Source population Before illustrating how we built the source population, we briefly recall the main formulas for the TDE gravitational emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The GWs associated with the disruption of a star can be approxi- mated as a monochromatic burst, with strain and frequency given by (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2022) ℎgw ≈ 2 × 10−22𝛽 × � M∗ M⊙ �4/3 × � 𝑀bh 106M⊙ �2/3 × � 𝑅∗ R⊙ �−1 × � 𝜒 16 Mpc �−1 , (18) 𝑓gw ≈ 𝛽3/2 × 10−4 Hz × � 𝑀∗ M⊙ �1/2 × � 𝑅∗ R⊙ �−3/2 , (19) where 𝑀∗ and 𝑅∗ are the stellar mass and radius, 𝑀bh the BH mass, M⊙ and R⊙ the solar mass and radius, and 𝛽 = 𝑅t/𝑅p is the pene- tration factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' the ratio between the maximum distance from the BH to have a full disruption, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' the tidal radius 𝑅t ≈ 7 × 1012 cm × � 𝑅∗ 𝑅⊙ � × � 𝑀bh 106M⊙ �1/3 × � 𝑀∗ M⊙ �−1/3 , (20) MNRAS 000, 1–11 (2022) Lensed TDEs 5 and the stellar pericenter 𝑅p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Requiring 𝑅p bigger than BH Schwarschild radius, we get the following limits for 𝛽 1 ≲ 𝛽 ≲ 𝛽max ≈ 20 × � 𝑅∗ R⊙ � × � 𝑀∗ M⊙ �−1/3 × � 𝑀bh 106M⊙ �−2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (21) The SNR 𝜌 for such a signal can be written as (Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2022, see also Appendix B) 𝜌 = ℎgw ℎc( 𝑓gw/(1 + 𝑧)) = = 𝛽 × 2 × 10−22 × � 𝑀∗ 𝑀⊙ �4/3 × � 𝑀bh 106M⊙ �2/3 × � 𝑅∗ R⊙ �−1 × � 𝜒 16 Mpc �−1 × 1 ℎc( 𝑓gw/(1 + 𝑧)) , (22) where ℎc is the characteristic noise of the instrument (Moore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Throughout this work, we consider the sensitivity curves of LISA (LISA Science Study Team 2018) and DECIGO (Sato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2017), illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In this study, when considering MS stars, we adopt the following approximated scaling relation (Kippenhahn & Weigert 1990) 𝑀ms 𝑀⊙ ≈ 𝑅ms R⊙ , (23) while for the WD case we assume fixed values for the mass and radius, 𝑀wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5M⊙, 𝑅wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='01R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Note that, from this point forward, we use 𝑀bh ≡ 𝑀•, when referring to massive BHs residing in galaxy cores, while we write 𝑀bh ≡ 𝑀h when referring to IMBHs located in GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1 Nuclear TDEs We build the population of MS stars tidally disrupted by massive BHs residing in galaxy cores following the same steps as in Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020, 𝑑Nms 𝑑𝑧𝑑𝑀•𝑑𝑀★𝑑𝛽 = 4𝜋𝑐𝜒(𝑧)2 𝐻0𝐸(𝑧) Φ(𝑀•)𝜓(𝛽)𝜙(𝑀∗) Γ(𝑀•) (1 + 𝑧)𝑇, (24) where we have the following terms: (i) the comoving volume term 4𝜋𝑐𝜒(𝑧)2/𝐻0𝐸(𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (ii) the distribution of nuclear massive BHs that we build from a Schechter mass function with z-dependence (see Gabasch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2006), expressed in terms of 𝑀• using the Faber-Jackson relation (Faber & Jackson 1976) and the 𝑀• − 𝜎 relation (McConnell & Ma 2013), as done in Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020, 𝜙(𝑀•) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='003Mpc−3 (1 + 𝑧)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='48108M⊙ × � 𝑀• 108M⊙ �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='24 (25) × exp � − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='59 (1 + 𝑧)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='7 � 𝑀• 108M⊙ �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='7� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (iii) the normalized distribution for 𝛽 (Stone & Metzger 2016) 𝜓(𝛽) = 𝛽max(𝑀•, 𝑀∗) 𝛽2(𝛽(𝑀•, 𝑀∗max) − 1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (26) (iv) the normalized Salpeter initial stellar mass function (Salpeter 1955) 𝜙(𝑀∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='35 𝑀−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='35 ∗min − 𝑀−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='35 ∗max 𝑀−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='35 ∗ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (27) (v) the galaxy rate for nuclear TDEs (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g, Stone & Metzger 2016) Γ(𝑀•) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='9 × 10−5 /(yr gal) � 𝑀• 108M⊙ �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='404 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (28) (vi) the observation time𝑇, that we take equal to the lifetime of the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Here we assume, both for LISA and DECIGO, 𝑇 = 10 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2 Globular TDEs We build the population of WDs tidally disrupted by IMBHs located in GCs in a similar way as done in Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020, 𝑑Nwd 𝑑𝑧𝑑𝑀•𝑑𝛽 = 4𝜋𝑐𝜒(𝑧)2 𝐻0𝐸(𝑧) Φ(𝑀•)𝑁gc gal(𝑀•)𝜓(𝛽) Π(𝑀h, 𝑀wd) (1 + 𝑧) 𝑇, (29) where we have the following terms: (i) the comoving volume term 4𝜋𝑐𝜒(𝑧)2/𝐻0𝐸(𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (ii) the distribution of nuclear massive BHs, Φ(𝑀•);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (iii) a scaling relation between the number of GCs per galaxy and the mass of the BH in the core (Harris & Harris 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Burkert & Tremaine 2010) 𝑁gc gal = 𝑀• 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='07 × 105𝑀⊙ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (30) (iv) the rate of globular TDEs per GCs (Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2004) Π ∼ 60Myr−1 × � 𝑅wd R⊙ �4/9 × � 𝑀wd M⊙ �−95/54 × × � 𝑀h 103M⊙ �61/27 × � 𝑛c pc−3 �−7/6 × � 𝑟c 1pc �−49/9 , (31) where we take the GC core density equal to 𝑛c = 105 pc−3 and the GC core radius equal to 𝑟c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We remind that we assume the WD mass and radius to be fixed, 𝑀wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5M⊙, 𝑅wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='01R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We assume that the mass distribution of IMBHs is a 𝛿 function at a fixed value of 𝑀h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In particular, we will build two populations, one with 𝑀h = 103M⊙, the other with 𝑀h = 104M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='3 Range of parameters for the TDE populations To derive the total number of observed TDEs in presence of mag- nification, Nobs, we start by building the nuclear and globular TDE populations according to the aforementioned description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In particu- lar we choose the following ranges: for the source redshift we take 𝑧s ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='001, 2], where the min- imum value of redshift corresponds to ≈ 20Mpc (average distance of the Virgo Cluster), while the maximum value corresponds to the redshift after which the GW emission from a TDE population is negligible (Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' for the central BH mass we take 𝑀• ∈ [104M⊙, 109M⊙], thus considering both dwarf and large galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' for the stellar mass we take 𝑀∗ ∈ [1M⊙, 100M⊙] for the MS star case, hence young stellar population, while for the WD case we assume fixed mass and radius 𝑀wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5M⊙, 𝑅wd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='01R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' for the penetration factor we take 𝛽 ∈ [1, 𝛽max], where the formula for 𝛽max, which in general will depend on the BH and star mass, is illustrated in Equation (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 0 2 4 6 8 10 10 1 101 103 105 107 d ms/d dzs LISA zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 0 1000 2000 3000 4000 5000 6000 10 1 101 103 105 107 d ms/d dzs DECIGO zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Number of nuclear TDEs per bin of SNR and bin of source redshift as a function of SNR 𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left panel we consider LISA, on the right panel we consider DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The pink horizontal dashed line represents 1 TDE per SNR and redshift bin in 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 0 100 200 300 400 500 101 102 103 104 d ms/dzs LISA z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 100 200 300 400 500 105 106 d ms/dzs DECIGO z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Number of visible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' above the threshold 𝜌lim/√𝜇) nuclear TDEs if the magnification is 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left panel we consider LISA, on the right panel we consider DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 4 RESULTS Once we have built the two TDE populations, we can study which is the probability that they are lensed by a foreground population of lenses, using the framework presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We illustrate the main results of our lensing study in the following, distinguishing between the case of nuclear and globular TDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1 Strong lensing of GWs from nuclear TDEs In Figure 4 we show the number of MS stars tidally disrupted by massive BH per bin of SNR and 𝑧s, 𝑑Nms/𝑑𝜌𝑑𝑧s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Equation 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left side we calculate the SNR considering LISA, on the right side DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Each colour represents the number of TDEs in a redshift bin 𝑧s ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='068, and in particular we show selected values of 𝑧s: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 (blue), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 (orange), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 (green), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In general, we MNRAS 000, 1–11 (2022) Lensed TDEs 7 0 1 2 3 4 5 6 7 8 10 1 100 101 102 103 104 d wd/d dzs DECIGO, Mh = 103M zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='0 10 1 100 101 102 103 104 105 106 d wd/d dzs DECIGO, Mh = 104M zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Number of globular TDEs per bin of SNR and bin of source redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left panel we consider 𝑀h = 103M⊙, on the right 𝑀h = 104M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We assume that the interferometer is DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The pink horizontal dashed line represents 1 TDE per SNR and redshift bin in 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 0 100 200 300 400 500 102 103 104 d wd/dzs DECIGO, Mh = 103M z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 0 100 200 300 400 500 105 106 d wd/dzs DECIGO, Mh = 104M z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 z=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Number of visible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' above the threshold 𝜌lim/√𝜇) globular TDEs if the magnification is 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left panel we consider 𝑀h = 103M⊙, on the right panel we consider 𝑀h = 103M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' see that 𝑑Nms/𝑑𝜌𝑑𝑧s diminishes for higher values of 𝜌 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In addition to this, the maximum SNR decreases for higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' As for the minimum SNR, this is always ≈ 0, which is a reasonable result since for each redshift bin we have TDEs below the instruments sensitivity curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In Figure 5, we show the number of visible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' above the threshold 𝜌lim/√𝜇) nuclear TDEs if the magnification is 𝜇, 𝑑Nms(𝜇, 𝑧s)/𝑑𝑧s, calculated through Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The layout and colour are the same as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' From these plots, we see that while for LISA 𝑑Nms(𝜇, 𝑧s)/𝑑𝑧s decreases for higher values of redshift, the same quantity for DECIGO first increases for higher values of 𝑧s, than it starts to lower again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This behavior can be explained in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Since DECIGO has a better sensitivity than LISA, there are two opposite trends that interplay between each other: i) the total number of TDEs increases for higher values of redshift (volume effect), ii) the number of visible TDEs decreases for high values of redshift MNRAS 000, 1–11 (2022) 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (SNR limitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the case of LISA however, which presents a worse sensitivity to TDEs, the ii) effect always prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In other words, LISA is always SNR limited and the very few detectable events decrease rapidly with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Finally, we have all the ingredients to calculate the total number of observed TDEs in presence of magnification, Nobs, through Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Restricting to magnification 𝜇 > 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' focusing on the stronger lensed image, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Section 2 and Appendix A) , we find that for LISA the number of lensed-magnified TDEs is 0, while for DECIGO we expect the detection of ∼130 magnified TDEs (𝜇 > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Yet, this number decreases quite rapidly as we increase the magnification threshold: it reduces to ∼13 for 𝜇 > 3, ∼3 for 𝜇 > 4 and goes to ∼ 0 for 𝜇 > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This fast drop is in agreement with the steep decreasing presented by the magnification PDF illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='2 Strong lensing of GWs from globular TDEs As for the case of globular TDEs, we consider two sub-populations: one where WDs are disrupted by IMBHs of mass 103M⊙, the other with an IMBH mass of 104M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We assume that in each GC there is an IMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' If we consider LISA, both these sub-populations of globular TDEs are below threshold and not even lensing can make part of these sources detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The situation is instead more interesting if we consider DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In Figure 6 we show the number of WDs tidally disrupted by IMBHs per bin of SNR and 𝑧s, 𝑑Nwd/𝑑𝜌𝑑𝑧s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' On the left side we consider 𝑀h = 103M⊙, while on the right 𝑀h = 104M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The colour scheme is the same as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In a similar way as for the nu- clear TDE scenario, we note that: i) 𝑑Nwd/𝑑𝜌𝑑𝑧s shows a general decreasing trend while 𝜌 increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' ii) the maximum SNR decreases for for higher 𝑧s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' iii) the minimum SNR is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In Figure 7, we show the number of observable globular TDEs if the magnification is 𝜇, 𝑑Nwd(𝜇, 𝑧s)/𝑑𝑧s, calculated through Equa- tion 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The layout and colour are the same as in the previous plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Also in this case, we note the interplay between the volume effect against the SNR effect already presented in Figure 5 for DECIGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Finally we have all the ingredients to calculate the number of ob- served magnified TDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For the case 𝑀h = 103M⊙, DECIGO will not observe any TDEs with 𝜇 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Thus, DECIGO will not detect any TDEs from this population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' As for the scenario 𝑀h = 104M⊙, the number of TDEs with 𝜇 > 2 is ∼10, with 𝜇 > 3 is ∼3, and it drops to 0 for higher magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 5 DISCUSSION AND CONCLUSIONS In this paper, we have investigated the effects of gravitational lensing of GW signals from TDEs of MS stars disrupted by massive BHs in galaxy cores (nuclear TDEs) and from TDEs of WDs disrupted by IMBHs in GCs (globular TDEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In order to follow a most realistic procedure as possible, we built the distribution of lenses following an hybrid approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The lenses (galaxies) are modeled as SIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To derive numerical results, we add the contribution of weak lensing from the Large Scale Structure using the results from ray-tracing simulations of Takahashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2011), which include both weak lensing contribution, and strong lensing tails in the magnification PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For the TDE population, we follow similar steps as in Toscani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Our work shows that, while LISA shall not be able to observe lensed-magnified TDEs, the situation will be different for 101 102 10 10 10 8 10 6 10 4 10 2 100 (zs, ) zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 zs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='49 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='10 zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Probability that, if DECIGO detect a nuclear TDE from redshift 𝑧s, this is magnified more than 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' an interferometer with a DECIGO-like sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' While this inter- ferometer will observe ∼10 magnified globular TDEs (𝜇 > 2), we expect to observe ∼130 lensed-magnified nuclear TDEs for 𝜇 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 8 we show a summary plot illustrating, for the most promising scenario of nuclear TDEs observed by DECIGO, the probability that a TDE from redshift 𝑧s is magnified more than 𝜇 (see Equation 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The probability that a TDE observed at redshift 𝑧s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='28 would be magnified more than 𝜇 > 2 is ∼10−4, and increases up to one order of magnitude if we go to higher redshift (𝑧s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' At fixed redshift bin, the probability to have higher magnification decreases steeply (roughly 9 to 10 orders of magnitudes in the interval 1 ≤ 𝜇 ≤ 100), which justifies why the number of magnified TDEs drops rapidly as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Our results point out that DECIGO will observe a non-negligible fraction (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='1%) of strongly lensed TDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Hence data analysis tech- niques need to be developed to be able to distinguish lensed TDEs from unlensed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This will be important to prevent a biased recon- struction of the parameters of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Lensed events will in fact have a (de-)magnified GW amplitude at the detector, which could bias the measurement of source parameters such as its distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Fur- thermore, we expect lensing to induce lensing selection effects on the study of the TDE population, in analogy with what found for example in Cusin & Tamanini (2021) for a population of supermassive black hole binaries visible by LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Indeed, a realistic GW detector has a finite sensitivity: magnified sources are on average easier to detect than de-magnified ones and this affects the distribution of lensing magnification of an observed source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' These lensing selection effects, which should disappear in the limit of a perfect instrument, are then expected to introduce a bias on the reconstruction of the source parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' the luminosity distance), independent of the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Hence, the characterisation of all the implications due to lensing, including selection effects due to the specifics of a given instrument, is necessary to accurately infer the source population as- MNRAS 000, 1–11 (2022) Lensed TDEs 9 trophysical properties across cosmic time, but also to be able to use high-redshift GW sources as a new cosmological probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We would like to remind that our study of lensing relies on the geo- metric optics approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We expect wave effects to become non- negligible in the mHz waveband only when dealing with diffusion off sub-galactic structures, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Takahashi & Nakamura (2003a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Nakamura (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Takahashi (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Dolan (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Cusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2019b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Cusin & Lagos (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Dalang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Diffraction on sub-galactic scales makes lenses on those scales effectively trans- parent to GW in the LISA band (Takahashi & Nakamura 2003a), in contrast with what happens for lensing of EM sources (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Fleury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2015) as the EM spectrum is at much lower wavelengths than any relevant astrophysical structure at cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We conclude by pointing out that TDEs may indeed constitute highly interesting multi-messenger sources, as they emit not only EM radiation, but also GWs and high-energy neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The obser- vation of both EM and GW signals from the same TDE, could in fact enable spectacular multi-messenger analyses which may well unveil new secrets on the intrinsic mechanisms behind these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Concretely for example, the GW signal would mark the moment of stellar disruption at the first pericenter passage, otherwise unde- tectable (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' A measurement of the time delay between the GW signal and the subsequent EM signals would decisively help discriminating between EM emission mechanisms, currently highly debated (Bonnerot & Stone 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We may compare this scenario with GW170817 which revolutionised our understanding of binary neutron star mergers (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2017a,c, 2019a), and triggered new tests of general relativity (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2019b) and new cosmo- logical measurements (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' TDEs may well be used for similar measurements in the future: for example they could allow us to probe the expansion of the universe if the luminosity distance is extracted from the GW signals and the host galaxy is identified from the EM emission, in analogy to massive BH binary mergers with LISA (Tamanini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Belgacem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2019) and double WD mergers with DECIGO (Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The observation of a multi-messenger lensed TDE would not only provide the data for the analyses outlined above, but the differences in the observed EM and GW lensed signals would yield unprecedented opportunities to study properties of both the source and the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For example the EM radiation will always fall within the geometric optics approximation, while as mentioned above the GW signal could show signs of wave optics effects, which may then be used to infer additional information on the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' A detailed analysis of lensed EM emission from TDEs is currently missing in the literature and this will be the subject for a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' ACKNOWLEDGEMENTS MT and NT acknowledge support form the French space agency CNES in the framework of LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The work of GC is supported by CNRS and by the Swiss National Science Foundation (Ambizione grant–Gravitational wave propagation in the clustered Universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' EMR acknowledges that this project has received funding from the European Research Council (ERC) under the European Union’s Hori- zon 2020 research and innovation programme (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 101002511 - VEGA P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Liptai D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', 2022, MNRAS, 510, 992 van Velzen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Holoien T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Onori F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Hung T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Arcavi I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', 2020, Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', 216, 124 APPENDIX A: STRONG LENSING IN A NUTSHELL In this Appendix we present details of the derivation of cross-section and optical depth for a SIS lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' A1 Basic strong lensing quantities A typical situation considered in gravitational lensing is the one illustrated in Figure 1, where a lens of mass 𝑀l at redshift 𝑧l deflects the signal from a source at redshift 𝑧s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The actual path followed by the signal3, which is smoothly curved in the space-time surrounding the lens, can be - as a first approximation - replaced by two straight rays with a kink near the deflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The difference between the angular position of the image and the angular position of the source is called deflection angle, and we denote it as ˆ𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The true position of the source is related to its lensed image on the sky through the lens equation, which reads (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1992) 𝜂𝜂𝜂 = 𝑑s 𝑑l 𝜉𝜉𝜉 − 𝑑ls ˆ𝛼ˆ𝛼ˆ𝛼(𝜉𝜉𝜉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A1) Taking into account the geometry illustrated in Figure 1, the source and lens impact parameters can be written as 𝜂𝜂𝜂 = 𝛾𝛾𝛾𝑑s, 𝜉𝜉𝜉 = 𝜃𝜃𝜃𝑑l, (A2) and thus the Equation (A1) becomes 𝛾𝛾𝛾 = 𝜃𝜃𝜃 − 𝑑ls 𝑑s ˆ𝛼ˆ𝛼ˆ𝛼(𝜃𝜃𝜃𝑑d) ≡ 𝜃𝜃𝜃 − 𝛼𝛼𝛼(𝜃𝜃𝜃), (A3) where we have introduced the scaled deflection angle 𝛼𝛼𝛼 = (𝑑ls/𝑑s) ˆ𝛼ˆ𝛼ˆ𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The scaled deflection angle can be expressed in terms of the conver- gence 𝜅, as 𝛼𝛼𝛼(𝜃𝜃𝜃) = 1 𝜋 ∫ 𝑑2𝜃′𝜅(𝜃′𝜃′𝜃′) 𝜃𝜃𝜃 − 𝜃′𝜃′𝜃′ |𝜃𝜃𝜃 − 𝜃′𝜃′𝜃′| , (A4) where 𝜅 is defined as 𝜅(𝜃𝜃𝜃) = Σ(𝑑l𝜃𝜃𝜃) Σcr , with Σcr = 𝑐2𝑑s 4𝜋𝐺𝑑l𝑑ls .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A5) 3 We recall that we are here working in the geometric optics limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Note that the surface mass density Σ is obtained by integrating the mean mass density of the lens along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We are in the strong lensing limit when Σ > Σcr, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' when the mass distribution of the lens allows the production of multiple images of the source (Kochanek 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Let us now focus on the SIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The mass density of a SIS is given by (Narayan & Bartelmann 1996) 𝜚(𝑟) = 𝜎2v 2𝜋𝐺𝑟2 , (A6) where 𝜎v denotes the velocity dispersion of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Despite the singularity in the center and the infinite total mass, this can be con- sidered as a rather realistic mass distribution for lensing by a galaxy (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1992), with 𝜎v velocity dispersion within the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Integrating along the line of sight we obtain the surface density Σ(𝜃) = 2 × 𝜎2v 2𝜋𝐺 ∫ +∞ 0 1 𝜉2 + 𝑧2 = 𝜎2𝑣 2𝐺 1 𝜉 = 𝜎2𝑣 2𝐺 1 𝑑l𝜃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A7) Thus, for a SIS the convergence reads 𝜅(𝜃) = Σ(𝜃) Σc = 2𝜋 𝜎2v 𝑐2 𝑑ls 𝑑s 1 𝜃 , (A8) with a constant deflection angle 𝛼(𝜃) = 4𝜋 𝜎2v 𝑐2 𝑑ls 𝑑s = 2𝜃𝜅(𝜃) ≡ 𝛼0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A9) This is usually called Einstein angle (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=', Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' 1992), and in order to have multiple images of the source the following condition needs to be fulfilled 𝛾 < 𝛼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' If we rescale our variables by 𝛼0, we can define the rescaled image and source positions as x = 𝜃𝜃𝜃/𝛼0 and y = 𝛾𝛾𝛾/𝛼0, hence Equation (A3) becomes y = x − x |x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A10) We can distinguish three cases: i) for y = 0 the solution is the Einstein ring |x| = 1, ii) for 𝑦 = |y| < 1 one solution is 𝑥1 = |x1| = 1 + 𝑦 (on the same side of the line of sight as the source), the other one is 𝑥2 = |x2| = 1 − 𝑦 (on the opposite side), iii) for 𝑦 > 1 the second solution no longer exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The Jacobian of the lens map is 𝐴𝑖 𝑗 = 𝛿𝑖 𝑗 � 1 − 1 |x| � + 𝑥𝑖𝑥 𝑗 |x|3 , det𝐴 = 1 − 1 |x| , 𝜇 = 1 |det𝐴| = |x| |1 − |x|| , (A11) where we have formally introduced the magnification 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Expressing the total magnification of a point source at position 𝑦 in terms of 𝑦 we find 𝜇(𝑦) = � 𝜇(x1) + 𝜇(x2) = 𝑦+1 𝑦 + 1−𝑦 𝑦 = 𝜇1 + 𝜇2 = 2 𝑦 , 𝑦 ≤ 1 , 𝑦+1 𝑦 = 1 + 1 𝑦 , 𝑦 ≥ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A12) We observe that the magnification is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This is a con- sequence of the fact that SIS is an overdensity, hence it cannot de- magnify the signal of a background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022) Lensed TDEs 11 A2 Cross-section and optical depth The impact parameter of the source (in the source plane) is given by |𝜂| = 𝜂 = 𝛾𝑑𝑠 = 𝑦𝛼0𝑑𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' A source with impact parameter smaller or equal to 𝜂 is amplified by at least a factor 𝜇(𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Hence, considering a SIS with velocity dispersion 𝜎𝑣, the cross section for amplification ≥ 𝜇1 of the stronger image is 𝜍(𝜇1, 𝑧ℓ, 𝑧𝑠, 𝜎𝑣) = 𝜋𝜂2 = 𝜋(𝑦𝛼0𝑑𝑠)2 = 𝜋(4𝜋)2𝜎4𝑣𝑑2 ls 𝑐4(𝜇1 − 1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A13) Note that this cross section gives the area, centered along the line of sight of the lens, within which a source at 𝑧𝑠 must lie so that it is amplified by a factor 𝜇1 or larger by the lens at 𝑧𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The expression (A13) remains valid also for 𝑦 ≥ 1, where we have only one image with magnification 𝜇1 which tends to 1 when 𝑦 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In our study of strong lensing of gravitational waves we consider only one image and not the sum of both, since we expect to see a short burst of GWs which comes only from one image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The second image is delayed in time, with typical time delay of the order of a few months (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Oguri 2018), much longer than the GW burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Since we are interested in magnification, we shall compute the cross section for the stronger image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This point has been raised in Cusin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (2021) but it was neglected in the previous literature: usually in Equation (A13) 𝑦−1 = 𝜇1 − 1 is replaced by 𝑦−1 = 𝜇/2 = (𝜇1 + 𝜇2)/2 which is the correct expression for a static situation where both images are seen together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For strong amplification, 𝜇1 ∼ 𝜇2 ≫ 1 this difference reduces the cross section by a factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To compute the corresponding optical depth, denoted 𝜏(𝜇, 𝑧𝑠), we need to know the physical density 𝑛(𝜎𝑣, 𝑧l) of lenses (galaxies) with a given velocity dispersion 𝜎v at redshift 𝑧l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' We define the density function 𝑛(𝜎v, 𝑧l) such that its integral ∫ 𝑛(𝜎v, 𝑧l)𝑑𝜎v simply gives the total density of lenses at redshift 𝑧l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' The optical depth for lensing with magnification ≥ 𝜇 (of the most strongly magnified image in the case of two images) for a source at redshift 𝑧s is 𝜏(𝜇, 𝑧s) = ∫ 𝑧s 0 𝑑𝑧l 𝑑𝑟 𝑑𝑧l � 𝑑l 𝑑s �2 ∫ 𝜍(𝜇, 𝑧l, 𝑧𝑠, 𝜎v)𝑛(𝜎v, 𝑧l)𝑑𝜎v , (A14) where 𝑑𝑟 is the physical length element at redshift 𝑧l, while 𝑛(𝜎v, 𝑧l) is the physical number density of lenses, which is related to the comoving number density by 𝑛(𝜎v, 𝑧l) = (1 + 𝑧)3𝑛com(𝜎v, 𝑧l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Note also that we have rescaled the cross section to the lens redshift, 𝜍 → 𝜍(𝑑l/𝑑s)2 since in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A14) we multiply by the lens density at 𝑧l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Inserting this and (A13) for the cross section in Equation (A14) we obtain 𝜏(𝜇, 𝑧𝑠) = = 𝜋(4𝜋)2 (𝜇 − 1)2 ∫ 𝑧𝑠 0 𝑑𝑧 𝜒2(𝑧, 𝑧𝑠)𝜒2(𝑧) 𝜒(𝑧𝑠)2(1 + 𝑧)3𝐻(𝑧) ∫ ∞ 0 𝑑𝜎𝑣𝜎4 𝑣𝑛(𝜎v, 𝑧l) , (A15) where 𝜒(𝑧1, 𝑧2) denotes the comoving distance from redshift 𝑧1 to 𝑧2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' (A15) is the optical depth for magnification larger than 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Often only the strong lensing case is considered and the magnifica- tion from the two images is added to give the total magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' To do this one has to replace 1/(𝜇 − 1)2 by 4/𝜇2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' As already mentioned above, here we cannot do this since in the case of strong magnifica- tion and double images we expect a considerable time delay, so that typically we observe only one image at one given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Here we as- sume this to be the stronger image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' In the strong magnification case, 𝜇 ≫ 2, this difference is roughly a factor 4, while in the limit 𝑦 → 1 where 𝜇 → 2 and the second image disappears, the two expressions agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' APPENDIX B: SNR CALCULATION To calculate the SNR 𝜌 of a TDE for a given interferometer, we need to express the signal in the detector frame of reference 𝜌 = ℎdgw ℎc( 𝑓 dgw) = (B1) = 𝛽𝜈 × 2 × 10−22 × � 𝑀d∗ 𝑀⊙ �1/3 × � 𝑀d• 106M⊙ �2/3 × � 𝐷 16Mpc �−1 × 1 ℎc( 𝑓gw/(1 + 𝑧)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' For sake of simplicity, in the above formula we have considered the MS star case and already applied the mass-radius scaling relation of Equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' Considering how to convert frequency and mass from detector to source-frame reference, 𝑀d = 𝑀(1 + 𝑧), (B2) 𝑓 d gw = 𝑓gw/(1 + 𝑧), (B3) and the relation between comoving and luminosity distance (Hogg 1999) 𝜒 = 𝐷 1 + 𝑧 , (B4) we can re-write 𝜌 as presented in Equation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAzT4oBgHgl3EQf1v6R/content/2301.01804v1.pdf'} +page_content=' MNRAS 000, 1–11 (2022)' metadata={'source': 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Korotin,1, ∗ Dmitry Y. Novoselov,1, 2 and Vladimir I. Anisimov1, 2 +1M.N. Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, +18 S. Kovalevskaya St., Yekaterinburg, 620137, Russia. +2Department of Theoretical Physics and Applied Mathematics, +Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia +(Dated: January 3, 2023) +The copper fluoride Cu2F5 is a compound with 2D-magnetic exchange interactions between the +Cu ions in the S = 1 and S = +1 +2 spin-states. +Using ab-initio calculations, we predict that the +existence of 5% vacancies in the fluoride sublattice of Cu2F5 results in the drastic transformation +of the spin-state of all copper ions and the final spin-states are S = 1 +2 and S = 0. Consequently, +the anisotropy of magnetic interactions increases, and the 1D linear chains of the Cu d9, S = 1 +2 ions +appear. We also propose a microscopic mechanism of such exchange interaction transformation via +CuO6 octahedra elongation. +I. +INTRODUCTION +Cuprates +are +well-known +objects +for +the +low- +dimensional magnetism appearance. +With such struc- +tural building blocks as CuO6 octahedra and CuO4 pla- +quettes, and due to the presence of copper ion in d9 elec- +tronic configuration, there is a variety of magnetic struc- +tures. In perovskite-like KCuF3 [1, 2], which structure +is formed with corner-sharing CuF6 octahedra, there is +the G-type antiferromagnetic ordering of moments with +existence of 1D magnetic chains of Cu ions. +2D anti- +ferromagnetic ladders are realized in Srn−1Cun+1O2n [3] +and SrCu2O3 [4] with edge-sharing CuO4 plaquettes. +And the corner-sharing plaquettes in Sr2CuO4 [5] and +AgCuVO4 [6] led to appearance of the one-dimensional +chain of S = 1 +2 Cu ions. +The plethora of spin lattices mentioned above ex- +ist in the copper-oxygen complexes. +The stability of +the copper-fluoride complex, Cu2F5, was predicted re- +cently [7]. Structurally it is formed by both blocks: the +octahedra (CuF6) and the plaquettes (CuF4). +Conse- +quently, one can expect that the magnetic anisotropy, +similar to the one seen in the cuprates, could appear in +the fluoride with or without additional doping with car- +riers. In our previous work [8], within the DFT+U cal- +culations, we show that in stoichiometric Cu2F5, the Cu +ions are in the S = 1 and S = 1 +2 spin-states (d8 and d9 +electronic configuration). Additionally, we obtained that +the Heisenberg exchange interaction along the a crystal +axis is tiny. +In the (100)-plane there is antiferromag- +netic superexchange between the half-filled d-orbitals of +the nearest Cu ions through fluoride p-states located in +between. As a consequence, the 2D spin-lattice exists in +stoichiometric Cu2F5. +One can assume that if an extra electron occupies the +half-filled d-orbital of Cu ion in Cu2F5, the corresponding +Cu-d ↔ F-p ↔ Cu-d superexchange interaction will be +∗ dmitry@korotin.name +destroyed. More extra electrons will cause a significant +modification of the exchange interactions. Our purpose +was to find conditions under which the one-dimensional +magnetic interactions prevail in copper fluoride. +We have produced additional electrons in the cell con- +sidering the doped compound – Cu2F5−x. Each fluoride +vacancy results in an extra electron that occupies one +of the copper d-orbitals. +As is shown in the following +sections, not only electronic and magnetic, but the crys- +tal structure of the compound also evolves following the +doping. +II. +METHODS +In this paper, we follow the methodology defined in the +previous one, describing Cu2F5 [8]. All calculations were +performed using Quantum-ESPRESSO [9] package with +pseudopotentials from pslibrary set [10]. The exchange- +correlation functional was chosen to be in Perdew-Burke- +Ernzerhof [11] form. The energy cutoff for plane wave +wave functions and charge density expansion was set to +50 Ry and 400 Ry, respectively. Integration in the recip- +rocal space was done on a regular 8×8×8 k-points mesh +in the irreducible part of the Brillouin zone. +Electronic +correlations +were +treated +within +the +DFT+U method [12] with the Hubbard U value equals +6 eV. As it was shown for the parent compound Cu2F5, +even the U value of 4 eV is enough for the band gap +to appear. +The variation of the U from 4 to 8 eV +doesn’t change the electronic structure qualitatively and +affects the Heisenberg exchange interaction parameters +only slightly. The Hund parameter J = 0.9 eV was set +to its typical value for cuprates [13, 14]. +The convergence criteria used for crystal cell relaxation +within DFT+U are: total energy < 10−6 Ry, total force +< 10−3 Ry/Bohr, pressure < 0.1 kbar. +arXiv:2301.00396v1 [cond-mat.str-el] 1 Jan 2023 + +2 +FIG. 1. Crystal structure of Cu2F5 and the three types of +fluoride ions that were removed to obtain the structure of +Cu2F5−x. Blue spheres denote Cu ions inside the ligand’s oc- +tahedron, red spheres denote Cu ions in the center of plaque- +ttes, and gray spheres – F ions. Visualized using VESTA [15]. +III. +RESULTS +There are three different structural sites for the fluo- +ride ion in Cu2F5 that could be substituted by a vacancy: +(a) the fluoride shared by CuF4 plaquette and CuF6 oc- +tahedron that are placed along the a axis of the cell; (b) +the ion shared by CuF4 plaquette and CuF6 octahedron +in the direction of the b axis; (c) the fluoride belonging +to the two CuF6 octahedra along the c axis. The Cu-F- +Cu bond angle is 180 for the (b) and (c) ions and only +129 degrees for the case-(a) fluoride ion, which results in +much weaker magnetic exchange interaction in the direc- +tion of the a axis. The absence of one of these F-ions +would naturally lead to a change in the electronic con- +figuration of the nearest Cu ions. With the destruction +of the Cu-F-Cu superexchange path, the disappearance +of one fluoride ion probably will change the magnetic ex- +change interaction pattern in Cu2F5. +In the Cu2F5 crystal cell containing 8 Cu ions (see +Fig. 1), we removed the F ion in (a), (b), or (c) po- +sition sequentially, as described above, and performed +full cell relaxation within the DFT+U approach to ob- +tain a ground state crystal structure. The used cell size +corresponds to a 5% concentration of fluoride vacancies +(Cu2F4.75), and we are focused on this simplest and the +most visual case. +After the relaxation, we compared the enthalpies of +the obtained structures. +The lowest enthalpy has the +cell where the vacancy is placed instead of the F ion in +the (b) position. If F is removed from the (c) position, +the enthalpy is +45 meV / formula unit higher and it +is +46 meV/formula unit higher for the vacancy in the +(a) position. +We conclude that the favorable vacancy +localization site in Cu2F5 is the F ion shared by CuF4 +Parameter +Cu2F5 Cu2F4.75 +Cell volume (˚A3) +369.3 +393.6 +Cuocta - Cuocta distance along a (˚A) +6.98 +6.96 +Cuocta - Cuocta distance along b (˚A) +7.61 +8.08 +Cuocta - Cuocta distance along c (˚A) +7.60 +7.52 +Average Cuplaq-F bond length (˚A) +1.90 +1.80 +Average Cuocta-F bond length (˚A) +1.93 +2.04 +Average Cuocta-F distance along b (˚A) +1.93 +2.26 +TABLE I. Transformation of crystal structure of Cu2F5 with +electrons doping via vacancies. +plaquette and CuF6 octahedron in the direction of the b +axis. Below under the Cu2F4.75 or Cu2F5−x, we mean +the corresponding crystal structure. +The vacancy creation results in the crystal structure +distortions presented in Table I. The unit cell volume +increase of 6.6% happens from the elongation of the cell +along the b lattice vector. At the same time, there is an +expansion of the CuF6 octahedra and a decrease of the +average Cu-F bond length within the CuF4 plaquette in +Cu2F5−x. +The two Cu ions, that had the F ion in between in +Cu2F5, and have the vacancy site instead in Cu2F4.75, +are slightly shifted to each other along the b crystal axis. +Hereafter, the Cu-F bonds of the ex-CuF4 plaquette and +the ex-CuF6 octahedron are distorted when they lose the +shared fluorine ion. For the next-nearest to the vacancy +Cu-F octahedra and plaquettes, one can still say that +the local environment for the Cu ion remains an octahe- +dron and a plaquette. The structure files for Cu2F5 and +Cu2F4.75 could be found here [16]. +The Cu2F4.75 is an antiferromagnetic insulator with +a 0.91 eV band gap. +The calculated partial densities +of states (pDOS) are presented in Fig. 2. +A plausible +assumption would be to say that all copper ions inside +the fluoride octahedra have very similar pDOSes despite +the various distortions of CuF6 structures regarding the +distance from the vacancy. The same is true for the CuF4 +plaquettes. +The corresponding pDOSes are marked in +Fig. 2 as Cuocta and Cuplaq respectively. Here and below, +we refer to the Cu 3z2 −r2, x2 −y2, etc. orbitals in terms +of the local coordinate system for each Cu ion, with the z +direction is perpendicular to the plaquette plane for the +Cuplaq ion and the z direction is along with the crystal b +vector for the Cuocta ion. +From the analysis of pDOS, one can see that Cu ions in +the octahedral environment obtained an additional elec- +tron and became Cu2+ ions in d9, S = 1 +2 electronic con- +figuration: all t2g and 3z2 − r2 orbitals are filled, and +the x2 − y2 orbital is half-filled. At the same time, the +plaquette-surrounded Cu ions are now in d8, S = 0 con- +figuration and have negligible magnetic moments. Both +Cuex−plaq and Cuex−octa ions are in d9, S = 1 +2 electronic +configuration with the hole at the x2 − y2 orbital. We +started our calculation from the cell containing 4 Cuocta +ions in d8, S = 1 configuration + 4 Cuplaq ions in d9, + +Cy +C +Cu +Cu +Cu +C +Cu +Cu +Cu +Cu +C! +Cu +Cu +b +Cu +Cu3 +Cu2F5: Cuplaq +Cu2F5: Cuocta +DOS (Arb. units) +Cuplaq +Cuocta +−4 +−2 +0 +2 +E-Ef (eV) +Cuex−plaq +−4 +−2 +0 +2 +E-Ef (eV) +Cuex−octa +Cu 3z2 − r2 +Cu x2 − y2 +Cu other d +F-p / 8 +FIG. 2. Partial densities of states for Cu2F5 (upper panel) +and Cu2F5−x (middle and lower panels). Positive/negative +pDOSes correspond to spin-up/down states respectively. +S = 1 +2 configuration + an electron from the vacancy. At +the end we’ve obtained 3 Cuplaq ions in d8, S = 0 + 5 +Cu ions (3 Cuocta, Cuex−plaq, Cuex−octa) in d9, S = 1 +2 +configuration. +In the Cu2F5 [8] the lower unoccupied state is the +3z2 − r2 orbital of the Cuocta ions (Fig. 2, upper panel, +right graph). We assumed that as a consequence of the +doping, the additional electron will occupy some of these +orbitals. Consequently, metallization of the Cu2F5 with +electron doping was expected due to the appearance of +the partially filled Cuocta 3z2 − r2 states. Surprisingly +we observe an electron transfer from the Cuplaq ions to +the Cuocta ions in Cu2F5 with doping. We interpret it in +the following way. Due to the partial occupation of the +Cuocta 3z2 − r2 orbital, the corresponding octahedron +elongates in the c direction, that results in the expan- +sion of the cell along the c axis. Consequently, all the +CuF6 octahedra are elongated. Since the Cu-F distances +in the b direction within the octahedron become large, +the Cuocta 3z2 − r2 orbital turns energetically more fa- +vorable than even the Cuplaq x2 − y2 orbital. Electrons +that occupied the Cuplaq x2 − y2 state in the stoichio- +metric Cu2F5 leave it in Cu2F5−x and fill the Cuocta ions +d-shell. As a result, Cuplaq ions that have the d9 config- +uration in Cu2F5, become d8 in the doped structure, and +Cuocta ions change their configuration in reverse way. +To confirm such evolution of crystal and electronic +structure arises from the electronic degrees of freedom, +not from interactions between the vacancy states, we +modeled the electrons doping in a stoichiometric Cu2F5. +The fluoride ion wasn’t removed. We just added one ex- +tra electron in the cell and relaxed the crystal structure. +As a result, the same effect was qualitatively repro- +FIG. 3. Two half-filled d-orbitals of Cu ions that provide the +superexchange interaction via F p-orbital along the b axis in +stoichiometric Cu2F5. In Cu2F5−x the d3z2−r2 become fully +filled and the superexchange is suppressed. +duced. The cell volume increase by 17.5% with its signif- +icant elongation along b direction (Cuocta-Cuocta distance +grows up to 8.08 ˚A). The CuF6 octahedra are stretched +along b direction too, with the corresponding average +Cuocta-F bond length equals 2.25 ˚A. The spin-state of +the copper ions was also changed as a result of the exis- +tence of additional electrons: the Cuocta ions became d9, +S = 1 +2. +We concluded that the electronic configuration and +spin state of the copper ions in Cu2F5 evolve in the same +manner if an extra electron appears regardless of the ori- +gin of the extra electron (vacancy or manual increase in +the number of carriers within the cell). We continue our +presentation and reasoning below for Cu2F5−x with the +vacancy. +The half-filled d3z2−r2-orbital of Cuocta ions in Cu2F5 +gives the superexchange interaction path between Cuocta +and Cuplaq ions along the b crystal axis as it is shown in +Figure 3. The electrons hoping between Cuocta d3z2−r2 +and Cuplaq dx2−y2 via the F-p orbital in between led to +the antiferromagnetic exchange. In Cu2F5−x the possi- +bility of such a hoping of electrons is suppressed, since +Cuocta d3z2−r2 becomes filled. Consequently, the corre- +sponding superexchange interaction along the b crystal +axis vanishes. +The described evolution of electronic structure has an +outcome that the 1D chains of Cuocta ions with a hole +on the dx2−y2-orbital appear. Taking into account the +existence of the fluoride ion between such copper ions +and following the Goodenough-Kanamori rule [17], the +antiferromagnetic superexchange interaction will emerge +along the c crystal axis. +Using Green’s function method based on magnetic- +force linear response theory [2], we computed the Heisen- +berg exchange interaction between Cu ions up to the 9th + +Cu +Cu +a4 +FIG. 4. Evolution of spin-states and patterns of the exchange interaction between the Cu ion from Cu2F5 (left) to Cu2F5−x +(right). Blue elements denote Cu ions in the octahedral surrounding, red elements denote Cu ions in the center of fluoride +plaquettes. The blue and red half-filled triangles correspond to Cuex−octa, S = 1 +2 and Cuex−plaq, S = 1 +2 ions respectively. The +filled, half-filled and empty circles denote S = 1, S = 1 +2, and S = 0 spin-states of the ions. The (100) lattice planes are shown +with a light green color for an eye guide. The vacancy position is shown with a black cross. The strongest exchange interactions +are J2D in Cu2F5 (green line) and J1D in Cu2F5−x (violet line). Other exchange interactions are negligible. Fluorine ions are +not shown for clarity. +nearest neighbor. The model Hamiltonian has the form: +H = − � +⟨ij⟩ Jijeiej, where ei are the unit vectors point- +ing in the direction of the ith site magnetization, and the +summation runs once over each ion pair. +Only one exchange interaction survived under doping. +The antiferromagnetic exchange between the Cuocta ions +along the c-axis is J1D = -29.5 meV. The second largest +interaction ≈ -3.3 meV is between Cuocta and Cuex−plaq +ions along the [101]-direction. +The pattern of the 1D +magnetic chains stems from such exchanges with the in- +terchain interaction being an order of magnitude smaller +than the intrachain one. It is shown in Fig. 4, right panel. +The figure illustrates also the evolution of the strongest +exchange interactions in Cu2F5 that arose from electron +doping. The absolute value of the exchange remains al- +most the same:J2D ≈ −33 meV in Cu2F5, but the di- +mension of the interaction decreases from 2D to 1D as a +result of doping. +We point here to an analogy to the exchange interac- +tion pattern that exists in KCuF3. The potassium copper +fluoride is formed with the CuF6 octahedra, and all cop- +per ions have a hole on the dx2−y2 orbital [14]. +Since +there is the Jahn-Teller distortions of the octahedra, the +lobes of the half-empty d-orbitals of the nearest Cu ions +are perpendicular to each other in the [001]-plane. Con- +sequently, the superexchange via the F p-orbitals in the +[001]-plane is negligible in KCuF3 and the 1D chains of +antiferromagnetically ordered moments appear along the +c crystal axis. Therefore, depsite the different building +blocks of the structure in Cu2F5−x (octahedra and pla- +quettes) and KCuF3 (octahedra only), there is a simi- +larity in the magnetic interactions picture between these +two compounds. +IV. +CONCLUSION +Using the DFT+U calculations, we explored the influ- +ence of the fluoride vacancy appearance on the crystal, +electronic and magnetic structure of Cu2F5−x. +Extra +electrons, which resulted from the absence of the fluorine +ion, are shown to result in the elongation of the CuF6 +octahedra along the b crystal axis and then the 3z2 − r2 +orbital of all Cuocta ions becomes occupied. +As a re- +sult, all the Cu ions in the center of CuF6 octahedra get +S = +1 +2 spin configuration, and the Cu ions inside the +CuF4 plaquettes become non-magnetic (S = 0). Such +significant effect become apparent even when one extra +electron per 4x formula unit is added. The antiferromag- +netic linear chains of copper ions appear along the c-axis +of the crystal. The interchain exchange interaction is ten +times smaller than the largest intrachain one. Our calcu- +lations show consistently that Cu2F5−x can be described +as a quasi-one-dimensional S = 1 +2 Heisenberg chain in a +good approximation. + +5 +ACKNOWLEDGMENTS +Calculation of the ground state crystal structure for +the doped Cu2F5−x was carried out within the state +assignment of Ministry of Science and Higher Educa- +tion of the Russian Federation (theme “Electron” No. +122021000039-4). +Results on the spin-lattice evolution +with doping were obtained with the support of the Rus- +sian Science Foundation (project No. 19-12-00012). +[1] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. +Rev. B 52, R5467 (1995). +[2] D. M. Korotin, V. V. Mazurenko, V. I. Anisimov, +and +S. V. Streltsov, Physical Review B 91, 224405 (2015). +[3] S. Gopalan, T. M. Rice, and M. Sigrist, Physical Review +B 49, 8901 (1994). +[4] T. F. A. M¨uller, V. Anisimov, T. M. Rice, I. Dasgupta, +and T. Saha-Dasgupta, Physical Review B 57, R12655 +(1998). +[5] H. Rosner, H. Eschrig, R. Hayn, S.-L. Drechsler, +and +J. M´alek, Phys. Rev. B 56, 3402 (1997). +[6] A. M¨oller, M. Schmitt, W. Schnelle, T. F¨orster, +and +H. Rosner, Phys. Rev. B 80, 125106 (2009). +[7] N. Rybin, D. Y. Novoselov, D. M. Korotin, V. I. Anisi- +mov, and A. R. Oganov, Phys. Chem. Chem. Phys. 23, +15989 (2021). +[8] D. M. Korotin, D. Y. Novoselov, V. I. Anisimov, +and +A. R. Oganov, Physical Review B 104, 064410 (2021). +[9] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, +C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- +cioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, +G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, +A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, +F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, +L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. +Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcov- +itch, Journal of Physics: Condensed Matter 21, 395502 +(2009). +[10] A. Dal Corso, Computational Materials Science 95, 337 +(2014). +[11] J. P. Perdew, K. Burke, +and M. Ernzerhof, Physical +Review Letters 77, 3865 (1996). +[12] M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, +035105 (2005). +[13] P. Blaha, K. Schwarz, and P. Nov´ak, International Jour- +nal of Quantum Chemistry 101, 550 (2005). +[14] I. Leonov, N. Binggeli, D. Korotin, V. I. Anisimov, +N. Stoji´c, +and D. Vollhardt, Physical Review Letters +101, 096405 (2008). +[15] K. Momma and F. Izumi, Journal of Applied Crystallog- +raphy 44, 1272 (2011). +[16] D. Korotin, “Crystal structure of the Cu2F4.75 and +Cu2F5 +copper fluorides,” Mendeley Data, +V1, +doi: +10.17632/xs2wywd7kd.1 (2022). +[17] J. Goodenough, Scholarpedia 3, 7382 (2008). + diff --git a/ZdAyT4oBgHgl3EQfifhm/content/tmp_files/load_file.txt b/ZdAyT4oBgHgl3EQfifhm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c002175f7cab92b1a49fd180c4e879c549c8e5a --- /dev/null +++ b/ZdAyT4oBgHgl3EQfifhm/content/tmp_files/load_file.txt @@ -0,0 +1,336 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf,len=335 +page_content='2D to 1D magnetic interactions evolution in Cu2F5−x through electron doping by fluoride non-stoichiometry Dmitry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Korotin,1, ∗ Dmitry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Novoselov,1, 2 and Vladimir I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Anisimov1, 2 1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Mikheev Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 18 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Kovalevskaya St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=', Yekaterinburg, 620137, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2Department of Theoretical Physics and Applied Mathematics, Ural Federal University, 19 Mira St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=', Yekaterinburg 620002, Russia (Dated: January 3, 2023) The copper fluoride Cu2F5 is a compound with 2D-magnetic exchange interactions between the Cu ions in the S = 1 and S = 1 2 spin-states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Using ab-initio calculations, we predict that the existence of 5% vacancies in the fluoride sublattice of Cu2F5 results in the drastic transformation of the spin-state of all copper ions and the final spin-states are S = 1 2 and S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Consequently, the anisotropy of magnetic interactions increases, and the 1D linear chains of the Cu d9, S = 1 2 ions appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We also propose a microscopic mechanism of such exchange interaction transformation via CuO6 octahedra elongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' INTRODUCTION Cuprates are well-known objects for the low- dimensional magnetism appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' With such struc- tural building blocks as CuO6 octahedra and CuO4 pla- quettes, and due to the presence of copper ion in d9 elec- tronic configuration, there is a variety of magnetic struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In perovskite-like KCuF3 [1, 2], which structure is formed with corner-sharing CuF6 octahedra, there is the G-type antiferromagnetic ordering of moments with existence of 1D magnetic chains of Cu ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2D anti- ferromagnetic ladders are realized in Srn−1Cun+1O2n [3] and SrCu2O3 [4] with edge-sharing CuO4 plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' And the corner-sharing plaquettes in Sr2CuO4 [5] and AgCuVO4 [6] led to appearance of the one-dimensional chain of S = 1 2 Cu ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The plethora of spin lattices mentioned above ex- ist in the copper-oxygen complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The stability of the copper-fluoride complex, Cu2F5, was predicted re- cently [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Structurally it is formed by both blocks: the octahedra (CuF6) and the plaquettes (CuF4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Conse- quently, one can expect that the magnetic anisotropy, similar to the one seen in the cuprates, could appear in the fluoride with or without additional doping with car- riers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In our previous work [8], within the DFT+U cal- culations, we show that in stoichiometric Cu2F5, the Cu ions are in the S = 1 and S = 1 2 spin-states (d8 and d9 electronic configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Additionally, we obtained that the Heisenberg exchange interaction along the a crystal axis is tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In the (100)-plane there is antiferromag- netic superexchange between the half-filled d-orbitals of the nearest Cu ions through fluoride p-states located in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As a consequence, the 2D spin-lattice exists in stoichiometric Cu2F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' One can assume that if an extra electron occupies the half-filled d-orbital of Cu ion in Cu2F5, the corresponding Cu-d ↔ F-p ↔ Cu-d superexchange interaction will be ∗ dmitry@korotin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='name destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' More extra electrons will cause a significant modification of the exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Our purpose was to find conditions under which the one-dimensional magnetic interactions prevail in copper fluoride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We have produced additional electrons in the cell con- sidering the doped compound – Cu2F5−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Each fluoride vacancy results in an extra electron that occupies one of the copper d-orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As is shown in the following sections, not only electronic and magnetic, but the crys- tal structure of the compound also evolves following the doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' METHODS In this paper, we follow the methodology defined in the previous one, describing Cu2F5 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' All calculations were performed using Quantum-ESPRESSO [9] package with pseudopotentials from pslibrary set [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The exchange- correlation functional was chosen to be in Perdew-Burke- Ernzerhof [11] form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The energy cutoff for plane wave wave functions and charge density expansion was set to 50 Ry and 400 Ry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Integration in the recip- rocal space was done on a regular 8×8×8 k-points mesh in the irreducible part of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Electronic correlations were treated within the DFT+U method [12] with the Hubbard U value equals 6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As it was shown for the parent compound Cu2F5, even the U value of 4 eV is enough for the band gap to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The variation of the U from 4 to 8 eV doesn’t change the electronic structure qualitatively and affects the Heisenberg exchange interaction parameters only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The Hund parameter J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='9 eV was set to its typical value for cuprates [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The convergence criteria used for crystal cell relaxation within DFT+U are: total energy < 10−6 Ry, total force < 10−3 Ry/Bohr, pressure < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='1 kbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='00396v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='str-el] 1 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Crystal structure of Cu2F5 and the three types of fluoride ions that were removed to obtain the structure of Cu2F5−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Blue spheres denote Cu ions inside the ligand’s oc- tahedron, red spheres denote Cu ions in the center of plaque- ttes, and gray spheres – F ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Visualized using VESTA [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' RESULTS There are three different structural sites for the fluo- ride ion in Cu2F5 that could be substituted by a vacancy: (a) the fluoride shared by CuF4 plaquette and CuF6 oc- tahedron that are placed along the a axis of the cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' (b) the ion shared by CuF4 plaquette and CuF6 octahedron in the direction of the b axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' (c) the fluoride belonging to the two CuF6 octahedra along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The Cu-F- Cu bond angle is 180 for the (b) and (c) ions and only 129 degrees for the case-(a) fluoride ion, which results in much weaker magnetic exchange interaction in the direc- tion of the a axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The absence of one of these F-ions would naturally lead to a change in the electronic con- figuration of the nearest Cu ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' With the destruction of the Cu-F-Cu superexchange path, the disappearance of one fluoride ion probably will change the magnetic ex- change interaction pattern in Cu2F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In the Cu2F5 crystal cell containing 8 Cu ions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 1), we removed the F ion in (a), (b), or (c) po- sition sequentially, as described above, and performed full cell relaxation within the DFT+U approach to ob- tain a ground state crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The used cell size corresponds to a 5% concentration of fluoride vacancies (Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75), and we are focused on this simplest and the most visual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' After the relaxation, we compared the enthalpies of the obtained structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The lowest enthalpy has the cell where the vacancy is placed instead of the F ion in the (b) position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' If F is removed from the (c) position, the enthalpy is +45 meV / formula unit higher and it is +46 meV/formula unit higher for the vacancy in the (a) position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We conclude that the favorable vacancy localization site in Cu2F5 is the F ion shared by CuF4 Parameter Cu2F5 Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75 Cell volume (˚A3) 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='3 393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='6 Cuocta - Cuocta distance along a (˚A) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='98 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='96 Cuocta - Cuocta distance along b (˚A) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='61 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='08 Cuocta - Cuocta distance along c (˚A) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='52 Average Cuplaq-F bond length (˚A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='80 Average Cuocta-F bond length (˚A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='04 Average Cuocta-F distance along b (˚A) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='93 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='26 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Transformation of crystal structure of Cu2F5 with electrons doping via vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' plaquette and CuF6 octahedron in the direction of the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Below under the Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75 or Cu2F5−x, we mean the corresponding crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The vacancy creation results in the crystal structure distortions presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The unit cell volume increase of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='6% happens from the elongation of the cell along the b lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' At the same time, there is an expansion of the CuF6 octahedra and a decrease of the average Cu-F bond length within the CuF4 plaquette in Cu2F5−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The two Cu ions, that had the F ion in between in Cu2F5, and have the vacancy site instead in Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75, are slightly shifted to each other along the b crystal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Hereafter, the Cu-F bonds of the ex-CuF4 plaquette and the ex-CuF6 octahedron are distorted when they lose the shared fluorine ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' For the next-nearest to the vacancy Cu-F octahedra and plaquettes, one can still say that the local environment for the Cu ion remains an octahe- dron and a plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The structure files for Cu2F5 and Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75 could be found here [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The Cu2F4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='75 is an antiferromagnetic insulator with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='91 eV band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The calculated partial densities of states (pDOS) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' A plausible assumption would be to say that all copper ions inside the fluoride octahedra have very similar pDOSes despite the various distortions of CuF6 structures regarding the distance from the vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The same is true for the CuF4 plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The corresponding pDOSes are marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2 as Cuocta and Cuplaq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Here and below, we refer to the Cu 3z2 −r2, x2 −y2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' orbitals in terms of the local coordinate system for each Cu ion, with the z direction is perpendicular to the plaquette plane for the Cuplaq ion and the z direction is along with the crystal b vector for the Cuocta ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' From the analysis of pDOS, one can see that Cu ions in the octahedral environment obtained an additional elec- tron and became Cu2+ ions in d9, S = 1 2 electronic con- figuration: all t2g and 3z2 − r2 orbitals are filled, and the x2 − y2 orbital is half-filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' At the same time, the plaquette-surrounded Cu ions are now in d8, S = 0 con- figuration and have negligible magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Both Cuex−plaq and Cuex−octa ions are in d9, S = 1 2 electronic configuration with the hole at the x2 − y2 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We started our calculation from the cell containing 4 Cuocta ions in d8, S = 1 configuration + 4 Cuplaq ions in d9, Cy C Cu Cu Cu C Cu Cu Cu Cu C!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Cu Cu b Cu Cu3 Cu2F5: Cuplaq Cu2F5: Cuocta DOS (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' units) Cuplaq Cuocta −4 −2 0 2 E-Ef (eV) Cuex−plaq −4 −2 0 2 E-Ef (eV) Cuex−octa Cu 3z2 − r2 Cu x2 − y2 Cu other d F-p / 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Partial densities of states for Cu2F5 (upper panel) and Cu2F5−x (middle and lower panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Positive/negative pDOSes correspond to spin-up/down states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' S = 1 2 configuration + an electron from the vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' At the end we’ve obtained 3 Cuplaq ions in d8, S = 0 + 5 Cu ions (3 Cuocta, Cuex−plaq, Cuex−octa) in d9, S = 1 2 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In the Cu2F5 [8] the lower unoccupied state is the 3z2 − r2 orbital of the Cuocta ions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 2, upper panel, right graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We assumed that as a consequence of the doping, the additional electron will occupy some of these orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Consequently, metallization of the Cu2F5 with electron doping was expected due to the appearance of the partially filled Cuocta 3z2 − r2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Surprisingly we observe an electron transfer from the Cuplaq ions to the Cuocta ions in Cu2F5 with doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We interpret it in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Due to the partial occupation of the Cuocta 3z2 − r2 orbital, the corresponding octahedron elongates in the c direction, that results in the expan- sion of the cell along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Consequently, all the CuF6 octahedra are elongated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Since the Cu-F distances in the b direction within the octahedron become large, the Cuocta 3z2 − r2 orbital turns energetically more fa- vorable than even the Cuplaq x2 − y2 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Electrons that occupied the Cuplaq x2 − y2 state in the stoichio- metric Cu2F5 leave it in Cu2F5−x and fill the Cuocta ions d-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As a result, Cuplaq ions that have the d9 config- uration in Cu2F5, become d8 in the doped structure, and Cuocta ions change their configuration in reverse way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' To confirm such evolution of crystal and electronic structure arises from the electronic degrees of freedom, not from interactions between the vacancy states, we modeled the electrons doping in a stoichiometric Cu2F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The fluoride ion wasn’t removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We just added one ex- tra electron in the cell and relaxed the crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As a result, the same effect was qualitatively repro- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Two half-filled d-orbitals of Cu ions that provide the superexchange interaction via F p-orbital along the b axis in stoichiometric Cu2F5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In Cu2F5−x the d3z2−r2 become fully filled and the superexchange is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The cell volume increase by 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='5% with its signif- icant elongation along b direction (Cuocta-Cuocta distance grows up to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='08 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The CuF6 octahedra are stretched along b direction too, with the corresponding average Cuocta-F bond length equals 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='25 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The spin-state of the copper ions was also changed as a result of the exis- tence of additional electrons: the Cuocta ions became d9, S = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We concluded that the electronic configuration and spin state of the copper ions in Cu2F5 evolve in the same manner if an extra electron appears regardless of the ori- gin of the extra electron (vacancy or manual increase in the number of carriers within the cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We continue our presentation and reasoning below for Cu2F5−x with the vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The half-filled d3z2−r2-orbital of Cuocta ions in Cu2F5 gives the superexchange interaction path between Cuocta and Cuplaq ions along the b crystal axis as it is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The electrons hoping between Cuocta d3z2−r2 and Cuplaq dx2−y2 via the F-p orbital in between led to the antiferromagnetic exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' In Cu2F5−x the possi- bility of such a hoping of electrons is suppressed, since Cuocta d3z2−r2 becomes filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Consequently, the corre- sponding superexchange interaction along the b crystal axis vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The described evolution of electronic structure has an outcome that the 1D chains of Cuocta ions with a hole on the dx2−y2-orbital appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Taking into account the existence of the fluoride ion between such copper ions and following the Goodenough-Kanamori rule [17], the antiferromagnetic superexchange interaction will emerge along the c crystal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Using Green’s function method based on magnetic- force linear response theory [2], we computed the Heisen- berg exchange interaction between Cu ions up to the 9th Cu Cu a4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Evolution of spin-states and patterns of the exchange interaction between the Cu ion from Cu2F5 (left) to Cu2F5−x (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Blue elements denote Cu ions in the octahedral surrounding, red elements denote Cu ions in the center of fluoride plaquettes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The blue and red half-filled triangles correspond to Cuex−octa, S = 1 2 and Cuex−plaq, S = 1 2 ions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The filled, half-filled and empty circles denote S = 1, S = 1 2, and S = 0 spin-states of the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The (100) lattice planes are shown with a light green color for an eye guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The vacancy position is shown with a black cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The strongest exchange interactions are J2D in Cu2F5 (green line) and J1D in Cu2F5−x (violet line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Other exchange interactions are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Fluorine ions are not shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' nearest neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The model Hamiltonian has the form: H = − � ⟨ij⟩ Jijeiej, where ei are the unit vectors point- ing in the direction of the ith site magnetization, and the summation runs once over each ion pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Only one exchange interaction survived under doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The antiferromagnetic exchange between the Cuocta ions along the c-axis is J1D = -29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The second largest interaction ≈ -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content='3 meV is between Cuocta and Cuex−plaq ions along the [101]-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The pattern of the 1D magnetic chains stems from such exchanges with the in- terchain interaction being an order of magnitude smaller than the intrachain one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' It is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 4, right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The figure illustrates also the evolution of the strongest exchange interactions in Cu2F5 that arose from electron doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The absolute value of the exchange remains al- most the same:J2D ≈ −33 meV in Cu2F5, but the di- mension of the interaction decreases from 2D to 1D as a result of doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' We point here to an analogy to the exchange interac- tion pattern that exists in KCuF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The potassium copper fluoride is formed with the CuF6 octahedra, and all cop- per ions have a hole on the dx2−y2 orbital [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Since there is the Jahn-Teller distortions of the octahedra, the lobes of the half-empty d-orbitals of the nearest Cu ions are perpendicular to each other in the [001]-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Con- sequently, the superexchange via the F p-orbitals in the [001]-plane is negligible in KCuF3 and the 1D chains of antiferromagnetically ordered moments appear along the c crystal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Therefore, depsite the different building blocks of the structure in Cu2F5−x (octahedra and pla- quettes) and KCuF3 (octahedra only), there is a simi- larity in the magnetic interactions picture between these two compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' CONCLUSION Using the DFT+U calculations, we explored the influ- ence of the fluoride vacancy appearance on the crystal, electronic and magnetic structure of Cu2F5−x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Extra electrons, which resulted from the absence of the fluorine ion, are shown to result in the elongation of the CuF6 octahedra along the b crystal axis and then the 3z2 − r2 orbital of all Cuocta ions becomes occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' As a re- sult, all the Cu ions in the center of CuF6 octahedra get S = 1 2 spin configuration, and the Cu ions inside the CuF4 plaquettes become non-magnetic (S = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Such significant effect become apparent even when one extra electron per 4x formula unit is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The antiferromag- netic linear chains of copper ions appear along the c-axis of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' The interchain exchange interaction is ten times smaller than the largest intrachain one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Our calcu- lations show consistently that Cu2F5−x can be described as a quasi-one-dimensional S = 1 2 Heisenberg chain in a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 5 ACKNOWLEDGMENTS Calculation of the ground state crystal structure for the doped Cu2F5−x was carried out within the state assignment of Ministry of Science and Higher Educa- tion of the Russian Federation (theme “Electron” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 122021000039-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Results on the spin-lattice evolution with doping were obtained with the support of the Rus- sian Science Foundation (project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' 19-12-00012).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Bonini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Calandra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Car, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Cavazzoni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Ceresoli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdAyT4oBgHgl3EQfifhm/content/2301.00396v1.pdf'} +page_content=' Chiarotti, M.' 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b/_tAyT4oBgHgl3EQfqvix/content/tmp_files/2301.00549v1.pdf.txt @@ -0,0 +1,1766 @@ +Prepared for submission to JHEP +CERN-TH-2022-176 +UV and IR Effects in Axion Quality Control +C.P. Burgess,1,2 Gongjun Choi3 and F. Quevedo4 +1 Department of Physics & Astronomy, McMaster University +1280 Main Street West, Hamilton ON, Canada. +2 Perimeter Institute for Theoretical Physics +31 Caroline Street North, Waterloo ON, Canada. +3 CERN, Theoretical Physics Department, Gen`eve 23, Switzerland. +4 DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK. +Abstract: Motivated by recent discussions and the absence of exact global symmetries in UV com- +pletions of gravity we re-examine the axion quality problem (and naturalness issues more generally) +using antisymmetric Kalb-Ramond (KR) fields rather than their pseudoscalar duals, as suggested by +string and higher dimensional theories. Two types of axions can be identified: a model independent +S-type axion dual to a two form Bµν in 4D and a T-type axion coming directly as 4D scalar Kaluza- +Klein (KK) components of higher-dimensional tensor fields. For T-type axions our conclusions largely +agree with earlier workers for the axion quality problem, but we also reconcile why T-type axions can +couple to matter localized on 3-branes with Planck suppressed strength even when the axion decay +constants are of order the KK scale. For S-type axions, we review the duality between form fields +and massive scalars and show how duality impacts naturalness arguments about the UV sensitivity +of the scalar potential. In particular UV contributions on the KR side suppress contributions on the +scalar side by powers of m/M with m the axion mass and M the UV scale. We re-examine how the +axion quality problem is formulated on the dual side and compare to recent treatments. We study +how axion quality is affected by the ubiquity of p-form gauge potentials (for both p = 2 and p = 3) +in string vacua and identify two criteria that can potentially lead to a problem. We also show why +most fields do not satisfy these criteria, but when they do the existence of multiple fields also provides +mechanisms for resolving it. We conclude that the quality problem is easily evaded. +arXiv:2301.00549v1 [hep-th] 2 Jan 2023 + +Contents +1 +Introduction +1 +1.1 +Types of UV axion pedigree +2 +1.2 +Non-propagating low-energy forms +3 +2 +Axions and duality +4 +2.1 +Axion/2-form duality +4 +2.2 +A Higgs mechanism for scalar masses +5 +3 +Naturalness issues for dual systems +7 +3.1 +QCD and the dual PQ mechanism +7 +3.2 +The Quality Problem +10 +3.3 +The dual Quality Problem +12 +4 +UV completion and matter couplings +18 +4.1 +Extra-dimensional UV completion +18 +4.2 +Coupling strengths +20 +5 +Conclusions +22 +1 +Introduction +String theory giveth and string theory taketh away, at least where axions1 are concerned. On one hand +axions are said to be ubiquitous in the spectrum of particles predicted around most string vacua [4, 5]. +This observation motivates the study of their phenomenological consequences [6], with a particular +focus of late on their possible role as a light form of dark matter [7]. +On the other hand, string theory equally generally forbids2 the existence of exact rigid (or global) +symmetries [8], in principle including the rigid shift symmetries on which low-energy axion properties +are founded. For Goldstone bosons this breaking can keep them from being light, and can interfere +with any mechanisms that rely on the survival of axions down to the low-energy theory. As applied +to the QCD axion this has come to be known as the axion ‘quality’ problem [10]. +So which is it? Are axions as abundant as dirt or as diamonds in low-energy string vacua? The +resolution (which has long been known) is that there is a sense they are both. The absence of global +symmetries really does mean that one never really directly finds scalar axions a with shift symmetries +in string vacua. Instead these scalars arise indirectly as Kaluza-Klein (KK) modes from fields that +1We follow the string literature and broadly refer to any low-energy Goldstone boson enjoying a rigid compact shift +symmetry as an ‘axion’ (as opposed to the ‘dilatons’ associated with rigid scaling symmetries), something that would be +called an ALP (axion-like particle) by particle phenomenologists. Our later focus is on those Goldstone bosons whose +symmetries have a QCD anomaly and so can take part in the strong-CP problem [1–3] (which is what a particle physicist +would usually mean by an ‘axion’). +2Although there are known ways out [9] the conclusion is nonetheless broadly true and global symmetries tend to be +both rare and approximate. +– 1 – + +not themselves scalars; commonly arising3 as components of 2-form Kalb-Ramond gauge fields [11], +B = +1 +2 BMN dzM ∧ dzN, subject to the gauge symmetries B → dλ for some arbitrary field λM(x). +Fields like BMN arise so frequently in string vacua because they are related to other fields (notably +the metric) by supersymmetry in higher dimensions. +1.1 +Types of UV axion pedigree +Low-energy scalars typically emerge in the 4D effective theory from such fields in one of two ways: +• T-type axions: b(x) are specific cases of Kaluza-Klein (KK) modes arising when dimension- +ally reducing the extra-dimensional components Bmn(x, y) = b(x) ωmn(y), where xµ denote the +observed 4 dimensions, ym are extra-dimensional coordinates and ωmn(y) is a harmonic 2-form +field within the extra dimensions. +• S-type axions: a(x) arise directly as the 4-dimensional components Bµν(x, y) = bµν(x) ω(y), +which in four dimensions are known to be dual to scalar fields with shift symmetries [12] through +relations of the form ∂µa ∝ ϵµνλρ∂νBλρ (much more about which below). Here ω(y) is a harmonic +0-form field – typically a y-independent constant that can depend on extra-dimensional moduli. +This UV provenance is of course relevant to the axion quality problem, which is in essence an issue +of UV sensitivity. One of our goals with this paper is to explore the ways that it helps, for both T- +and S-type axions. Some of our conclusions are similar to earlier discussions of this issue [13, 14], in +particular that the problem gets rephrased in dual form (for S-type axions) in terms of the existence +of multiple 3-form gauge potentials. +Since these issues have recently been revisited anew [15, 16] we clarify what properties these +fields must have to actually cause a quality problem and use this to argue why gravitational examples +specifically (and the great abundance of such potentials in string vacua more generally) need not pose +a problem in themselves. The dual formulation also suggests how the presence of multiple axions (as is +common in string theory) can help alleviate the quality problem. The upshot is that the UV can, but +need not, cause a quality problem. Whether or not it does cannot be decided purely at low energies +because it depends on what happens in the UV.4 +But our discussion has implications that apply more broadly than just to the quality problem for +the QCD axion. Along the way we identify more generally how dimensional ‘naturalness’ arguments +for the scalar potential give very different estimates depending on whether they are made directly for +the scalar or are first done for its dual and then mapped to the scalar using duality. In particular +terms involving n powers of the canonically normalized scalar arise additionally suppressed by powers +of (m/M)n where M is the UV scale and m is the axion mass (an observation also made in the past +for inflationary models [18]). +We find a number of other ways that axion properties suggested by string-motivated extra- +dimensional physics can be informative. For instance we describe a simple model for which T-type +axions have physical axion-matter couplings gaff that are dramatically smaller than the naive value +1/f read off from the axion kinetic term. In the example given (motivated by the models of [19]) gaff +is order 1/Mp despite f being an ordinary particle-physics scale. Decoupling these scales from one +another could have practical implications for axion phenomenology. +34D axions can also arise as KK modes from other types of extra-dimensional fields, but we focus on the Kalb-Ramond +field because it allows a unified treatment of two different types of 4D axion. +4The same is is also true of other naturalness problems; they arise because of strong dependence on physical masses +for states that actually appear in the UV theory and not a dependence on cutoffs, as is sometimes mistakenly asserted +(for a summary of these issues see e.g. [17]). +– 2 – + +We show why the same hierarchy does not arise in these models for S-type axions and we clarify +why not. Physical couplings of S-type axions really are of order 1/f and we identify which interactions +in the UV completion are responsible for the breakdown of the E/f expansion at energies E > +∼ f. S- +type axions illustrate how the scalar and dual representations can provide instances of weak/strong +coupling duality, for which both the scalar and the dual cannot be within the weakly coupled regime. +In the extra-dimensional example studied it is the Kalb-Ramond formulation that is weakly coupled. +This could also have phenomenological implications to the extent that an axion that is dual to a weakly +coupled system is unlikely to be well-described by the semiclassical methods that are universally used +when exploring its physical implications. +Some of these observations imply that the use of the scalar (rather than Kalb-Ramond) vari- +able can be misleading in some circumstances. This can seem surprising at first sight because the +duality between axions and Kalb-Ramond fields is in essence a field redefinition and so scalar and +dual formulations should be completely equivalent; it shouldn’t matter that string theory hands you +Kalb-Ramond fields if scalar axions are equivalent and are much simpler to work with. Why should +one care that a more complicated framework exists if it only obscures implications drawn using more +transparent methods? We argue here that phenomena like weak/strong coupling duality are special +cases of Weinberg’s Third Law of Progress in Theoretical Physics [20]: You can use any degrees of +freedom you like to describe a physical system, but if you use the wrong ones you’ll be sorry. +1.2 +Non-propagating low-energy forms +These duality arguments touch on a related rich vein of physics with broader significance: the im- +portance of keeping non-propagating entities like auxiliary and/or topological fields when formulating +Wilsonian effective theories. These are fields that can be integrated out without changing the types of +particles that propagate, and so it is tempting to think one should do so once and for all and simply +ignore them thereafter. However such fields bring to the low-energy effective theory information about +how its UV completion responds, e.g. to environments with nontrivial topology. They arise in concrete +situations (such as in EFTs for 3-dimensional Quantum Hall systems, where the presence of emergent +non-propagating gauge fields is essential for capturing the fractional quantization of Hall plateaux and +the unusual charge and statistics of some excitations [21, 22]). +Evidence is building that a similar role is played more widely by 3-form gauge potentials in four +spacetime dimensions, C := +1 +6 Cµνλ dxµdxνdxλ subject to the gauge freedom C → C + dΛ where +Λµν(x) = −Λνµ(x) is an arbitrary 2-form field. +These are known to bring to the low-energy 4D +effective theory topological information coming from integrated out extra dimensions [23, 24], and +more generally provide the origin for the auxiliary fields that appear in the 4D supergravities that +are the low-energy limits of string vacua [25, 26]. They appear in the QCD quality story because +they can give masses to Kalb-Ramond fields [27] through a Higgs mechanism that is dual to more +mundane methods of axion mass generation. Because the field strength H = dC often appears in the +action with a definite sign (often as a square), its presence can alter the implications of naturalness +arguments for the scalar potential [28]. Indeed such terms provide the 4D understanding of why 6D +SLED models [29] can in some circumstances suppress the 4D vacuum energy, but also why they +struggle to do so enough to solve the cosmological constant problem [24, 30–32]. Their interplay with +accidental scaling symmetries lies behind a recent attempt to find a dynamical relaxation mechanism +for vacuum energies in four dimensions [19]. +In what follows we build our case for the above story using concrete examples. We first, in §2, +briefly review the duality construction – in particular its extension to massive axions [27], which +provides a way to think about scalar masses arising through a Higgs mechanism. +§3 then briefly +– 3 – + +reviews and clarifies its use to dualize the axion solution to the strong-CP problem [14], highlighting +in particular how the quality problem gets rephrased in the dual language and how the apparent UV +sensitivity of terms in the EFT differs between the axion formulation and its dual. Finally §4 provides +a concrete extra-dimensional example – inspired by a UV completion of [19] – that illustrates both +how axion/Kalb-Ramond duality can map weak to strong couplings, and how enormous hierarchies +can arise with gaff, gaγγ ∼ 1/Mp even with f as low as eV scales. +2 +Axions and duality +We start with a review of why 2-form gauge potentials like Bµν are dual [12, 33] to scalar fields, +both in the standard shift-symmetric massless case and for massive scalars, following the discussion +of [27] (which in turn generalizes earlier arguments [34] aimed at describing particle/vortex duality in +Kosterlitz-Thouless transitions [35]). +2.1 +Axion/2-form duality +Consider the following path integral +Ξ[J] = +� +DB eiS1[B] +(2.1) +where S1 = +� +d4x L1 with L1 chosen (at least to start) to be +L1(B) = − Z +2 · 3! GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ , +(2.2) +with G = dB the exterior derivative of a 2-form field Bµν and Z and Jρ possibly depending on other +fields (collectively denoted ψ). B = +1 +2 Bµν dxµ ∧ dxν is only defined up to the gauge redundancy +B → B + dλ for an arbitrary 1-form λ = λµ dxµ. +The duality starts by trading the integration over Bµν for an integral over Gµνλ subject to a +constraint that imposes the Bianchi identity dG = 0. +These are equivalent because the Bianchi +identity is sufficient to guarantee the local existence of a field Bµν with G = dB. The constraint is +imposed by integrating over a scalar Lagrange-multiplier field a, and so writing +Ξ[J] = +� +DG Da eiS0 +(2.3) +where S0 = +� +d4x L0 with +L0(G, a) = − Z +2 · 3! GµνλGµνλ − 1 +3!a ϵµνλρ∂µGνλρ − 1 +3! ϵµνλρGµνλJρ , +(2.4) +Integrating out a imposes the Bianchi identity dG = 0 and allows the integral over G to be replaced +with the integral over B, leading back to (2.2). +The dual version is obtained from (2.4) by instead integrating out Gµνλ so that a is the remaining +field. The result inherits a shift symmetry a → a + constant because L0 transforms into a total +derivative. The G integration is gaussian, whose saddle point is Gµνλ = Gµνλ where +Gµνλ = −Z−1ϵµνλρ +� +∂ρa + Jρ� +, +(2.5) +and so the integration gives the new lagrangian density +L2(a) = − 1 +2Z (∂µa + Jµ)(∂µa + Jµ) . +(2.6) +– 4 – + +If Z = 1 then a is a canonically normalized massless scalar derivatively coupled to the same local +current Jµ as in the original formulation. Because (2.2) and (2.6) are both obtained from (2.4) they +must describe equivalent physics. Although the implied field redefinition from Bµν to a is in principle +nonlocal the physics on both sides is nonetheless local because this is true of the relation between the +field strengths given in (2.5). +Significance of Z ↔ Z−1 +In reality the above gaussian action is always supplemented by other non-gaussian interactions Lint +within a low-energy Wilsonian effective field theory (EFT). To the extent that both Bµν and a are +derivatively coupled perturbative semiclassical methods in the presence of nongaussian terms like +(GµνλGµνλ)2 ∈ Lint are ultimately justified by a low-energy derivative expansion that applies equally +well on both sides of a duality relationship because relationships like (2.5) involve equal numbers of +derivatives on both sides. +The inversion of Z → Z−1 as one passes from (2.2) to (2.6) is a noteworthy feature of duality. +When Z ≫ 1 this implies 2-point correlators of Gµνλ are order Z−1 in size while those of ∂µa are +instead order Z. The significance of the change Z → Z−1 depends on whether or not Bµν and a can +be freely rescaled to remove Z by going to canonically normalized variables. If this is so then Z in +any case drops out of observables. For instance, when Jµ ̸= 0 this rescaling shows that Ξ is really only +a function of �Jµ := Z−1/2Jµ rather than depending on Z and Jµ separately. Although Z ↔ Z−1 is +sometimes called weak/strong coupling duality, Ξ[ �J] is the same on both sides of the duality and so +both sides agree on its functional dependence if expanded order-by-order in powers of Z−1 (say). +One situation where this kind of rescaling is not possible is when Z depends on other fields and the +target-space metric in field space is not flat. Another case where physics can depend explicitly on Z is +when the field Bµν or a is quantized5, perhaps satisfying a boundary condition like +� +W dxµ∂µa = 2πnf +for some curve W, integer n and mass scale f, or perhaps +� +C B = 2π˜nf −1 for some 2-cycle C and +possibly different integer ˜n and mass scale ˜f. In these situations physical results can depend on Z +(i.e. on f and/or ˜f) and Jµ separately, and the relation Z → Z−1 can carry physical significance. +2.2 +A Higgs mechanism for scalar masses +Although the above makes the shift symmetry (and so also masslessness) of a seem automatic, we +next summarize how duality extends to massive scalars, following [27]. A scalar potential is achieved +in the dual framing through a Higgs mechanism in which the field Bµν ‘eats’ (or is eaten by) a non- +propagating gauge potential6 Cµνλ. Because Cµνλ does not propagate this meal does not change the +number of propagating degrees of freedom. +To this end consider the following gaussian path integral +Ξ[J] = +� +DC DB eiS1 +(2.7) +where S1 = +� +d4x L1 and +L1(C, B) = − +1 +2 · 4!HµνλρHµνλρ − +1 +2 · 3!(Gµνλ + mCµνλ)(Gµνλ + mCµνλ) +− 1 +3! ϵµνλρ(Gµνλ + mCµνλ)Jρ . +(2.8) +5This is generic the case in string theory for which the symmetries associated to antisymmetric tensors and axions +are compact (meaning there always exist magnetic-like branes). For a general discussion see [36]. +6Known string vacua can also contain a large number of these 3-form gauge potentials. +– 5 – + +Here Cµνλ is a 3-form gauge potential with field strength H = dC and Bµν is a 2-form gauge potential +with G = dB while m is a parameter with dimension mass. +This lagrangian has the gauge symmetry C → C + dΛ and B → B − m Λ for an arbitrary 2-form +Λ. So when m ̸= 0 we can choose a gauge B = 0. The field equation for C that follows from this +action then is +DµHµνλρ + m2Cνλρ + m ϵνλρµJµ = 0 . +(2.9) +This describes a single spin state propagating with mass m once all the gauge symmetries are used, +as can be seen by counting the massless states from which it is built. (In 4D Bµν is shown above to +be equivalent to a massless scalar and Cµνλ contains no propagating degrees of freedom at all because +one can always write Hµνλρ = h ϵµνλρ with field equation ∂µHµνλρ = 0 in the massless limit, which +implies h is a constant and so does not propagate.) +The dual should therefore be a massive scalar and this can be verified by trading the integral over +B for an integral over G and introducing (as before) a lagrange multiplier a to impose the Bianchi +identity7 dG = 0, leading to the lagrangian density +L0(C, G, a) = − +1 +2 · 4!HµνλρHµνλρ − +1 +2 · 3!(Gµνλ + mCµνλ)(Gµνλ + mCµνλ) +− 1 +3! a ϵµνλρ∂µGνλρ − 1 +3! ϵµνλρ(Gµνλ + mCµνλ)Jρ . +(2.10) +Integrating out a returns us to the above formulation, but instead performing the integration over G +leads to the saddle point +Gµνλ = −mCµνλ − ϵµνλρ +� +∂ρa + Jρ� +, +(2.11) +and so to the lagrangian +L2(C, a) = − +1 +2 · 4!HµνλρHµνλρ − m +4! a ϵµνλρHµνλρ − 1 +2∂µa ∂µa − Jµ∂µa − 1 +2JµJµ . +(2.12) +Next we perform the integral over Cµνλ, and this is equivalent to simply performing the gaussian +integral over Hµνλρ because the integrability condition for writing H = dC is dH = 0 which is always +true (in 4D). The saddle point for the H integral occurs for Hµνλρ = Hµνλρ where +Hµνλρ = −m a ϵµνλρ +(2.13) +and so leads to the scalar lagrangian +L2(a) = −1 +2(∂a)2 − m2 +2 a2 − Jµ∂µa − 1 +2JµJµ . +(2.14) +This is the expected massive scalar. +2.2.1 +Scalar potential +For future reference notice that it is only this last step that would differ if we’d had higher-dimension +terms like δL = W(X) in the lagrangian with X = 1 +4!ϵµνλρHµνλρ and so X2 = − 1 +4!HµνλρHµνλρ and +so on. The above discussion is the special case W = 1 +2X2 but one could entertain, for example, +W = c1M 2X + 1 +2 X2 + 2c3 +3M 2 X3 + +c4 +4M 4 X4 + · · · +(2.15) +7One can equivalently omit the mCµνλ terms everywhere and instead impose the modified Bianchi identity dG = mH. +– 6 – + +where the coefficients ci are dimensionless and M is a UV scale inserted everywhere on dimensional +grounds (with Hµνλρ canonically normalized). +For non-quadratic W the integral over H is no longer gaussian, but we can proceed assuming a +semiclassical saddle-point approximation is valid, in which case the saddle point (2.13) is modified to +�∂W +∂X +� +H=H += m a , +(2.16) +which agrees with (2.13) when W = 1 +2X2. For example, for the choice (2.15) this becomes +c1M 2 + X +� +1 + 2c3 +M 2 X + c4 +M 4 X2 + · · · +� +≃ m a +(2.17) +and so +X ≃ m a − c1M 2 − 2c3 +M 2 +� +m a − c1M 2�2 + O +�� +m a − c1M 2�3 /M 4� +. +(2.18) +Once used in the lagrangian this shows how non-quadratic pieces of W map over to non-quadratic +contributions to the scalar potential for a in the dual lagrangian L2. In particular the axion potential +becomes +V (a) = −W(X) + maX = 1 +2 +� +ma − c1M 2�2 − 2c3 +3M 2 +� +ma − c1M 2�3 + · · · . +(2.19) +Two features are noteworthy about this potential: +• First, notice it shares the usual Legendre property +∂V +∂a = mX + +� +−∂W +∂X + ma +� ∂X +∂a = mX , +(2.20) +where the last equality uses (2.16). Even if new non-quadratic terms introduce new stationary +points for V (a) (or shifts the positions of old ones) eq. (2.20) ensures X = 0 for all of them. +• Second, once a is shifted so that the minimum is at a = 0 the potential depends on m and a only +through the combination ma. Consequently, a term proportional to an comes suppressed by a +power of (m/M)n relative to what would naively be expected on dimensional grounds for V (a). +This is how the dual theory reproduces the same M-dependence as found for higher powers of +Hµνλρ given that a has canonical dimension mass while H has dimension (mass)2. This shows +how a dimensional assessment of how UV scales appear in the low-energy theory can care about +the existence of a dual formulation. +3 +Naturalness issues for dual systems +This section examines how naturalness arguments look for T- and S-type axions, and for S-type axions +how they depend on which side of the duality relation they are made. We do so using the axion quality +problem as a representative example. +3.1 +QCD and the dual PQ mechanism +To this end we extend the above reasoning to the main event: QCD and the θ-term. The idea is to +dualize the coupling of the axion to QCD to see how the strong-CP problem gets formulated, along +the general lines of [14]. We then ask how UV physics might complicate the story in the dual theory. +– 7 – + +Consider then adding a gauge potential Aµ (with field strength Fµν) to represent the QCD gauge +sector8 and this time consider the path integral +Ξ[J] = +� +DG DA Da eiS0 +(3.1) +where S0 = +� +d4x L0 and +L0(G, A, a) = − +1 +2 · 3!GµνλGµνλ − a +3! ϵµνλρ +� +∂µGνλρ − 1 +4 Ωµνλρ +� +− 1 +3! ϵµνλρGµνλJρ +−1 +4FµνF µν − θ +2 ϵµνλρFµνFλρ . +(3.2) +We suppress both gauge-group indices and traces over them to avoid notational clutter. Fµν is the +field strength for the gauge potential Aµ but Gµνλ is an arbitrary 3-form until the integral over a is +performed. +Integrating out a imposes the Bianchi identity dG = Ω where Ω is a gauge-invariant quantity built +from the gauge field that on grounds of consistency must satisfy dΩ = 0, for which we take +1 +12 ϵµνλρΩµνλρ = 1 +f ϵµνλρFµνFλρ +(3.3) +The mass scale f is here required on dimensional grounds. Doing this allows the G integral to be +traded for one over B as before and gives the lagrangian +L1(B, A) = − +1 +2 · 3!GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ − 1 +4FµνF µν − θ +2 ϵµνλρFµνFλρ . +(3.4) +where G = dB + S where dΩ = 0 implies there locally exists an Sµνλ – the Chern-Simons 3-form – +that satisfies Ω = dS. +The dual formulation instead integrates out G and leaves a as the dual variable. Integrating out +G leads to the lagrangian density +L2(A, a) = −1 +2(∂a)2 − Jµ∂µa − 1 +2JµJµ + a +4! ϵµνλρΩµνλρ − 1 +4FµνF µν − θ +2 ϵµνλρFµνFλρ += −1 +2(∂a)2 − Jµ∂µa − 1 +2JµJµ − 1 +4FµνF µν + 1 +2 +� a +f − θ +� +ϵµνλρFµνFλρ . +(3.5) +This shows that the standard axion-gauge coupling is the dual of the 2-form/QCD coupling given in +L1 and that f can be interpreted as its decay constant. +Below the QCD scale +In the standard axion-QCD story integrating out QCD leaves a residual axion potential due its anoma- +lous coupling to F ∧ F. This minimum is argued to be minimized where a = θ f (where θ is the usual +combination of θ and phases in the quark mass matrices) which ensures that the CP-odd contribution +turns off. We seek to express how physics below the QCD scale works in the dual language involving +Bµν. +Below ΛQCD the gauge degrees of freedom are integrated out, naively leaving only hadrons coupled +to Bµν. The key thought is that this is not quite right: the QCD EFT below ΛQCD contains a path +integral over low-energy hadrons and an integration over a low-energy field Cµνλ, whose emergent +8We do not write quarks explicitly but flag the few places where their implicit presence affects what is written. +– 8 – + +presence the strongly coupled vacuum of QCD makes mandatory. The field Cµνλ ∝ ⟨Sµνλ⟩ is the low- +energy counterpart of the Chern-Simons field appearing in the topological susceptibility [37] above the +QCD scale, where dS = F ∧ F. +Having this field in the low-energy theory below the QCD scale does not affect the existence of a +gap or the spectrum of the known hadrons because Cµνλ does not propagate. It is an auxiliary field +that is required in order for the low-energy theory to capture properly the response of QCD to any +topology in its environment. Similar fields are known to arise in this way in other concrete systems +like the EFTs describing Quantum Hall systems [21, 22]. This 3-form potential differs from many of +the others that often arise in string vacua because it arises from the IR properties of QCD rather than +from the physics of UV compactification. +On dimensional grounds we write H = dC with +1 +12 +˜Λ2 +QCDϵµνλρHµνλρ = ϵµνλρ⟨FµνFλρ⟩ , +(3.6) +where ˜ΛQCD denotes a parameter of order the QCD scale that ensures that H has canonical dimension +(mass)2. The lagrangian (3.4) above the QCD scale is then replaced with its low-energy counterpart +L1(C, B) = − +1 +2 · 3!GµνλGµνλ− 1 +3! ϵµνλρGµνλJρ− θ +4! +˜Λ2 +QCDϵµνλρHµνλρ− +1 +2 · 4!HµνλρHµνλρ+· · · , (3.7) +where the explicit term proportional to θX combines with quark mass phases – that also enter as +terms linear in X, as in the c1 term of (2.15) – to produce θX. The ellipses in (3.7) are at least cubic +in X (or involve derivatives of X). +Combining eq. (3.3) (and the discussion just above it) with (3.6) implies +dG = ⟨Ω⟩ = +˜Λ2 +QCD +f +H , +(3.8) +and so comparing this to dG = mH (as would follow from G = dB + mC) allows us to read off the +mass relation m = ˜Λ2 +QCD/f. We see that the mC term captures the expectation value ⟨S⟩/f of the +Chern-Simons term in the UV theory above the QCD scale if m scales with f in the same way that +the usual axion mass depends on its decay constant. +We expect the low-energy presence of such a 4-form field H to give B a nonzero mass, as we check +by introducing the lagrange multiplier a in the usual way and integrating out G. This leads to the +result +L2(C, a) = −1 +2(∂a)2 −Jµ∂µa− 1 +2JµJµ + 1 +4!(ma−θ˜Λ2 +QCD)ϵµνλρHµνλρ − +1 +2 · 4!HµνλρHµνλρ +· · · . (3.9) +Integrating out H leads to the saddle point Hµνλρ = Hµνλρ with +Hµνλρ = +� +ma − θ˜Λ2 +QCD +� +ϵµνλρ , +(3.10) +and so gives the axion lagrangian +L2(a) = −1 +2(∂a)2 − Jµ∂µa − 1 +2JµJµ − 1 +2 +� +ma − θ˜Λ2 +QCD +�2 +, +(3.11) +showing that the minimum indeed occurs where a = θ˜Λ2 +QCD/m = θf, which turns off the CP-violating +term of (3.9). +– 9 – + +In general integrating out the UV QCD sector also generates more complicated low-energy in- +teractions involving C, such as the function W(X) of X = +1 +4!ϵµνλρHµνλρ. +As above, such terms +semiclassically change the saddle point to +�∂W +∂X +� +H=H += m a − θ˜Λ2 +QCD , +(3.12) +and so leads to the axion potential +V (a) = −W(X) + (ma − θ˜Λ2 +QCD)X . +(3.13) +This satisfies +∂V +∂a = mX + +� +−∂W +∂X + ma − θ˜Λ2 +QCD +� ∂X +∂a = mX , +(3.14) +and so again ensures that X = 0 at any of the stationary points of V . We see that the presence of +interactions like W(X) show that V is minimized at ma = θ˜Λ2 +QCD if ∂W/∂X vanishes when X = 0. +3.2 +The Quality Problem +We now have the tools required to explore UV sensitivity and the axion quality problem. We start +by restating the original formulaton of the quality problem and then how it is rephrased in 2-form +language for both T-type (this section) and S-type (next section) axions. +The axion quality problem asks two related questions [10]: +1. Do corrections to the QCD axion potential change its minimum in a way that preserves a +sufficiently small effective vacuum angle: ¯θeff ≲ 10−10? +2. Do corrections to the QCD axion potential change the usual expression for the axion mass (that +assumes it is dominantly generated by the ‘IR-dominated’ QCD instanton with size ρ ∼ Λ−1 +QCD)? +The first of these essentially asks if the QCD axion remains a good solution to the strong CP problem +when perturbed by new physics, whereas the second asks the same of our understanding of axion mass. +The axion mass question can apply more generally to ALPs as well, whereas the first one is specific +to the QCD axion. +Any UV completion must decide what happens at energies above the axion decay constant f +above which the low-energy expansion in powers of E/f breaks down. +We consider in turn the +original formulation and the T- and S-type axions that arise within an extra-dimensional context. +3.2.1 +Original formulation +In the initial formulation the UV completion for scales above f was assumed to involve a second scalar +that combines with the axion to linearly realize the PQ symmetry as a complex scalar Φ. In this +picture the modulus of Φ acquires a mass proportional to f ∼ ⟨Φ⟩ and the axion starts life as the +phase of Φ ∝ eia/f. +Motivated by string theory and black-hole thought experiments it is then assumed that UV physics +cannot support an unbroken global symmetry, and so at some large scale M the form of the scalar +potential for Φ cannot be assumed to be invariant under re-phasings of Φ. As an expansion in powers +of Φ, the generic potential form would be +VUV (Φ) = M 4 +2 +∞ +� +n=1 +� +cn +Φn +M n + h.c. +� +, +(3.15) +– 10 – + +where the cn’s are in general complex. This is true even if the UV physics is assumed to be CP- +invariant because cn will inherit the phase of the fermion mass matrix after chiral PQ rotations. In +the initial formulation M is assumed to be the Planck mass Mp, and although we can see that such a +choice would dominate smaller M for the terms with n < 4 it is likely that M < Mp would be more +dangerous for n > 4. Early workers typically assumed that the renormalizable part of the potential +would be tuned to make the axion potential sufficiently shallow and so effectively started the sum in +(3.15) at n = 5. +Freezing the modulus field at ⟨Φ⟩ = f and integrating it out at the classical level leads to the +following effective axion potential +VUV (a) = M 4 +2 +� +n=1 +|cn| f n +M n +� +eiδn eina/f + h.c. +� += M 4 � +n=5 +|cn| f n +M n cos +�na +f + δn +� +, +(3.16) +where we shift fields so that the standard QCD solution is a = 0. The QCD minimum therefore +remains unchanged if V ′ +UV (0) = 0 and this would be true if all of the δn’s were to vanish. Although +the axion potential height (and therefore possibly axion mass) might still change due to the presence +of VUV (a), evasion of the strong CP problem requires only that the minimum for a remains unmoved. +Stability of the minimum: +For δn , |cn| ∼ O(1) we can estimate the size of the effective value of ¯θeff by perturbing around the +QCD minimum at a = aQCD: +¯θeff ≃ − V ′ +UV (aQCD) +fV ′′ +QCD(aQCD) ∼ VUV (aQCD) +VQCD(aQCD) ∼ M 4 +Λ4 +QCD +� f +M +�n0 +, +(3.17) +where n0 represents the first power appearing in the sum. For example, requiring ¯θeff < 10−10 for the +example f = 1012 GeV, M = Mp = 1018 GeV and ΛQCD ≃ 0.2 GeV in (3.17) requires n0 > +∼ 15. +Stability of the axion mass: +The change to the axion mass induced by the UV axion potential is given by +δm2 +a = ∂2VUV (a) +∂a2 +���� +a=0 += M 2 � +n=1 +n2|cn| +� f +M +�n−2 +cos δn , +(3.18) +which can be significant unless the coefficients |cn|’s are extremely small even if all the δn’s could be +contrived to vanish. When significant such contributions spoil the relation maf ∼ mπFπ that holds +for the low-energy QCD contribution and on which most axion phenomenology is based. Because the +mass is not inversely proportional to f this expression shows that the relation between ma and f need +not be inversely proportional to one other, for example allowing a very heavy axion to be still very +weakly coupled to matter – a drastic change relative to standard axion phenomenology. +3.2.2 +T-type axions +The story is similar for T-type axions, at least below the Kaluza-Klein scale where they are 4D scalars. +No quality issue arises above the KK scale because here the relevant fields are higher-dimensional form +fields HMNP and the only symmetries involved are gauge symmetries like B → B + dλ [13]. +Recalling that T-type axions, b, arise as extra-dimensional reductions of the form Bmn(x, y) = +b(x) ωmn(y), with ωmn a harmonic form in the extra dimensions, the origin of the low-energy shift +– 11 – + +symmetry b → b + c (for constant c) has its origins as the extra-dimensional transformation Bmn → +Bmn + c ωmn. This is a symmetry of H = dB because harmonic forms are closed: dω = 0. It is +strictly speaking a ‘large’ gauge transformation because harmonic forms are not exact: there does not +globally9 exist a λm such that ω = dλ. +The quality problem arises because the shift symmetry in the low-energy 4D theory is not a local +gauge symmetry and so it in principle need not be respected by UV effects. One consequently cannot +completely preclude the generation of a scalar potential, +VUV (b) ∼ M 4 � +n +cn +� b +M +�n +, +(3.19) +where cn are dimensionless order-unity coefficients. But its failure to be a local gauge symmetry is +a global obstruction rather than a local one and this means that UV effects cannot generate VUV (b) +until scales are integrated out that ‘see’ the topology that can distinguish ω from dλ. This implies +two sorts of changes to the standard quality-problem argument. First, the scale M where problems +first arise cannot be higher than the KK scale M ∼ 1/RKK corresponding to the size of the 2D cycle +in the extra dimensions whose presence is associated with the existence of the harmonic form ωmn(y). +Second, the physics at scale M that generates the potential must itself be sensitive to the nontrivial +topology, often leading to additional suppressions. +For instance, an example of physics that can generate PQ-violating operators in (3.15) identified +in [13, 38] is wormhole [41]. For these the coefficients cn in (3.16) are exponentially suppressed, given +by [13] +cn ∼ e−S ∼ e−(MpL)2 +(3.20) +where S is a wormhole action and L the size of its throat. Maintaining the success of the PQ mechanism +requires S > +∼ 190. More complicated configurations are possible for extra-dimensional theories, for +which Mp can be replaced by another UV gravity scale Mg, that might be the string scale or the +extra-dimensional Planck scale Mg in specific examples. +Similarly L can be one of the geometric +scales of the background, that could (but need not) be approximately a compactification scale RKK. +All known semiclassical arguments of this type must assume MgL ≫ 1 for the calculation to be under +control, because semiclassical methods are justified within an expansion in powers of (MgL)−1 within +any gravitational EFT. MgL ∼ 14 suffices to ensure S > +∼ 190 and so satisfying this constraint seems +not that difficult within the semiclassical regime. These kinds of arguments were used in [18] to argue +for the absence of large gravitational correction to the inflaton potential. +3.3 +The dual Quality Problem +For S-type axions the representation directly obtained from UV physics is the field bµν dual to the +scalar axion. And as alluded to earlier – c.f. §2.2.1 – issues of UV sensitivity can look very different +in dual formulations to scalar theories, with for example the existence of a dual implying that the +effective couplings for terms like an ∈ VUV (a) come suppressed by powers of the axion mass (m/M)n +relative to generic scalar estimates. Such suppressions can be enormous given the small size of m +relative to UV scales. +We therefore revisit earlier discussions of how the axion quality problem arises in the dual formu- +lation, partly motivated by recent discussions [15, 16] that argue that gravity causes new problems. +9The situation resembles a gauge field Am(x, y) dimensionally reduced on a circle, so Am(x, y + L) = Am(x, y). In +this case the massless scalar would be Am(x, y) = a(x)ω(y) where ω(y) is independent of y, for which the shift symmetry +a → a+c locally corresponds to a gauge transformation Am → Am +∂mζ if ∂ζ/∂y = c, but this cannot be done globally +because the solution cannot satisfy ζ(y + L) = ζ(y). +– 12 – + +Although we confirm the important role played by multiple 3-form potentials [14] in the framing of +the dual quality problem, we also show that the many 3-forms found in string vacua do not generically +pose a problem. Problems are only caused where strongly interacting systems make instanton-like +effects important and this is not the case for the many ‘elementary’ 3-forms that descend from extra +dimensional vacua. We argue that for similar reasons 4D gravitational Chern-Simons forms also need +not cause problems (such as for string vacua where the UV completion of gravity is described by +weakly coupled physics). +To the extent that the shape of the axion potential V (a) is dual to interactions like W(X) involving +the 4-form field strength X = 1 +4!Hµνλρϵµνλρ, one might think that the dual version of the axion quality +issue should hinge on the detailed form of UV contributions to W(X). This proves not to be right, +as we now argue. The central point turns on the Legendre transformation relating V (a) to W(X); in +particular on (3.12) and (3.14), that state +�∂W +∂X +� +H=H += m a − θ˜Λ2 +QCD +and +∂V +∂a = mX . +(3.21) +On the scalar side the strong-CP problem is not solved unless ma = θ˜Λ2 at the minimum of V , +and the quality problem is the statement that corrections to V can perturb the minimum so that this +relation fails. Although X always vanishes at a minimum for V , eq. (3.21) suggests that on the dual +side the criterion for satisfying the strong-CP problem is that ∂W/∂X = 0 is satisfied when X = 0. +So the quality problem seems to hinge on whether or not UV physics can introduce a linear term +δW = ηX whose inclusion would modify (3.21) in a way that obstructs having m a = θ˜Λ2 +QCD be a +solution to ∂V/∂a = 0. +Suppose, then, that one finds after integrating out the UV physics an EFT below the QCD scale +of the form (3.7), but with a linear term in X whose coefficient is not proportional to the CP violating +parameter θ: +L1(C, B) = − +1 +2 · 3!(Gµνλ + mCµνλ)(Gµνλ + mCµνλ) − 1 +3! ϵµνλρ(Gµνλ + mCµνλ)Jρ +− 1 +4!(θ + η) ˜Λ2 +QCDϵµνλρHµνλρ − +1 +2 · 4!HµνλρHµνλρ + · · · , +(3.22) +with two low-energy CP-violating parameters θ and η. Dualizing this system as above then shows that +scalar potential on the scalar side is given by a function of ma−(θ+η)˜Λ2, in which θ and η only appear +as a sum. The arguments of §3.1 now show that this potential is minimized when ma − (θ + η)˜Λ2 = 0. +Repeating the calculation of the neutron electric dipole moment (edm) in this case – for a recent +review, see for example [42] – then shows that the neutron edm also depends only on the sum θ + η +and so would continue to vanish when a is evaluated at the potential’s minimum. Interestingly, just +introducing new terms linear in Hµνλρ in (3.7) appears not to cause a quality problem. +3.3.1 +A second strong sector +Just introducing a linear term in Hµνλρ in (3.7) does not cause a quality problem because doing so +below the QCD scale is like introducing the new CP-violating parameter η only in the F ∧ F term of +(3.4) above the QCD scale (i.e. shifting θ → θ + η). This also would not cause a quality problem on +the scalar side. For there to be a problem requires there to be a CP-violating contribution to V (a) +that is independent of the CP-violation in the θ-term. +What might this look like on the dual side? One way to proceed is to imagine a specific type +of CP-violating UV completion and ask what happens in this case. +One such an example would +– 13 – + +add another strongly interacting nonabelian gauge sector that also contributes to the axion anomaly. +In this case VUV (a) is obtained by integrating out the new gauge sector and this is by construction +independent of the QCD-generated part. A dual formulation of this type of system would involve a +new Chern-Simons form Eµνλ for the new sector in addition to the QCD field Cµνλ, since both gauge +sectors have their own Chern-Simons fields and either of these can be the field that is eaten by Bµν. +Instead of (3.22) below the QCD scale one would find the following low-energy action +L1(C, E, B) = − +1 +2 · 3!GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ − 1 +4!ϵµνλρ� +θ˜Λ2 +QCDHµνλρ + η˜Λ2 +XKµνλρ +� +− +1 +2 · 4! +� +HµνλρHµνλρ + KµνλρKµνλρ� ++ · · · , +(3.23) +where K = dE and H = dC and G = dB + mC + ˜mE. +Proceeding as before we introduce a Lagrange multiplier a to enforce the G Bianchi identity and +then semiclassically integrate out G, H and K to find +L2(a) = −1 +2(∂a)2 − Jµ∂µa − 1 +2JµJµ − V (a) , +(3.24) +where defining X = 1 +4!ϵµνλρHµνλρ and Y = 1 +4!ϵµνλρKµνλρ we find +V (a) = −W(X, Y ) + (ma − θ˜Λ2 +QCD)X + ( ˜ma − η˜Λ2 +X)Y , +(3.25) +where W = 1 +2(X2 + Y 2)+(higher powers). At the saddle point (H, K) = (H, K) we have +�∂W +∂X +� +Y += m a − θ˜Λ2 +QCD +and +�∂W +∂Y +� +X += ˜m a − η˜Λ2 +X , +(3.26) +where the subscripts indicate what is held fixed in the derivative. Differentiating (3.25) implies +∂V +∂a = mX + ˜mY . +(3.27) +This does have a quality problem because the competition between the two gauge sectors drives the +axion away from the minimum for which the neutron electric dipole moment vanishes. For the simplest +example – where W = 1 +2(X2 + Y 2) – we can see explicitly how the shift of the global minimum of the +axion potential is induced. From (3.27) we learn that ∂V (a)/∂a = 0 takes place at Y = −(m/ ˜m)X. +From (3.26), we obtain +X = ma − ¯θ˜Λ2 +QCD +and +Y = − +�m +˜m +� +X = ˜ma − η˜Λ2 +X +(3.28) +Equating these two expressions for X and solving for a, we obtain +amin = m¯θ˜Λ2 +QCD + ˜mη˜Λ2 +X +m2 + ˜m2 += aQCD + ( ˜mη˜Λ2 +X/m2) +1 + ( ˜m/m)2 +≃ aQCD + ˜mη˜Λ2 +X +m2 +, +(3.29) +which denotes the global minimum before introducing an extra three form gauge field by aQCD = +¯θ˜Λ2 +QCD/m. The approximate equality assumes m ≫ ˜m so as not to spoil the QCD axion solution the +strong CP problem. +Finally, defining the UV contribution to the effective vacuum angle by θeff := (amin − aQCD)/f +where mf ≃ ˜Λ2 +QCD, we obtain the constraint +¯θeff ∼ η +� ˜m +m +� � +˜ΛX +˜ΛQCD +�2 +< +∼ 10−10 . +(3.30) +– 14 – + +Although this derivation assumed the simplest form W = 1 +2(X2 + Y 2), the reasoning presented here +can be applied to a more complicated W(X, Y ). In such a case (3.27) remains unchanged while (3.26) +and (3.28) are modified. But amin remains connected to the value for (X, Y ) that makes ∂V/∂a vanish +via (3.26) and (3.27). Once amin is expressed in terms of aQCD, one can always infer ¯θeff as above and +impose the constraint ¯θeff < 10−10. +The upshot is this: the requirement of multiple strongly coupled sectors on the dual side to +generate a quality problem is much more explicit because the contribution of each sector is described by +a separate 3-form potential, rather than having everything all be rolled into the same scalar potential. +3.3.2 +Multiple fundamental 3-forms +At first sight the previous section makes it sound like string theory should typically have a huge +quality problem, because of the generic appearance there of multiple 3-form potentials. We identify +the circumstances under which these potentials could cause a quality problem and argue why such a +problem generically does not happen. We also discuss how these criteria bear on a recent realization +of these issues [15, 16]. +To start consider how the EFT (3.23) above the QCD scale would be modified by the presence of +many 3-form potentials CA +µνλ (where A = 1, . . . , N distinguishes the different UV potentials): +L1(B, A, C) = − +1 +2 · 3!GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ − 1 +4FµνF µν − θ +2 ϵµνλρFµνFλρ +− 1 +4! ηAH +A +µνλρϵµνλρ − +1 +2 · 4!H +A +µνλρHµνλρ +A ++ · · · . +(3.31) +where HA = dCA and G = dB + S for the QCD Chern-Simons 3-form that satisfies Ω = dS with Ω as +given in (3.3). To the extent that none of the new fields HA +µνλρ appear in the Bianchi identity dG = Ω +they do not couple to QCD or to Bµν and so play no role in the duality transformation from Bµν to +a. One then arrives below the QCD scale with the lagrangian +L1(C, B) = − +1 +2 · 3!GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ − θ +4! +˜Λ2 +QCDϵµνλρHµνλρ − +1 +2 · 4!HµνλρHµνλρ +− 1 +4! ηAH +A +µνλρϵµνλρ − +1 +2 · 4!H +A +µνλρHµνλρ +A ++ · · · . +(3.32) +Dualization proceeds as before, with the introduction of the scalar a to enforce dG = Ω, and the +saddle point in the integral over the 3-form potentials becomes +�∂W +∂X +� +Y += m a − θ˜Λ2 +QCD +and +� ∂W +∂Y A +� +X += −ηA , +(3.33) +where +W = 1 +2X2 + 1 +2Y +AYA + (higher powers) , +(3.34) +and we define as before X = 1 +4!ϵµνλρHµνλρ and Y A = 1 +4!ϵµνλρHA +µνλρ. The dual lagrangian is +L2(a) = −1 +2(∂a)2 − Jµ∂µa − 1 +2JµJµ − V (a) , +(3.35) +where +V (a) = −W(X, Y +A) + (ma − θ˜Λ2 +QCD)X − ηAY +A += −1 +2 X2 + (ma − θ˜Λ2 +QCD)X + 1 +2ηAη +A , +(3.36) +– 15 – + +and so +∂V +∂a = mX . +(3.37) +We see that X = 0 in the vacuum and this implies from (3.33) and (3.34) that the strong-CP problem +remains solved. +These arguments also show that two ingredients are required for additional 3-form potentials to +cause a problem: +1. The additional 3-form potential CA0 must contribute to the Bianchi identity for G, and so κA0 ̸= 0 +in the expression dG = Ω + κAHA, where HA = dCA; and +2. The additional 3-form potential must appear linearly in W, so ηA0 ̸= 0 in (3.32). +When both of these are satisfied then a couples to HA and leads to the competition of minima as in +(3.26) along the lines described in §3.3.1. The need for both of these conditions to be true is why the +bound (3.30) is proportional to both η and ˜Λ2 +X. The good news is that the vanishing of κA can be +enforced by a gauge symmetry, since κA can only be nonzero if B transforms as B → B − κAΛA under +the 3-form gauge transformations CA → CA + dΛA. +There is at least one example of a 3-form potential which we know must exist and which also +contributes to the Bianchi identity dG: the gravitational Chern Simons 3-form, Sg. The existence of +a PQ-Lorentz-Lorentz anomaly requires this form to appear in G and so have a nonzero coefficient κg +in the same way that the PQ-QCD-QCD anomaly requires the QCD Chern Simons form to appear +there. Ref. [14] argues that this is real trouble whose evasion requires model-building, such as that +done in [16]. +Whether the existence of this form is a problem or not depends on whether it also satisfies item +2 above: i.e. whether or not it appears linearly in the lagrangian with coefficient ηg ̸= 0. How big +should ηg be expected to be? Because any 4-form field strength H = dC is locally a total derivative it +wants to drop out of perturbative physics when it appears linearly in the action (much as does F ∧F). +Consequently its appearance in a low-energy action requires some sort of nonperturbative process (like +an instanton) to contribute to physical processes. This is indeed what happens for QCD for which the +linear term in Ω appears with coefficient +˜Λ2 +QCD ∝ M 2 e−2πb/α +(3.38) +with M a UV scale, b a pure number and α = g2/4π the QCD coupling. The tell-tale nonperturbative +dependence on α is a semiclassical consequence of the topological character of +� +F ∧ F and +� +H. +This suggests that for gravity a linear term in Hg should similarly be of size +η ∝ M 2 e−(ML)2 +(3.39) +for a characteristic instanton length scale L and gravitational UV scale M given that (ML)−2 plays +the role of the semiclassical expansion parameter (compare to (3.20)). This can be extremely small +within the domain of validity of semiclassical reasoning, for which ML ≫ 1 (as would presumably +apply when the UV completion is weakly coupled, such as for perturbative string vacua). +Examples of three forms characterized by η in (3.39) include Eguchi-Hanson instantons [39, 40] and +the gravitational Chern-Simons 3-form made up of gravitational connection. For the Chern-Simons +3-form ref. [14] argues that gravity indeed becomes strong in the UV, as would be required for η to +be significant. This could well be true, but the evidence for there being a problem hinges on how +convinced one is about gravitational interactions becoming strong in the UV. +– 16 – + +3.3.3 +Multiple-axion solution +We close this section by remarking that having multiple axion candidates (as is often true for string +vacua) can alleviate the above problem associated with multiple 3-form fields, even if the above two +conditions are satisfied.10 This observation points to an equally general quality control mechanism on +the scalar side of the duality as well. +To see why, we introduce a second Kalb-Ramond field Bµν to the model of §3.3.1, and supple- +menting the lagrangian of (3.23) with the appropriate additional kinetic term gives +L1(C, E, B, B) = − +1 +2 · 3!GµνλGµνλ − +1 +2 · 3!GµνλGµνλ − 1 +3! ϵµνλρGµνλJρ +(3.40) +− 1 +4!ϵµνλρ� +θ˜Λ2 +QCDHµνλρ + η˜Λ2 +XKµνλρ +� +− +1 +2 · 4! +� +HµνλρHµνλρ + KµνλρKµνλρ� ++ · · · , +where as before K = dE and H = dC and G = dB + mC + ˜mE, but now also +G := dB + m⋆E . +(3.41) +This system dualizes much as before: we introduce Lagrange multipliers a and b to enforce the G +and G Bianchi identities dG = mH + ˜mK and dG = m⋆K and then integrate out G, G, H and K to +find +L2(a) = −1 +2(∂b)2 − 1 +2(∂a + J)2 − V (a, b) , +(3.42) +with +V (a, b) = −W(X, Y ) + (ma − θ˜Λ2 +QCD)X + ( ˜ma + m⋆b − η˜Λ2 +X)Y , +(3.43) +and we define as before X = +1 +4!ϵµνλρHµνλρ and Y = +1 +4!ϵµνλρKµνλρ. +For the simplest example of +W = 1 +2(X2 + Y 2), at the saddle point (H, K) = (H, K) gives the following relation between (X, Y ) +and (a, b): +�∂W +∂X +� +Y += m a − θ˜Λ2 +QCD +and +�∂W +∂Y +� +X += ˜m a + m⋆ b − η˜Λ2 +X . +(3.44) +Differentiating (3.43) with respect to a and b implies +∂V +∂a = mX + ˜mY, +∂V +∂b = m⋆Y . +(3.45) +and so shows that all extrema of the potential satisfy X = Y = 0 (provided m, ˜m and m⋆ are nonzero). +Because ∂W/∂X vanishes at X = 0 it follows that the dynamics chooses amin to satisfy θ˜Λ2 +QCD/m = θf +through (3.44); the axion quality problem essentially disappears. +What happened? Why does introducing another axion resolve the quality problem? The crux of +the mechanism lies in the difference between eq. (3.45) and (3.27). The derivative of the potential +always sets a linear combination of 4-form field strengths to zero and if there are as many equations +as there are fields the only solution is generically to have all 4-form field strengths vanish. Once this +is true then the first of eqs. (3.44) ensures that this solution solves the strong-CP problem. Trouble +only arises – as it did in §3.3.1 – when there are fewer equations than unknowns (i.e. fewer axions +than 3-form potentials), since then X need not vanish and eqs. (3.44) become competing conditions +on the same axion variable. +10The use of multiple axions to solve the quality problem is mentioned also in [43], who have different but related +motivations for there being a plethora of form fields present in the UV. +– 17 – + +A similar mechanism also exists on the scalar side of the duality. If two sectors generate con- +tributions to the QCD axion potential then the problem arises because these compete in the value +they imply for the axion expectation value. Introducing a second anomalous U(1) symmetry that also +has anomalies with the same two sectors provides enough latitude to minimize each sector’s potential +separately, thereby removing the troublesome competition. +For instance, suppose there was a new non-Abelian gauge sector G and suppose the usual PQ +symmetry has both a QCD anomaly and an anomaly in the G sector. This is the kind of thing that +can cause a quality problem because of the contradictory conditions the two sectors impose on the +QCD axion. But also introducing another global U(1) with only a G-sector anomaly can help because +there is a linear combination of the PQ symmetry and the new U(1) that is anomaly free in the G +sector and the PQ mechanism then goes through using this new symmetry.11 +4 +UV completion and matter couplings +Since the motivations both for considering Kalb-Ramond fields and for the absence of global symmetries +come from the UV, it is useful to ask whether there are other potential surprises for axion physics +having their roots in the UV. This section examines two such examples; one each for T-type and +for S-type axions. For T-type axions we provide simple examples for which physical axion-matter +couplings like gaff can be much smaller than the naive value 1/fb read off from the axion kinetic term. +In the example shown here gauge invariant matter couplings like gaff are order 1/Mp despite fb being +an ordinary particle physics scale, while anomalous gauge couplings remain order 1/fb in size (if they +exist at all). +For S-type axions we show that the corresponding physical couplings indeed are of order 1/fa +and we identify the UV physics to which couplings of size E/fa match at energies E > +∼ fa. We also +show how S-type axions can be examples of weak/strong duality, and that it is the Kalb-Ramond +side of the duality that is usually weakly coupled. Weak/strong coupling interchange due to duality +could be relevant to applications for which the effects of scalar axions are explored using semiclassical +reasoning, and if so would provide a further motivation for taking the Kalb-Ramond formulation as +primary. +4.1 +Extra-dimensional UV completion +To this end suppose that both Kalb-Ramond field and the standard model arise within an extra- +dimensional model. For concreteness’ sake we take the higher-dimensional kinetic term for the 2-form +field and the Einstein-Hilbert part of the action to be12 +Skin = −1 +2 M 2+d +� +d4x ddy +� +−˜g(D) +� +�R + 1 +3! e−λφ GMNPG +MNP +� +, +(4.1) +where there are D = 4+d spacetime dimensions and M is a UV scale – the higher-dimensional Planck +scale. +�R here denotes the Ricci scalar and ˜g(D) is the determinant of the full D-dimensional metric +˜gMN. As above H = dB+· · · is the Kalb-Ramond field strength and φ is the extra-dimensional dilaton +that often arises within the higher-dimensional gravity supermultiplet. The parameter λ depends on +11Ref. [16] uses a special case of this general mechanism by introducing an extra U(1) symmetry in the leptonic sector +to resolve the problem raised by the assumption that gravity is strongly coupled. +12For simplicity we ignore extra-dimensional warping in this discussion. We also do not canonically normalize BMN, +which here is taken to be dimensionless. +– 18 – + +higher-dimensional details, with (d, λ) = (2, 2) for chiral 6D supergravity [44], (d, λ) = (6, 1) for Neveu- +Schwarz 2-forms in 10D supergravity and (d, λ = −1) for Ramond 2-forms in 10D supergravity [45] +(for example). +The derivation of this type of lagrangian as the low-energy limit of a string vacuum usually relies +on two approximations: the low-energy approximation (or α′ expansion) where energies are well below +the string scale E ≪ Ms; and the weak string coupling approximation, which involves expanding in +powers of eφ ≪ 1. For simplicity we restrict ourselves to this limit as well, and specialize to the +simplest case (d, λ) = (2, 2) corresponding to 6D chiral supergravity. +Dimensional reduction to 4D proceeds by integrating out the two extra dimensions and putting +the 4D Einstein-Hilbert term into standard form (4D Einstein frame) by appropriately rescaling the +4D part of the metric +˜gµν = +�V2⋆ +V2 +� +gµν = 1 +V2 +� +M 2 +p +M 2 +� +gµν +where +Vd := M d +� +ddy +� +˜g(d) +(4.2) +is the dimensionless extra-dimensional volume and the subscript ‘⋆’ on a field denotes its present-day +value13 and the 4D Planck massis is defined by +M 2 +p = V2⋆ M 2 . +(4.3) +S-type axion +The kinetic term for bµν in 4D Einstein frame that is obtained by dimensional reduction is +Lkin = − 1 +12 M 2V2 +� +−˜g(4) e−2φ˜gµν˜gβρ˜gξζ HµβξHνρζ = − M 4 +12M 2p +√−g e−2φ V2 +2 hµνβhµνβ , +(4.4) +where hµνλ = ∂µbνλ + (cyclic). This last form can be written in terms of a scalar by dualizing as in +earlier sections, imposing the Bianchi identity14 dh = Ω/M 2, leading to the dual result +Ldual = −√−g +� +M 2 +pe2φ +V2 +2 +∂µa ∂µa + 1 +3! a ϵµνβρΩµνβρ +� +(4.5) +which suggests its decay constant can be written fa = (Mp/Vd⋆) eφ⋆. +Two things are noteworthy here. First, notice that the volume dependence means that fa can +be very much smaller than Planckian size. In the extreme case of two large extra dimensions (and +working in the weak-coupling regime for which eφ is moderately small) the size of the extra dimensions +can be as large as MRKK < +∼ 1014 and so V2 ∼ (MRKK)2 < +∼ 1028 can be enormous (potentially allowing +fa ≪ Mp to be as small as eV energies). +Second, notice that although (4.4) has large coefficients when e2φ ≪ 1, the same is not true of the +kinetic term in (4.5). This reflects how Kalb-Ramond/axion duality is a weak-strong coupling duality +from the point of view of the string coupling gs ∼ eφ. To the extent that semiclassical expansions +rely on the leading action being proportional to the inverse of a small coupling15 – L0 = L0/g2 +s – +semiclassical methods should fail for the scalar representation but hold for its dual. +13The factor of V2⋆ ensures the rescaling is trivial at present, as required to not change present-day units of length. +14The factors of M here are chosen so that Ω has dimension (mass)4. +15When this is true then powers of g2 +s and powers of ℏ are equivalent when evaluating a path integral over eiS0. +– 19 – + +T-type axion +For T-type axions we use Bmn(x, y) = b(x) ωmn(y) where in six dimensions the harmonic form can +be taken to be proportional to the extra-dimensional volume form εmn(y). Typically ωmn satisfies +a quantization condition that states the integral of ωmn over the two extra dimensions +� +C ω is a +pure number, proportional to an integer. Because this result is volume independent it follows that +ωmn = V−1 +2 +εmn. +The kinetic term for the T-type scalar b obtained in this way is therefore proportional to +Lkin = −1 +2 M 2V2 +� +−˜g(4) e−2φ˜gµν∂µb ∂νb V−2 +2 += −√−g M 2 +pe−2φV−2 +2 gµν∂µb ∂νb . +(4.6) +Notice that the kinetic term, both here and in (4.5), takes the form +Lkin = −1 +2 +√−g M 2 +p +�(∂b)2 +τ 2 ++ (∂a)2 +σ2 +� +(4.7) +with τ = V2 eφ in (4.6), and σ = V2 e−φ in (4.5). +4.2 +Coupling strengths +What matters for phenomenology is the couplings of the fields b and a to matter. This is controlled +by the size of F for axion couplings of the form +Lax = −1 +2 ∂µa ∂µa − 1 +F ∂µaJµ , +(4.8) +where a is the canonically normalized axion field and Jµ is a matter current. F−1 = gaff is the +axion-fermion current if Jµ is built from fermion bilinears and F−1 ≃ gagg or gaγγ if Jµ is the Hodge +dual of the QCD or QED Chern-Simons 3-form. +For concreteness’ sake we evaluate the size of this coupling in the perturbative semiclassical regime +where V2 is large and the UV physics is weakly coupled (and so eφ small). In this limit we have +fb > +∼ Mp/V2 > +∼ fa and both are much smaller than Mp. In both cases we will see that F can (but need +not) be simply given by the corresponding decay constant fa or fb. +In higher dimensional constructions very often ordinary matter is localized on a space-filling brane, +Σ, within the extra dimensions. Σ could be a four-dimensional 3-brane or a higher-dimensional p- +brane with 3 ≤ p ≤ 3 + d. If p > 3 then the extra-dimensional part of the brane typically wraps some +topological cycle within the extra dimensions, and if this were a two-cycle (e.g. if p = 5) it would +also have an associated harmonic 2-form ωmn(y) required to ensure that T-type axions appear in the +low-energy 4D theory. We here explore the simplest case p = 3. +S-type axion +A generally covariant low-dimension interaction between HMNP and matter fields living on the brane, +that is linear in BMN is16 +Sint = −ˆc +� +Σ +e−βφH ∧ J = − c +3! +� +d4x√−g e−βφ ϵµνλρhµνλJρ , +(4.9) +where Jρ is a current built from brane-localized matter fields and β is a parameter – like λ in (4.1) – that +is predicted by any specific extra-dimensional UV completion. The matter current Jρ has dimension +16The first equality shows that this interaction is independent of the metric, and this can also be seen after the second +equality from the observation that ϵµνλρ = ±√−g and so ϵµνλρ = ±(−g)−1/2. +– 20 – + +(mass)3 – making the coupling parameters ˆc and c dimensionless – and so could be a fermion bilinear +or the Hodge dual of a gauge boson Chern-Simons term (though for the Chern Simons term gauge +invariance would require β = 0). Because this term is covariant without use of the metric it does not +acquire factors of V2 or Mp/M when going to 4D Einstein frame. +The dual effective theory for a is then found by adding (4.9) to the kinetic term (4.4), imposing +the Bianchi identity dG = Ω/M 2 and integrating out hµνλ, modifying (4.5) to +Ldual = −√−g +� +M 2 +pe2φ +2V2 +2 +Dµa Dµa + 1 +3! a ϵµνβρΩµνβρ +� +(4.10) +where +Dµa := ∂µa + +c +M 2p +e−(β+2)φ V2 +2 Jµ = ∂µa + c +f 2a +e−βφ Jµ . +(4.11) +As before we use the kinetic term to identify fa = (Mp/V2⋆) eφ⋆. Using M 2 +p = M 2V2⋆ with V2⋆ = +(MRKK)2 for a Kaluza-Klein length scale RKK, this implies fa ∼ MV−1/2 +2⋆ +eφ⋆ ∼ (1/RKK) eφ⋆, and so +fa ∼ mKK ∼ 1/RKK when eφ⋆ is not that much smaller than order unity. +The physical coupling that comes from comparing the kinetic and ∂µa Jµ term to (4.8) is +gagg ∼ gaff = +1 +Faff +∼ cV2⋆e−(β+1)φ⋆ +Mp +∼ c e−βφ⋆ +fa +for couplings to J . +(4.12) +In the special case where Jµ is the Hodge dual of a gauge-field Chern Simons term, gauge invariance +also requires we take β = 0, and once this is done the coupling in (4.12) agrees (up to numerical factors) +with the physical coupling to Ω implied by (4.10). This coupling becomes strong when E ∼ fa, which +we’ve seen is of order the Kaluza-Klein scale in the special case of two extra dimensions. +T-type axion +The lowest-dimension generally covariant and gauge invariant interaction that couples HMNP to matter +localized on a space-filling 3-brane and that is linear in the components Hµmn has the form +Sint = +� +Σ +e−2φ ⋆H ∧ J ∋ M 2 +p +M 2 +� +d4x√−g e−2φ +V2 +2 +gµν∂µb(x)Jν(x) , +(4.13) +where ⋆H denotes the 6D Hodge dual and we choose the φ coupling to be the same as the kinetic +term. +The kinetic and interactions terms combine to give the effective action (in 4D Einstein frame) +Seff = +� +d4x √−g M 2 +p +τ 2 +� +(∂b)2 + ∂µbJµ +M 2 +� +(4.14) +where τ := V2 eφ as before. Inspection of the kinetic term identifies the decay constant as fb ≃ Mp/τ⋆ +with τ⋆ ∝ V2⋆ denoting the present value of τ. Because M 2 +p ≃ M 2V2⋆ we see that fb ∼ MV−1/2 +2⋆ +∼ R−1 +KK +is of order the KK scale in size. +Canonically normalizing by rescaling b = Mp b/τ⋆ —- then produces a lagrangian of the form (4.8) +with +F ∼ M 2τ⋆ +Mp +∼ Mp ≫ fb ∼ Mp +τ⋆ +. +(4.15) +As is typical for KK modes the field b ∈ Bmn couples with gravitational strength. Notice that the +ratio F/fb ∝ V2⋆ can be enormous, since V2⋆ can be as large as 1028 in the extreme case of two large +– 21 – + +eV-scale dimensions. In this case gauge invariance precludes choosing J to be the dual of the Chern +Simons form of a brane localized gauge sector, even in the absence of any φ-dependence in (4.13). +It is not that surprising to have a breakdown of 4D EFT methods at the KK scale, but the above +discussion shows there is a difference between what happens at this scale for T- and S-type axions. +For T-type axions the coupling to matter is order 1/Mp and this remains true above the KK scale. +The breakdown of the 4D EFT is about the appearance of a multitude of new KK modes, all of which +couple with gravitational strength. But the S-type axion’s coupling to matter is proportional to E/fa +and so actually grows to become order unity at the KK scale; what does this order unity coupling +match to in the UV theory? It matches to a dimensionless extra-dimensional coupling in the UV +theory: either to the coupling c appearing in (4.9) or to the coupling gcs of BMN to the Chern-Simons +term SMNP that is implied by the field strength G = dB +gcsS. Although gcs is order 1/M 2 when BMN +is dimensionless (as above), it is dimensionless once BMN is canonically normalized in six dimensions. +Since gauge invariance prevents the coupling (4.13) from containing a coupling between T-type +axions and a gauge sector localized on a 3-brane, one can ask whether such couplings are more generally +forbidden. The answer to this is ‘no’ if we allow ourselves to consider gauge sectors localized on higher- +dimensional branes. For instance for a six-dimensional 5-brane Σ6 they can arise from an interaction +of the form +Sint,g = cM 2 +� +Σ6 +B ∧ F ∧ F ∝ c +� +d4x√−g b ϵµνλρFµνFλρ , +(4.16) +where the explicit factor of V2 coming from the integration over the additional two dimensions cancels +the normalization of the harmonic form ωmn ∝ V−1 +2 εmn. For a canonically normalized scalar this +would imply gaγγ ∼ 1/fb. +The upshot is this: the model-dependent T-type axions can couple surprisingly weakly to non- +gauge matter compared to the scale set by their decay constant: 1/F ∼ 1/Mp ≪ 1/fb ∼ RKK. By +contrast, the model-independent S-type axion couples to matter with strength 1/F ∼ 1/fa ∼ 1/fb +and the same is true of T-type couplings to gauge fields on higher-dimensional branes. From the point +of view of the underlying string coupling eφ the duality that maps bµν to a is also a strong/weak +coupling duality. +5 +Conclusions +Axions (or ALPs) are often motivated by appealing to string theory, which seems to provide them +with abundance. But string theory also provides strong concrete evidence for the assertion that exact +global symmetries cannot survive contact with quantum gravity; the observation that underlies the +UV quality problem for attempts to solve the strong-CP problem using a global PQ symmetry. +We here reconsider some of the implications that follow from the observation that axions arise +as antisymmetric tensor fields in higher dimensions and that Standard Model fields usually live in +localised objects like D-branes within the extra dimensions. Axions arise in two general types in this +way: the model independent S-type axion originating from a two-form field in 4-dimensions after +compactification; and the model dependent T-type axion such as arises as a Kaluza-Klein mode for +an extra-dimensional tensor field (of which we focus for simplicity on two-form potentials in two extra +dimensions).17 +These allow for a rich structure of axion phenomenology and each type of axion can be adapted +to realize the PQ solution to the strong CP problem. It has been known for a while that UV effects +17Axions may come from other forms such as three or four forms in ten dimensions depending of which type of string +theory. +– 22 – + +can affect the original PQ proposal by generating effective interactions that violate the global PQ +symmetry and modify the prediction for the axion mass: the axion quality problem. We revisit how +this problem arises for the two types of axion using the UV tools at hand. +We find that for T-type axions the quality problem resembles the form originally studied, since the +UV theory directly provides a pseudoscalar field once compactified to four dimensions. Our framework +differs from early versions of the quality problem that imagine the PQ symmetry to be linearly realized +by a complex scalar at energies E > f, but generally agree with estimates based on the contributions +due to wormholes or gravitational instantons below a compactification scale. To the extent that these +contributions are exponentially suppressed their constraints are mild. +The S-type case is more interesting since both the strong CP and axion quality problems must +first be reformulated in terms of the two-form field and its field strength. The PQ mechanism involves +giving a mass to the axion and so on the dual side involves the ‘eating’ of a 3-form potential along +the lines proposed in [27]. The required 3-form potential is generated by the QCD sector itself as a +non-propagating topological field in the EFT below the QCD scale. As applied to QCD our re-analysis +broadly agrees with that of [14] in concluding that the quality problem gets recast as an issue that +arises when there are multiple 3-form fields present in the low-energy theory. This might be imagined +to be a problem for string theory, for which 3-form potentials are as ubiquitous as axions. +Prompted by recent discussions of this problem [15, 16] we formulate the two properties which +new 3-form fields must have if they are to threaten the PQ solution to the strong CP problem, arguing +why string-generated 3-form fields are not generically a problem, largely because these fields need not +couple to QCD (in string theory it depends on how bulk fields couple to brane fields and usually only +one couples to QCD). The gravitational Chern-Simons term does couple to QCD but whether or not it +sparks a new strong CP problem depends on whether or not gravity is strongly coupled in the UV. The +discussions of [14, 16] assume that it is, but we argue that if it is not (such as if the UV completion is a +weakly coupled string vacuum) then the estimates for the size of the problem are again exponentially +suppressed and so would not pose a quality problem. +Finally we also explore other UV implications for axion physics. We found that depending on the +brane configuration hosting the Standard Model, extra dimensions can dramatically suppress physical +couplings between the axion and Standard Model sector relative to the axion decay constant appearing +in the axion kinetic term, especially if the volume of the extra dimensions is very large. This is possible +for T-type axions but in the the examples examined does so only for non-gauge couplings (making +this observation more pertinent for ALPs, whose properties would tend to be ‘fermiophobic’). +For S-type fields both kinds of couplings have similar size.18 For this case though, we argue that +the duality relating the 2-form to the axion field swaps weak and strong couplings, and suggests a +semiclassical description of 2-form response need not correspond to the usual semiclassical description +of a scalar axion. This again motivates better exploring the 2-form side of the theory. +It is an old argument that UV information can have important implications for low-energy natu- +ralness questions such as the strong CP problem. The observation that this could be informative in +situations where the questions are solved using features like global symmetries that apparently should +not be present at very high energies has sparked a revival of studies of generalised and non-invertible +symmetries. Many of these ideas resonate well with string-motivated constructions, such as those we +explore here. +18For supersymmetric realizations this can be seen because both the K¨ahler potential and gauge kinetic function +depend directly on the S field. +– 23 – + +Acknowledgements +We thank Philippe Brax and Junwu Huang for helpful conversations. CB’s research was partially +supported by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada. +Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC +and by the Province of Ontario through MRI. 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Sezgin, World Scientific +(Singapore) 1989. +– 27 – + diff --git a/_tAyT4oBgHgl3EQfqvix/content/tmp_files/load_file.txt b/_tAyT4oBgHgl3EQfqvix/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..251c1c2bbd0ce5e46d893a9af8e86fda31a27649 --- /dev/null +++ b/_tAyT4oBgHgl3EQfqvix/content/tmp_files/load_file.txt @@ -0,0 +1,1085 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf,len=1084 +page_content='Prepared for submission to JHEP CERN-TH-2022-176 UV and IR Effects in Axion Quality Control C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Burgess,1,2 Gongjun Choi3 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Quevedo4 1 Department of Physics & Astronomy, McMaster University 1280 Main Street West, Hamilton ON, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2 Perimeter Institute for Theoretical Physics 31 Caroline Street North, Waterloo ON, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3 CERN, Theoretical Physics Department, Gen`eve 23, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 4 DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Abstract: Motivated by recent discussions and the absence of exact global symmetries in UV com- pletions of gravity we re-examine the axion quality problem (and naturalness issues more generally) using antisymmetric Kalb-Ramond (KR) fields rather than their pseudoscalar duals, as suggested by string and higher dimensional theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Two types of axions can be identified: a model independent S-type axion dual to a two form Bµν in 4D and a T-type axion coming directly as 4D scalar Kaluza- Klein (KK) components of higher-dimensional tensor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For T-type axions our conclusions largely agree with earlier workers for the axion quality problem, but we also reconcile why T-type axions can couple to matter localized on 3-branes with Planck suppressed strength even when the axion decay constants are of order the KK scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For S-type axions, we review the duality between form fields and massive scalars and show how duality impacts naturalness arguments about the UV sensitivity of the scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In particular UV contributions on the KR side suppress contributions on the scalar side by powers of m/M with m the axion mass and M the UV scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We re-examine how the axion quality problem is formulated on the dual side and compare to recent treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We study how axion quality is affected by the ubiquity of p-form gauge potentials (for both p = 2 and p = 3) in string vacua and identify two criteria that can potentially lead to a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We also show why most fields do not satisfy these criteria, but when they do the existence of multiple fields also provides mechanisms for resolving it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We conclude that the quality problem is easily evaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='00549v1 [hep-th] 2 Jan 2023 Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Types of UV axion pedigree 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 Non-propagating low-energy forms 3 2 Axions and duality 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Axion/2-form duality 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 A Higgs mechanism for scalar masses 5 3 Naturalness issues for dual systems 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 QCD and the dual PQ mechanism 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 The Quality Problem 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3 The dual Quality Problem 12 4 UV completion and matter couplings 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Extra-dimensional UV completion 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 Coupling strengths 20 5 Conclusions 22 1 Introduction String theory giveth and string theory taketh away, at least where axions1 are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' On one hand axions are said to be ubiquitous in the spectrum of particles predicted around most string vacua [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This observation motivates the study of their phenomenological consequences [6], with a particular focus of late on their possible role as a light form of dark matter [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' On the other hand, string theory equally generally forbids2 the existence of exact rigid (or global) symmetries [8], in principle including the rigid shift symmetries on which low-energy axion properties are founded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For Goldstone bosons this breaking can keep them from being light, and can interfere with any mechanisms that rely on the survival of axions down to the low-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' As applied to the QCD axion this has come to be known as the axion ‘quality’ problem [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' So which is it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Are axions as abundant as dirt or as diamonds in low-energy string vacua?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The resolution (which has long been known) is that there is a sense they are both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The absence of global symmetries really does mean that one never really directly finds scalar axions a with shift symmetries in string vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Instead these scalars arise indirectly as Kaluza-Klein (KK) modes from fields that 1We follow the string literature and broadly refer to any low-energy Goldstone boson enjoying a rigid compact shift symmetry as an ‘axion’ (as opposed to the ‘dilatons’ associated with rigid scaling symmetries), something that would be called an ALP (axion-like particle) by particle phenomenologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Our later focus is on those Goldstone bosons whose symmetries have a QCD anomaly and so can take part in the strong-CP problem [1–3] (which is what a particle physicist would usually mean by an ‘axion’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2Although there are known ways out [9] the conclusion is nonetheless broadly true and global symmetries tend to be both rare and approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 1 – not themselves scalars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' commonly arising3 as components of 2-form Kalb-Ramond gauge fields [11], B = 1 2 BMN dzM ∧ dzN, subject to the gauge symmetries B → dλ for some arbitrary field λM(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Fields like BMN arise so frequently in string vacua because they are related to other fields (notably the metric) by supersymmetry in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Types of UV axion pedigree Low-energy scalars typically emerge in the 4D effective theory from such fields in one of two ways: T-type axions: b(x) are specific cases of Kaluza-Klein (KK) modes arising when dimension- ally reducing the extra-dimensional components Bmn(x, y) = b(x) ωmn(y), where xµ denote the observed 4 dimensions, ym are extra-dimensional coordinates and ωmn(y) is a harmonic 2-form field within the extra dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' S-type axions: a(x) arise directly as the 4-dimensional components Bµν(x, y) = bµν(x) ω(y), which in four dimensions are known to be dual to scalar fields with shift symmetries [12] through relations of the form ∂µa ∝ ϵµνλρ∂νBλρ (much more about which below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Here ω(y) is a harmonic 0-form field – typically a y-independent constant that can depend on extra-dimensional moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This UV provenance is of course relevant to the axion quality problem, which is in essence an issue of UV sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One of our goals with this paper is to explore the ways that it helps, for both T- and S-type axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Some of our conclusions are similar to earlier discussions of this issue [13, 14], in particular that the problem gets rephrased in dual form (for S-type axions) in terms of the existence of multiple 3-form gauge potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Since these issues have recently been revisited anew [15, 16] we clarify what properties these fields must have to actually cause a quality problem and use this to argue why gravitational examples specifically (and the great abundance of such potentials in string vacua more generally) need not pose a problem in themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The dual formulation also suggests how the presence of multiple axions (as is common in string theory) can help alleviate the quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The upshot is that the UV can, but need not, cause a quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Whether or not it does cannot be decided purely at low energies because it depends on what happens in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4 But our discussion has implications that apply more broadly than just to the quality problem for the QCD axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Along the way we identify more generally how dimensional ‘naturalness’ arguments for the scalar potential give very different estimates depending on whether they are made directly for the scalar or are first done for its dual and then mapped to the scalar using duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In particular terms involving n powers of the canonically normalized scalar arise additionally suppressed by powers of (m/M)n where M is the UV scale and m is the axion mass (an observation also made in the past for inflationary models [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We find a number of other ways that axion properties suggested by string-motivated extra- dimensional physics can be informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For instance we describe a simple model for which T-type axions have physical axion-matter couplings gaff that are dramatically smaller than the naive value 1/f read off from the axion kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In the example given (motivated by the models of [19]) gaff is order 1/Mp despite f being an ordinary particle-physics scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Decoupling these scales from one another could have practical implications for axion phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 34D axions can also arise as KK modes from other types of extra-dimensional fields, but we focus on the Kalb-Ramond field because it allows a unified treatment of two different types of 4D axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 4The same is is also true of other naturalness problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' they arise because of strong dependence on physical masses for states that actually appear in the UV theory and not a dependence on cutoffs, as is sometimes mistakenly asserted (for a summary of these issues see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 2 – We show why the same hierarchy does not arise in these models for S-type axions and we clarify why not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Physical couplings of S-type axions really are of order 1/f and we identify which interactions in the UV completion are responsible for the breakdown of the E/f expansion at energies E > ∼ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' S- type axions illustrate how the scalar and dual representations can provide instances of weak/strong coupling duality, for which both the scalar and the dual cannot be within the weakly coupled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In the extra-dimensional example studied it is the Kalb-Ramond formulation that is weakly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This could also have phenomenological implications to the extent that an axion that is dual to a weakly coupled system is unlikely to be well-described by the semiclassical methods that are universally used when exploring its physical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Some of these observations imply that the use of the scalar (rather than Kalb-Ramond) vari- able can be misleading in some circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This can seem surprising at first sight because the duality between axions and Kalb-Ramond fields is in essence a field redefinition and so scalar and dual formulations should be completely equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' it shouldn’t matter that string theory hands you Kalb-Ramond fields if scalar axions are equivalent and are much simpler to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Why should one care that a more complicated framework exists if it only obscures implications drawn using more transparent methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We argue here that phenomena like weak/strong coupling duality are special cases of Weinberg’s Third Law of Progress in Theoretical Physics [20]: You can use any degrees of freedom you like to describe a physical system, but if you use the wrong ones you’ll be sorry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 Non-propagating low-energy forms These duality arguments touch on a related rich vein of physics with broader significance: the im- portance of keeping non-propagating entities like auxiliary and/or topological fields when formulating Wilsonian effective theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' These are fields that can be integrated out without changing the types of particles that propagate, and so it is tempting to think one should do so once and for all and simply ignore them thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' However such fields bring to the low-energy effective theory information about how its UV completion responds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' to environments with nontrivial topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' They arise in concrete situations (such as in EFTs for 3-dimensional Quantum Hall systems, where the presence of emergent non-propagating gauge fields is essential for capturing the fractional quantization of Hall plateaux and the unusual charge and statistics of some excitations [21, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Evidence is building that a similar role is played more widely by 3-form gauge potentials in four spacetime dimensions, C := 1 6 Cµνλ dxµdxνdxλ subject to the gauge freedom C → C + dΛ where Λµν(x) = −Λνµ(x) is an arbitrary 2-form field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' These are known to bring to the low-energy 4D effective theory topological information coming from integrated out extra dimensions [23, 24], and more generally provide the origin for the auxiliary fields that appear in the 4D supergravities that are the low-energy limits of string vacua [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' They appear in the QCD quality story because they can give masses to Kalb-Ramond fields [27] through a Higgs mechanism that is dual to more mundane methods of axion mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because the field strength H = dC often appears in the action with a definite sign (often as a square), its presence can alter the implications of naturalness arguments for the scalar potential [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Indeed such terms provide the 4D understanding of why 6D SLED models [29] can in some circumstances suppress the 4D vacuum energy, but also why they struggle to do so enough to solve the cosmological constant problem [24, 30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Their interplay with accidental scaling symmetries lies behind a recent attempt to find a dynamical relaxation mechanism for vacuum energies in four dimensions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In what follows we build our case for the above story using concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We first, in §2, briefly review the duality construction – in particular its extension to massive axions [27], which provides a way to think about scalar masses arising through a Higgs mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' §3 then briefly – 3 – reviews and clarifies its use to dualize the axion solution to the strong-CP problem [14], highlighting in particular how the quality problem gets rephrased in the dual language and how the apparent UV sensitivity of terms in the EFT differs between the axion formulation and its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Finally §4 provides a concrete extra-dimensional example – inspired by a UV completion of [19] – that illustrates both how axion/Kalb-Ramond duality can map weak to strong couplings, and how enormous hierarchies can arise with gaff, gaγγ ∼ 1/Mp even with f as low as eV scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2 Axions and duality We start with a review of why 2-form gauge potentials like Bµν are dual [12, 33] to scalar fields, both in the standard shift-symmetric massless case and for massive scalars, following the discussion of [27] (which in turn generalizes earlier arguments [34] aimed at describing particle/vortex duality in Kosterlitz-Thouless transitions [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Axion/2-form duality Consider the following path integral Ξ[J] = � DB eiS1[B] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1) where S1 = � d4x L1 with L1 chosen (at least to start) to be L1(B) = − Z 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2) with G = dB the exterior derivative of a 2-form field Bµν and Z and Jρ possibly depending on other fields (collectively denoted ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' B = 1 2 Bµν dxµ ∧ dxν is only defined up to the gauge redundancy B → B + dλ for an arbitrary 1-form λ = λµ dxµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The duality starts by trading the integration over Bµν for an integral over Gµνλ subject to a constraint that imposes the Bianchi identity dG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' These are equivalent because the Bianchi identity is sufficient to guarantee the local existence of a field Bµν with G = dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The constraint is imposed by integrating over a scalar Lagrange-multiplier field a, and so writing Ξ[J] = � DG Da eiS0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3) where S0 = � d4x L0 with L0(G, a) = − Z 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='a ϵµνλρ∂µGνλρ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) Integrating out a imposes the Bianchi identity dG = 0 and allows the integral over G to be replaced with the integral over B, leading back to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The dual version is obtained from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) by instead integrating out Gµνλ so that a is the remaining field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The result inherits a shift symmetry a → a + constant because L0 transforms into a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The G integration is gaussian, whose saddle point is Gµνλ = Gµνλ where Gµνλ = −Z−1ϵµνλρ � ∂ρa + Jρ� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5) and so the integration gives the new lagrangian density L2(a) = − 1 2Z (∂µa + Jµ)(∂µa + Jµ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) – 4 – If Z = 1 then a is a canonically normalized massless scalar derivatively coupled to the same local current Jµ as in the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) are both obtained from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) they must describe equivalent physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Although the implied field redefinition from Bµν to a is in principle nonlocal the physics on both sides is nonetheless local because this is true of the relation between the field strengths given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Significance of Z ↔ Z−1 In reality the above gaussian action is always supplemented by other non-gaussian interactions Lint within a low-energy Wilsonian effective field theory (EFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To the extent that both Bµν and a are derivatively coupled perturbative semiclassical methods in the presence of nongaussian terms like (GµνλGµνλ)2 ∈ Lint are ultimately justified by a low-energy derivative expansion that applies equally well on both sides of a duality relationship because relationships like (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5) involve equal numbers of derivatives on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The inversion of Z → Z−1 as one passes from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) is a noteworthy feature of duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' When Z ≫ 1 this implies 2-point correlators of Gµνλ are order Z−1 in size while those of ∂µa are instead order Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The significance of the change Z → Z−1 depends on whether or not Bµν and a can be freely rescaled to remove Z by going to canonically normalized variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' If this is so then Z in any case drops out of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For instance, when Jµ ̸= 0 this rescaling shows that Ξ is really only a function of �Jµ := Z−1/2Jµ rather than depending on Z and Jµ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Although Z ↔ Z−1 is sometimes called weak/strong coupling duality, Ξ[ �J] is the same on both sides of the duality and so both sides agree on its functional dependence if expanded order-by-order in powers of Z−1 (say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One situation where this kind of rescaling is not possible is when Z depends on other fields and the target-space metric in field space is not flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Another case where physics can depend explicitly on Z is when the field Bµν or a is quantized5, perhaps satisfying a boundary condition like � W dxµ∂µa = 2πnf for some curve W, integer n and mass scale f, or perhaps � C B = 2π˜nf −1 for some 2-cycle C and possibly different integer ˜n and mass scale ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In these situations physical results can depend on Z (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' on f and/or ˜f) and Jµ separately, and the relation Z → Z−1 can carry physical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 A Higgs mechanism for scalar masses Although the above makes the shift symmetry (and so also masslessness) of a seem automatic, we next summarize how duality extends to massive scalars, following [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' A scalar potential is achieved in the dual framing through a Higgs mechanism in which the field Bµν ‘eats’ (or is eaten by) a non- propagating gauge potential6 Cµνλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because Cµνλ does not propagate this meal does not change the number of propagating degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To this end consider the following gaussian path integral Ξ[J] = � DC DB eiS1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) where S1 = � d4x L1 and L1(C, B) = − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (Gµνλ + mCµνλ)(Gµνλ + mCµνλ) − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρ(Gµνλ + mCµνλ)Jρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='8) 5This is generic the case in string theory for which the symmetries associated to antisymmetric tensors and axions are compact (meaning there always exist magnetic-like branes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For a general discussion see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 6Known string vacua can also contain a large number of these 3-form gauge potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 5 – Here Cµνλ is a 3-form gauge potential with field strength H = dC and Bµν is a 2-form gauge potential with G = dB while m is a parameter with dimension mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This lagrangian has the gauge symmetry C → C + dΛ and B → B − m Λ for an arbitrary 2-form Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' So when m ̸= 0 we can choose a gauge B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The field equation for C that follows from this action then is DµHµνλρ + m2Cνλρ + m ϵνλρµJµ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9) This describes a single spin state propagating with mass m once all the gauge symmetries are used, as can be seen by counting the massless states from which it is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (In 4D Bµν is shown above to be equivalent to a massless scalar and Cµνλ contains no propagating degrees of freedom at all because one can always write Hµνλρ = h ϵµνλρ with field equation ∂µHµνλρ = 0 in the massless limit, which implies h is a constant and so does not propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=') The dual should therefore be a massive scalar and this can be verified by trading the integral over B for an integral over G and introducing (as before) a lagrange multiplier a to impose the Bianchi identity7 dG = 0, leading to the lagrangian density L0(C, G, a) = − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (Gµνλ + mCµνλ)(Gµνλ + mCµνλ) − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' a ϵµνλρ∂µGνλρ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρ(Gµνλ + mCµνλ)Jρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='10) Integrating out a returns us to the above formulation, but instead performing the integration over G leads to the saddle point Gµνλ = −mCµνλ − ϵµνλρ � ∂ρa + Jρ� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='11) and so to the lagrangian L2(C, a) = − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ − m 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' a ϵµνλρHµνλρ − 1 2∂µa ∂µa − Jµ∂µa − 1 2JµJµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='12) Next we perform the integral over Cµνλ, and this is equivalent to simply performing the gaussian integral over Hµνλρ because the integrability condition for writing H = dC is dH = 0 which is always true (in 4D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The saddle point for the H integral occurs for Hµνλρ = Hµνλρ where Hµνλρ = −m a ϵµνλρ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) and so leads to the scalar lagrangian L2(a) = −1 2(∂a)2 − m2 2 a2 − Jµ∂µa − 1 2JµJµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='14) This is the expected massive scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Scalar potential For future reference notice that it is only this last step that would differ if we’d had higher-dimension terms like δL = W(X) in the lagrangian with X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHµνλρ and so X2 = − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The above discussion is the special case W = 1 2X2 but one could entertain, for example, W = c1M 2X + 1 2 X2 + 2c3 3M 2 X3 + c4 4M 4 X4 + · · · (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) 7One can equivalently omit the mCµνλ terms everywhere and instead impose the modified Bianchi identity dG = mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 6 – where the coefficients ci are dimensionless and M is a UV scale inserted everywhere on dimensional grounds (with Hµνλρ canonically normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For non-quadratic W the integral over H is no longer gaussian, but we can proceed assuming a semiclassical saddle-point approximation is valid, in which case the saddle point (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) is modified to �∂W ∂X � H=H = m a , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='16) which agrees with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) when W = 1 2X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For example, for the choice (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) this becomes c1M 2 + X � 1 + 2c3 M 2 X + c4 M 4 X2 + · · · � ≃ m a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='17) and so X ≃ m a − c1M 2 − 2c3 M 2 � m a − c1M 2�2 + O �� m a − c1M 2�3 /M 4� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='18) Once used in the lagrangian this shows how non-quadratic pieces of W map over to non-quadratic contributions to the scalar potential for a in the dual lagrangian L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In particular the axion potential becomes V (a) = −W(X) + maX = 1 2 � ma − c1M 2�2 − 2c3 3M 2 � ma − c1M 2�3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='19) Two features are noteworthy about this potential: First, notice it shares the usual Legendre property ∂V ∂a = mX + � −∂W ∂X + ma � ∂X ∂a = mX , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='20) where the last equality uses (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Even if new non-quadratic terms introduce new stationary points for V (a) (or shifts the positions of old ones) eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='20) ensures X = 0 for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Second, once a is shifted so that the minimum is at a = 0 the potential depends on m and a only through the combination ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Consequently, a term proportional to an comes suppressed by a power of (m/M)n relative to what would naively be expected on dimensional grounds for V (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is how the dual theory reproduces the same M-dependence as found for higher powers of Hµνλρ given that a has canonical dimension mass while H has dimension (mass)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This shows how a dimensional assessment of how UV scales appear in the low-energy theory can care about the existence of a dual formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3 Naturalness issues for dual systems This section examines how naturalness arguments look for T- and S-type axions, and for S-type axions how they depend on which side of the duality relation they are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We do so using the axion quality problem as a representative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 QCD and the dual PQ mechanism To this end we extend the above reasoning to the main event: QCD and the θ-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The idea is to dualize the coupling of the axion to QCD to see how the strong-CP problem gets formulated, along the general lines of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We then ask how UV physics might complicate the story in the dual theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 7 – Consider then adding a gauge potential Aµ (with field strength Fµν) to represent the QCD gauge sector8 and this time consider the path integral Ξ[J] = � DG DA Da eiS0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1) where S0 = � d4x L0 and L0(G, A, a) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − a 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρ � ∂µGνλρ − 1 4 Ωµνλρ � − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ −1 4FµνF µν − θ 2 ϵµνλρFµνFλρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2) We suppress both gauge-group indices and traces over them to avoid notational clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Fµν is the field strength for the gauge potential Aµ but Gµνλ is an arbitrary 3-form until the integral over a is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Integrating out a imposes the Bianchi identity dG = Ω where Ω is a gauge-invariant quantity built from the gauge field that on grounds of consistency must satisfy dΩ = 0, for which we take 1 12 ϵµνλρΩµνλρ = 1 f ϵµνλρFµνFλρ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3) The mass scale f is here required on dimensional grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Doing this allows the G integral to be traded for one over B as before and gives the lagrangian L1(B, A) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ − 1 4FµνF µν − θ 2 ϵµνλρFµνFλρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) where G = dB + S where dΩ = 0 implies there locally exists an Sµνλ – the Chern-Simons 3-form – that satisfies Ω = dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The dual formulation instead integrates out G and leaves a as the dual variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Integrating out G leads to the lagrangian density L2(A, a) = −1 2(∂a)2 − Jµ∂µa − 1 2JµJµ + a 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρΩµνλρ − 1 4FµνF µν − θ 2 ϵµνλρFµνFλρ = −1 2(∂a)2 − Jµ∂µa − 1 2JµJµ − 1 4FµνF µν + 1 2 � a f − θ � ϵµνλρFµνFλρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5) This shows that the standard axion-gauge coupling is the dual of the 2-form/QCD coupling given in L1 and that f can be interpreted as its decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Below the QCD scale In the standard axion-QCD story integrating out QCD leaves a residual axion potential due its anoma- lous coupling to F ∧ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This minimum is argued to be minimized where a = θ f (where θ is the usual combination of θ and phases in the quark mass matrices) which ensures that the CP-odd contribution turns off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We seek to express how physics below the QCD scale works in the dual language involving Bµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Below ΛQCD the gauge degrees of freedom are integrated out, naively leaving only hadrons coupled to Bµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The key thought is that this is not quite right: the QCD EFT below ΛQCD contains a path integral over low-energy hadrons and an integration over a low-energy field Cµνλ, whose emergent 8We do not write quarks explicitly but flag the few places where their implicit presence affects what is written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 8 – presence the strongly coupled vacuum of QCD makes mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The field Cµνλ ∝ ⟨Sµνλ⟩ is the low- energy counterpart of the Chern-Simons field appearing in the topological susceptibility [37] above the QCD scale, where dS = F ∧ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Having this field in the low-energy theory below the QCD scale does not affect the existence of a gap or the spectrum of the known hadrons because Cµνλ does not propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It is an auxiliary field that is required in order for the low-energy theory to capture properly the response of QCD to any topology in its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Similar fields are known to arise in this way in other concrete systems like the EFTs describing Quantum Hall systems [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This 3-form potential differs from many of the others that often arise in string vacua because it arises from the IR properties of QCD rather than from the physics of UV compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' On dimensional grounds we write H = dC with 1 12 ˜Λ2 QCDϵµνλρHµνλρ = ϵµνλρ⟨FµνFλρ⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) where ˜ΛQCD denotes a parameter of order the QCD scale that ensures that H has canonical dimension (mass)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The lagrangian (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) above the QCD scale is then replaced with its low-energy counterpart L1(C, B) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ− 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ− θ 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ˜Λ2 QCDϵµνλρHµνλρ− 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ+· · · , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) where the explicit term proportional to θX combines with quark mass phases – that also enter as terms linear in X, as in the c1 term of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) – to produce θX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The ellipses in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) are at least cubic in X (or involve derivatives of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Combining eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3) (and the discussion just above it) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) implies dG = ⟨Ω⟩ = ˜Λ2 QCD f H , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='8) and so comparing this to dG = mH (as would follow from G = dB + mC) allows us to read off the mass relation m = ˜Λ2 QCD/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We see that the mC term captures the expectation value ⟨S⟩/f of the Chern-Simons term in the UV theory above the QCD scale if m scales with f in the same way that the usual axion mass depends on its decay constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We expect the low-energy presence of such a 4-form field H to give B a nonzero mass, as we check by introducing the lagrange multiplier a in the usual way and integrating out G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This leads to the result L2(C, a) = −1 2(∂a)2 −Jµ∂µa− 1 2JµJµ + 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (ma−θ˜Λ2 QCD)ϵµνλρHµνλρ − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ +· · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9) Integrating out H leads to the saddle point Hµνλρ = Hµνλρ with Hµνλρ = � ma − θ˜Λ2 QCD � ϵµνλρ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='10) and so gives the axion lagrangian L2(a) = −1 2(∂a)2 − Jµ∂µa − 1 2JµJµ − 1 2 � ma − θ˜Λ2 QCD �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='11) showing that the minimum indeed occurs where a = θ˜Λ2 QCD/m = θf, which turns off the CP-violating term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 9 – In general integrating out the UV QCD sector also generates more complicated low-energy in- teractions involving C, such as the function W(X) of X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHµνλρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' As above, such terms semiclassically change the saddle point to �∂W ∂X � H=H = m a − θ˜Λ2 QCD , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='12) and so leads to the axion potential V (a) = −W(X) + (ma − θ˜Λ2 QCD)X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) This satisfies ∂V ∂a = mX + � −∂W ∂X + ma − θ˜Λ2 QCD � ∂X ∂a = mX , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='14) and so again ensures that X = 0 at any of the stationary points of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We see that the presence of interactions like W(X) show that V is minimized at ma = θ˜Λ2 QCD if ∂W/∂X vanishes when X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 The Quality Problem We now have the tools required to explore UV sensitivity and the axion quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We start by restating the original formulaton of the quality problem and then how it is rephrased in 2-form language for both T-type (this section) and S-type (next section) axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The axion quality problem asks two related questions [10]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Do corrections to the QCD axion potential change its minimum in a way that preserves a sufficiently small effective vacuum angle: ¯θeff ≲ 10−10?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Do corrections to the QCD axion potential change the usual expression for the axion mass (that assumes it is dominantly generated by the ‘IR-dominated’ QCD instanton with size ρ ∼ Λ−1 QCD)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The first of these essentially asks if the QCD axion remains a good solution to the strong CP problem when perturbed by new physics, whereas the second asks the same of our understanding of axion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The axion mass question can apply more generally to ALPs as well, whereas the first one is specific to the QCD axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Any UV completion must decide what happens at energies above the axion decay constant f above which the low-energy expansion in powers of E/f breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We consider in turn the original formulation and the T- and S-type axions that arise within an extra-dimensional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Original formulation In the initial formulation the UV completion for scales above f was assumed to involve a second scalar that combines with the axion to linearly realize the PQ symmetry as a complex scalar Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In this picture the modulus of Φ acquires a mass proportional to f ∼ ⟨Φ⟩ and the axion starts life as the phase of Φ ∝ eia/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Motivated by string theory and black-hole thought experiments it is then assumed that UV physics cannot support an unbroken global symmetry, and so at some large scale M the form of the scalar potential for Φ cannot be assumed to be invariant under re-phasings of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' As an expansion in powers of Φ, the generic potential form would be VUV (Φ) = M 4 2 ∞ � n=1 � cn Φn M n + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) – 10 – where the cn’s are in general complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is true even if the UV physics is assumed to be CP- invariant because cn will inherit the phase of the fermion mass matrix after chiral PQ rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In the initial formulation M is assumed to be the Planck mass Mp, and although we can see that such a choice would dominate smaller M for the terms with n < 4 it is likely that M < Mp would be more dangerous for n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Early workers typically assumed that the renormalizable part of the potential would be tuned to make the axion potential sufficiently shallow and so effectively started the sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) at n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Freezing the modulus field at ⟨Φ⟩ = f and integrating it out at the classical level leads to the following effective axion potential VUV (a) = M 4 2 � n=1 |cn| f n M n � eiδn eina/f + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' � = M 4 � n=5 |cn| f n M n cos �na f + δn � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='16) where we shift fields so that the standard QCD solution is a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The QCD minimum therefore remains unchanged if V ′ UV (0) = 0 and this would be true if all of the δn’s were to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Although the axion potential height (and therefore possibly axion mass) might still change due to the presence of VUV (a), evasion of the strong CP problem requires only that the minimum for a remains unmoved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Stability of the minimum: For δn , |cn| ∼ O(1) we can estimate the size of the effective value of ¯θeff by perturbing around the QCD minimum at a = aQCD: ¯θeff ≃ − V ′ UV (aQCD) fV ′′ QCD(aQCD) ∼ VUV (aQCD) VQCD(aQCD) ∼ M 4 Λ4 QCD � f M �n0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='17) where n0 represents the first power appearing in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For example, requiring ¯θeff < 10−10 for the example f = 1012 GeV, M = Mp = 1018 GeV and ΛQCD ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 GeV in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='17) requires n0 > ∼ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Stability of the axion mass: The change to the axion mass induced by the UV axion potential is given by δm2 a = ∂2VUV (a) ∂a2 ���� a=0 = M 2 � n=1 n2|cn| � f M �n−2 cos δn , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='18) which can be significant unless the coefficients |cn|’s are extremely small even if all the δn’s could be contrived to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' When significant such contributions spoil the relation maf ∼ mπFπ that holds for the low-energy QCD contribution and on which most axion phenomenology is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because the mass is not inversely proportional to f this expression shows that the relation between ma and f need not be inversely proportional to one other, for example allowing a very heavy axion to be still very weakly coupled to matter – a drastic change relative to standard axion phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 T-type axions The story is similar for T-type axions, at least below the Kaluza-Klein scale where they are 4D scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' No quality issue arises above the KK scale because here the relevant fields are higher-dimensional form fields HMNP and the only symmetries involved are gauge symmetries like B → B + dλ [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Recalling that T-type axions, b, arise as extra-dimensional reductions of the form Bmn(x, y) = b(x) ωmn(y), with ωmn a harmonic form in the extra dimensions, the origin of the low-energy shift – 11 – symmetry b → b + c (for constant c) has its origins as the extra-dimensional transformation Bmn → Bmn + c ωmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is a symmetry of H = dB because harmonic forms are closed: dω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It is strictly speaking a ‘large’ gauge transformation because harmonic forms are not exact: there does not globally9 exist a λm such that ω = dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The quality problem arises because the shift symmetry in the low-energy 4D theory is not a local gauge symmetry and so it in principle need not be respected by UV effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One consequently cannot completely preclude the generation of a scalar potential, VUV (b) ∼ M 4 � n cn � b M �n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='19) where cn are dimensionless order-unity coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' But its failure to be a local gauge symmetry is a global obstruction rather than a local one and this means that UV effects cannot generate VUV (b) until scales are integrated out that ‘see’ the topology that can distinguish ω from dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This implies two sorts of changes to the standard quality-problem argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' First, the scale M where problems first arise cannot be higher than the KK scale M ∼ 1/RKK corresponding to the size of the 2D cycle in the extra dimensions whose presence is associated with the existence of the harmonic form ωmn(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Second, the physics at scale M that generates the potential must itself be sensitive to the nontrivial topology, often leading to additional suppressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For instance, an example of physics that can generate PQ-violating operators in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) identified in [13, 38] is wormhole [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For these the coefficients cn in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='16) are exponentially suppressed, given by [13] cn ∼ e−S ∼ e−(MpL)2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='20) where S is a wormhole action and L the size of its throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Maintaining the success of the PQ mechanism requires S > ∼ 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' More complicated configurations are possible for extra-dimensional theories, for which Mp can be replaced by another UV gravity scale Mg, that might be the string scale or the extra-dimensional Planck scale Mg in specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Similarly L can be one of the geometric scales of the background, that could (but need not) be approximately a compactification scale RKK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' All known semiclassical arguments of this type must assume MgL ≫ 1 for the calculation to be under control, because semiclassical methods are justified within an expansion in powers of (MgL)−1 within any gravitational EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' MgL ∼ 14 suffices to ensure S > ∼ 190 and so satisfying this constraint seems not that difficult within the semiclassical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' These kinds of arguments were used in [18] to argue for the absence of large gravitational correction to the inflaton potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3 The dual Quality Problem For S-type axions the representation directly obtained from UV physics is the field bµν dual to the scalar axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' And as alluded to earlier – c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 – issues of UV sensitivity can look very different in dual formulations to scalar theories, with for example the existence of a dual implying that the effective couplings for terms like an ∈ VUV (a) come suppressed by powers of the axion mass (m/M)n relative to generic scalar estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Such suppressions can be enormous given the small size of m relative to UV scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We therefore revisit earlier discussions of how the axion quality problem arises in the dual formu- lation, partly motivated by recent discussions [15, 16] that argue that gravity causes new problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 9The situation resembles a gauge field Am(x, y) dimensionally reduced on a circle, so Am(x, y + L) = Am(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In this case the massless scalar would be Am(x, y) = a(x)ω(y) where ω(y) is independent of y, for which the shift symmetry a → a+c locally corresponds to a gauge transformation Am → Am +∂mζ if ∂ζ/∂y = c, but this cannot be done globally because the solution cannot satisfy ζ(y + L) = ζ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 12 – Although we confirm the important role played by multiple 3-form potentials [14] in the framing of the dual quality problem, we also show that the many 3-forms found in string vacua do not generically pose a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Problems are only caused where strongly interacting systems make instanton-like effects important and this is not the case for the many ‘elementary’ 3-forms that descend from extra dimensional vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We argue that for similar reasons 4D gravitational Chern-Simons forms also need not cause problems (such as for string vacua where the UV completion of gravity is described by weakly coupled physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To the extent that the shape of the axion potential V (a) is dual to interactions like W(X) involving the 4-form field strength X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='Hµνλρϵµνλρ, one might think that the dual version of the axion quality issue should hinge on the detailed form of UV contributions to W(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This proves not to be right, as we now argue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The central point turns on the Legendre transformation relating V (a) to W(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' in particular on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='14), that state �∂W ∂X � H=H = m a − θ˜Λ2 QCD and ∂V ∂a = mX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='21) On the scalar side the strong-CP problem is not solved unless ma = θ˜Λ2 at the minimum of V , and the quality problem is the statement that corrections to V can perturb the minimum so that this relation fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Although X always vanishes at a minimum for V , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='21) suggests that on the dual side the criterion for satisfying the strong-CP problem is that ∂W/∂X = 0 is satisfied when X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' So the quality problem seems to hinge on whether or not UV physics can introduce a linear term δW = ηX whose inclusion would modify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='21) in a way that obstructs having m a = θ˜Λ2 QCD be a solution to ∂V/∂a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Suppose, then, that one finds after integrating out the UV physics an EFT below the QCD scale of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7), but with a linear term in X whose coefficient is not proportional to the CP violating parameter θ: L1(C, B) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (Gµνλ + mCµνλ)(Gµνλ + mCµνλ) − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρ(Gµνλ + mCµνλ)Jρ − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (θ + η) ˜Λ2 QCDϵµνλρHµνλρ − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ + · · · , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='22) with two low-energy CP-violating parameters θ and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Dualizing this system as above then shows that scalar potential on the scalar side is given by a function of ma−(θ+η)˜Λ2, in which θ and η only appear as a sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The arguments of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 now show that this potential is minimized when ma − (θ + η)˜Λ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Repeating the calculation of the neutron electric dipole moment (edm) in this case – for a recent review, see for example [42] – then shows that the neutron edm also depends only on the sum θ + η and so would continue to vanish when a is evaluated at the potential’s minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Interestingly, just introducing new terms linear in Hµνλρ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) appears not to cause a quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 A second strong sector Just introducing a linear term in Hµνλρ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) does not cause a quality problem because doing so below the QCD scale is like introducing the new CP-violating parameter η only in the F ∧ F term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) above the QCD scale (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' shifting θ → θ + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This also would not cause a quality problem on the scalar side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For there to be a problem requires there to be a CP-violating contribution to V (a) that is independent of the CP-violation in the θ-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' What might this look like on the dual side?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One way to proceed is to imagine a specific type of CP-violating UV completion and ask what happens in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One such an example would – 13 – add another strongly interacting nonabelian gauge sector that also contributes to the axion anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In this case VUV (a) is obtained by integrating out the new gauge sector and this is by construction independent of the QCD-generated part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' A dual formulation of this type of system would involve a new Chern-Simons form Eµνλ for the new sector in addition to the QCD field Cµνλ, since both gauge sectors have their own Chern-Simons fields and either of these can be the field that is eaten by Bµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='22) below the QCD scale one would find the following low-energy action L1(C, E, B) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρ� θ˜Λ2 QCDHµνλρ + η˜Λ2 XKµνλρ � − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' � HµνλρHµνλρ + KµνλρKµνλρ� + · · · , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='23) where K = dE and H = dC and G = dB + mC + ˜mE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Proceeding as before we introduce a Lagrange multiplier a to enforce the G Bianchi identity and then semiclassically integrate out G, H and K to find L2(a) = −1 2(∂a)2 − Jµ∂µa − 1 2JµJµ − V (a) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='24) where defining X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHµνλρ and Y = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρKµνλρ we find V (a) = −W(X, Y ) + (ma − θ˜Λ2 QCD)X + ( ˜ma − η˜Λ2 X)Y , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='25) where W = 1 2(X2 + Y 2)+(higher powers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' At the saddle point (H, K) = (H, K) we have �∂W ∂X � Y = m a − θ˜Λ2 QCD and �∂W ∂Y � X = ˜m a − η˜Λ2 X , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='26) where the subscripts indicate what is held fixed in the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='25) implies ∂V ∂a = mX + ˜mY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='27) This does have a quality problem because the competition between the two gauge sectors drives the axion away from the minimum for which the neutron electric dipole moment vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For the simplest example – where W = 1 2(X2 + Y 2) – we can see explicitly how the shift of the global minimum of the axion potential is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='27) we learn that ∂V (a)/∂a = 0 takes place at Y = −(m/ ˜m)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='26), we obtain X = ma − ¯θ˜Λ2 QCD and Y = − �m ˜m � X = ˜ma − η˜Λ2 X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='28) Equating these two expressions for X and solving for a, we obtain amin = m¯θ˜Λ2 QCD + ˜mη˜Λ2 X m2 + ˜m2 = aQCD + ( ˜mη˜Λ2 X/m2) 1 + ( ˜m/m)2 ≃ aQCD + ˜mη˜Λ2 X m2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='29) which denotes the global minimum before introducing an extra three form gauge field by aQCD = ¯θ˜Λ2 QCD/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The approximate equality assumes m ≫ ˜m so as not to spoil the QCD axion solution the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Finally, defining the UV contribution to the effective vacuum angle by θeff := (amin − aQCD)/f where mf ≃ ˜Λ2 QCD, we obtain the constraint ¯θeff ∼ η � ˜m m � � ˜ΛX ˜ΛQCD �2 < ∼ 10−10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='30) – 14 – Although this derivation assumed the simplest form W = 1 2(X2 + Y 2), the reasoning presented here can be applied to a more complicated W(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In such a case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='27) remains unchanged while (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='28) are modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' But amin remains connected to the value for (X, Y ) that makes ∂V/∂a vanish via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Once amin is expressed in terms of aQCD, one can always infer ¯θeff as above and impose the constraint ¯θeff < 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The upshot is this: the requirement of multiple strongly coupled sectors on the dual side to generate a quality problem is much more explicit because the contribution of each sector is described by a separate 3-form potential, rather than having everything all be rolled into the same scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 Multiple fundamental 3-forms At first sight the previous section makes it sound like string theory should typically have a huge quality problem, because of the generic appearance there of multiple 3-form potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We identify the circumstances under which these potentials could cause a quality problem and argue why such a problem generically does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We also discuss how these criteria bear on a recent realization of these issues [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To start consider how the EFT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='23) above the QCD scale would be modified by the presence of many 3-form potentials CA µνλ (where A = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' , N distinguishes the different UV potentials): L1(B, A, C) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ − 1 4FµνF µν − θ 2 ϵµνλρFµνFλρ − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ηAH A µνλρϵµνλρ − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='H A µνλρHµνλρ A + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='31) where HA = dCA and G = dB + S for the QCD Chern-Simons 3-form that satisfies Ω = dS with Ω as given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To the extent that none of the new fields HA µνλρ appear in the Bianchi identity dG = Ω they do not couple to QCD or to Bµν and so play no role in the duality transformation from Bµν to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' One then arrives below the QCD scale with the lagrangian L1(C, B) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ − θ 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ˜Λ2 QCDϵµνλρHµνλρ − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='HµνλρHµνλρ − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ηAH A µνλρϵµνλρ − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='H A µνλρHµνλρ A + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='32) Dualization proceeds as before, with the introduction of the scalar a to enforce dG = Ω, and the saddle point in the integral over the 3-form potentials becomes �∂W ∂X � Y = m a − θ˜Λ2 QCD and � ∂W ∂Y A � X = −ηA , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='33) where W = 1 2X2 + 1 2Y AYA + (higher powers) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='34) and we define as before X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHµνλρ and Y A = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHA µνλρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The dual lagrangian is L2(a) = −1 2(∂a)2 − Jµ∂µa − 1 2JµJµ − V (a) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='35) where V (a) = −W(X, Y A) + (ma − θ˜Λ2 QCD)X − ηAY A = −1 2 X2 + (ma − θ˜Λ2 QCD)X + 1 2ηAη A , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='36) – 15 – and so ∂V ∂a = mX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='37) We see that X = 0 in the vacuum and this implies from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='34) that the strong-CP problem remains solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' These arguments also show that two ingredients are required for additional 3-form potentials to cause a problem: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The additional 3-form potential CA0 must contribute to the Bianchi identity for G, and so κA0 ̸= 0 in the expression dG = Ω + κAHA, where HA = dCA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The additional 3-form potential must appear linearly in W, so ηA0 ̸= 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' When both of these are satisfied then a couples to HA and leads to the competition of minima as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='26) along the lines described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The need for both of these conditions to be true is why the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='30) is proportional to both η and ˜Λ2 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The good news is that the vanishing of κA can be enforced by a gauge symmetry, since κA can only be nonzero if B transforms as B → B − κAΛA under the 3-form gauge transformations CA → CA + dΛA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' There is at least one example of a 3-form potential which we know must exist and which also contributes to the Bianchi identity dG: the gravitational Chern Simons 3-form, Sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The existence of a PQ-Lorentz-Lorentz anomaly requires this form to appear in G and so have a nonzero coefficient κg in the same way that the PQ-QCD-QCD anomaly requires the QCD Chern Simons form to appear there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' [14] argues that this is real trouble whose evasion requires model-building, such as that done in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Whether the existence of this form is a problem or not depends on whether it also satisfies item 2 above: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' whether or not it appears linearly in the lagrangian with coefficient ηg ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' How big should ηg be expected to be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because any 4-form field strength H = dC is locally a total derivative it wants to drop out of perturbative physics when it appears linearly in the action (much as does F ∧F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Consequently its appearance in a low-energy action requires some sort of nonperturbative process (like an instanton) to contribute to physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is indeed what happens for QCD for which the linear term in Ω appears with coefficient ˜Λ2 QCD ∝ M 2 e−2πb/α (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='38) with M a UV scale, b a pure number and α = g2/4π the QCD coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The tell-tale nonperturbative dependence on α is a semiclassical consequence of the topological character of � F ∧ F and � H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This suggests that for gravity a linear term in Hg should similarly be of size η ∝ M 2 e−(ML)2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='39) for a characteristic instanton length scale L and gravitational UV scale M given that (ML)−2 plays the role of the semiclassical expansion parameter (compare to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This can be extremely small within the domain of validity of semiclassical reasoning, for which ML ≫ 1 (as would presumably apply when the UV completion is weakly coupled, such as for perturbative string vacua).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Examples of three forms characterized by η in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='39) include Eguchi-Hanson instantons [39, 40] and the gravitational Chern-Simons 3-form made up of gravitational connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For the Chern-Simons 3-form ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' [14] argues that gravity indeed becomes strong in the UV, as would be required for η to be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This could well be true, but the evidence for there being a problem hinges on how convinced one is about gravitational interactions becoming strong in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 16 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3 Multiple-axion solution We close this section by remarking that having multiple axion candidates (as is often true for string vacua) can alleviate the above problem associated with multiple 3-form fields, even if the above two conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='10 This observation points to an equally general quality control mechanism on the scalar side of the duality as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To see why, we introduce a second Kalb-Ramond field Bµν to the model of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1, and supple- menting the lagrangian of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='23) with the appropriate additional kinetic term gives L1(C, E, B, B) = − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 2 · 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='GµνλGµνλ − 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' ϵµνλρGµνλJρ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='40) − 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρ� θ˜Λ2 QCDHµνλρ + η˜Λ2 XKµνλρ � − 1 2 · 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' � HµνλρHµνλρ + KµνλρKµνλρ� + · · · , where as before K = dE and H = dC and G = dB + mC + ˜mE, but now also G := dB + m⋆E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='41) This system dualizes much as before: we introduce Lagrange multipliers a and b to enforce the G and G Bianchi identities dG = mH + ˜mK and dG = m⋆K and then integrate out G, G, H and K to find L2(a) = −1 2(∂b)2 − 1 2(∂a + J)2 − V (a, b) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='42) with V (a, b) = −W(X, Y ) + (ma − θ˜Λ2 QCD)X + ( ˜ma + m⋆b − η˜Λ2 X)Y , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='43) and we define as before X = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρHµνλρ and Y = 1 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='ϵµνλρKµνλρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For the simplest example of W = 1 2(X2 + Y 2), at the saddle point (H, K) = (H, K) gives the following relation between (X, Y ) and (a, b): �∂W ∂X � Y = m a − θ˜Λ2 QCD and �∂W ∂Y � X = ˜m a + m⋆ b − η˜Λ2 X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='44) Differentiating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='43) with respect to a and b implies ∂V ∂a = mX + ˜mY, ∂V ∂b = m⋆Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='45) and so shows that all extrema of the potential satisfy X = Y = 0 (provided m, ˜m and m⋆ are nonzero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because ∂W/∂X vanishes at X = 0 it follows that the dynamics chooses amin to satisfy θ˜Λ2 QCD/m = θf through (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='44);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' the axion quality problem essentially disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' What happened?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Why does introducing another axion resolve the quality problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The crux of the mechanism lies in the difference between eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The derivative of the potential always sets a linear combination of 4-form field strengths to zero and if there are as many equations as there are fields the only solution is generically to have all 4-form field strengths vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Once this is true then the first of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='44) ensures that this solution solves the strong-CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Trouble only arises – as it did in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 – when there are fewer equations than unknowns (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' fewer axions than 3-form potentials), since then X need not vanish and eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='44) become competing conditions on the same axion variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 10The use of multiple axions to solve the quality problem is mentioned also in [43], who have different but related motivations for there being a plethora of form fields present in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 17 – A similar mechanism also exists on the scalar side of the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' If two sectors generate con- tributions to the QCD axion potential then the problem arises because these compete in the value they imply for the axion expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Introducing a second anomalous U(1) symmetry that also has anomalies with the same two sectors provides enough latitude to minimize each sector’s potential separately, thereby removing the troublesome competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For instance, suppose there was a new non-Abelian gauge sector G and suppose the usual PQ symmetry has both a QCD anomaly and an anomaly in the G sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is the kind of thing that can cause a quality problem because of the contradictory conditions the two sectors impose on the QCD axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' But also introducing another global U(1) with only a G-sector anomaly can help because there is a linear combination of the PQ symmetry and the new U(1) that is anomaly free in the G sector and the PQ mechanism then goes through using this new symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='11 4 UV completion and matter couplings Since the motivations both for considering Kalb-Ramond fields and for the absence of global symmetries come from the UV, it is useful to ask whether there are other potential surprises for axion physics having their roots in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This section examines two such examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' one each for T-type and for S-type axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For T-type axions we provide simple examples for which physical axion-matter couplings like gaff can be much smaller than the naive value 1/fb read off from the axion kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In the example shown here gauge invariant matter couplings like gaff are order 1/Mp despite fb being an ordinary particle physics scale, while anomalous gauge couplings remain order 1/fb in size (if they exist at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For S-type axions we show that the corresponding physical couplings indeed are of order 1/fa and we identify the UV physics to which couplings of size E/fa match at energies E > ∼ fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We also show how S-type axions can be examples of weak/strong duality, and that it is the Kalb-Ramond side of the duality that is usually weakly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Weak/strong coupling interchange due to duality could be relevant to applications for which the effects of scalar axions are explored using semiclassical reasoning, and if so would provide a further motivation for taking the Kalb-Ramond formulation as primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1 Extra-dimensional UV completion To this end suppose that both Kalb-Ramond field and the standard model arise within an extra- dimensional model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For concreteness’ sake we take the higher-dimensional kinetic term for the 2-form field and the Einstein-Hilbert part of the action to be12 Skin = −1 2 M 2+d � d4x ddy � −˜g(D) � �R + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' e−λφ GMNPG MNP � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1) where there are D = 4+d spacetime dimensions and M is a UV scale – the higher-dimensional Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' �R here denotes the Ricci scalar and ˜g(D) is the determinant of the full D-dimensional metric ˜gMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' As above H = dB+· · · is the Kalb-Ramond field strength and φ is the extra-dimensional dilaton that often arises within the higher-dimensional gravity supermultiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The parameter λ depends on 11Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' [16] uses a special case of this general mechanism by introducing an extra U(1) symmetry in the leptonic sector to resolve the problem raised by the assumption that gravity is strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 12For simplicity we ignore extra-dimensional warping in this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We also do not canonically normalize BMN, which here is taken to be dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 18 – higher-dimensional details, with (d, λ) = (2, 2) for chiral 6D supergravity [44], (d, λ) = (6, 1) for Neveu- Schwarz 2-forms in 10D supergravity and (d, λ = −1) for Ramond 2-forms in 10D supergravity [45] (for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The derivation of this type of lagrangian as the low-energy limit of a string vacuum usually relies on two approximations: the low-energy approximation (or α′ expansion) where energies are well below the string scale E ≪ Ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' and the weak string coupling approximation, which involves expanding in powers of eφ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For simplicity we restrict ourselves to this limit as well, and specialize to the simplest case (d, λ) = (2, 2) corresponding to 6D chiral supergravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Dimensional reduction to 4D proceeds by integrating out the two extra dimensions and putting the 4D Einstein-Hilbert term into standard form (4D Einstein frame) by appropriately rescaling the 4D part of the metric ˜gµν = �V2⋆ V2 � gµν = 1 V2 � M 2 p M 2 � gµν where Vd := M d � ddy � ˜g(d) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2) is the dimensionless extra-dimensional volume and the subscript ‘⋆’ on a field denotes its present-day value13 and the 4D Planck massis is defined by M 2 p = V2⋆ M 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='3) S-type axion The kinetic term for bµν in 4D Einstein frame that is obtained by dimensional reduction is Lkin = − 1 12 M 2V2 � −˜g(4) e−2φ˜gµν˜gβρ˜gξζ HµβξHνρζ = − M 4 12M 2p √−g e−2φ V2 2 hµνβhµνβ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) where hµνλ = ∂µbνλ + (cyclic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This last form can be written in terms of a scalar by dualizing as in earlier sections, imposing the Bianchi identity14 dh = Ω/M 2, leading to the dual result Ldual = −√−g � M 2 pe2φ V2 2 ∂µa ∂µa + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' a ϵµνβρΩµνβρ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5) which suggests its decay constant can be written fa = (Mp/Vd⋆) eφ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Two things are noteworthy here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' First, notice that the volume dependence means that fa can be very much smaller than Planckian size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In the extreme case of two large extra dimensions (and working in the weak-coupling regime for which eφ is moderately small) the size of the extra dimensions can be as large as MRKK < ∼ 1014 and so V2 ∼ (MRKK)2 < ∼ 1028 can be enormous (potentially allowing fa ≪ Mp to be as small as eV energies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Second, notice that although (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4) has large coefficients when e2φ ≪ 1, the same is not true of the kinetic term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This reflects how Kalb-Ramond/axion duality is a weak-strong coupling duality from the point of view of the string coupling gs ∼ eφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To the extent that semiclassical expansions rely on the leading action being proportional to the inverse of a small coupling15 – L0 = L0/g2 s – semiclassical methods should fail for the scalar representation but hold for its dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 13The factor of V2⋆ ensures the rescaling is trivial at present, as required to not change present-day units of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 14The factors of M here are chosen so that Ω has dimension (mass)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 15When this is true then powers of g2 s and powers of ℏ are equivalent when evaluating a path integral over eiS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 19 – T-type axion For T-type axions we use Bmn(x, y) = b(x) ωmn(y) where in six dimensions the harmonic form can be taken to be proportional to the extra-dimensional volume form εmn(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Typically ωmn satisfies a quantization condition that states the integral of ωmn over the two extra dimensions � C ω is a pure number, proportional to an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because this result is volume independent it follows that ωmn = V−1 2 εmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The kinetic term for the T-type scalar b obtained in this way is therefore proportional to Lkin = −1 2 M 2V2 � −˜g(4) e−2φ˜gµν∂µb ∂νb V−2 2 = −√−g M 2 pe−2φV−2 2 gµν∂µb ∂νb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6) Notice that the kinetic term, both here and in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5), takes the form Lkin = −1 2 √−g M 2 p �(∂b)2 τ 2 + (∂a)2 σ2 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='7) with τ = V2 eφ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='6), and σ = V2 e−φ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='2 Coupling strengths What matters for phenomenology is the couplings of the fields b and a to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is controlled by the size of F for axion couplings of the form Lax = −1 2 ∂µa ∂µa − 1 F ∂µaJµ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='8) where a is the canonically normalized axion field and Jµ is a matter current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' F−1 = gaff is the axion-fermion current if Jµ is built from fermion bilinears and F−1 ≃ gagg or gaγγ if Jµ is the Hodge dual of the QCD or QED Chern-Simons 3-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For concreteness’ sake we evaluate the size of this coupling in the perturbative semiclassical regime where V2 is large and the UV physics is weakly coupled (and so eφ small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In this limit we have fb > ∼ Mp/V2 > ∼ fa and both are much smaller than Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In both cases we will see that F can (but need not) be simply given by the corresponding decay constant fa or fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In higher dimensional constructions very often ordinary matter is localized on a space-filling brane, Σ, within the extra dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Σ could be a four-dimensional 3-brane or a higher-dimensional p- brane with 3 ≤ p ≤ 3 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' If p > 3 then the extra-dimensional part of the brane typically wraps some topological cycle within the extra dimensions, and if this were a two-cycle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' if p = 5) it would also have an associated harmonic 2-form ωmn(y) required to ensure that T-type axions appear in the low-energy 4D theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We here explore the simplest case p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' S-type axion A generally covariant low-dimension interaction between HMNP and matter fields living on the brane, that is linear in BMN is16 Sint = −ˆc � Σ e−βφH ∧ J = − c 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' � d4x√−g e−βφ ϵµνλρhµνλJρ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9) where Jρ is a current built from brane-localized matter fields and β is a parameter – like λ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='1) – that is predicted by any specific extra-dimensional UV completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The matter current Jρ has dimension 16The first equality shows that this interaction is independent of the metric, and this can also be seen after the second equality from the observation that ϵµνλρ = ±√−g and so ϵµνλρ = ±(−g)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 20 – (mass)3 – making the coupling parameters ˆc and c dimensionless – and so could be a fermion bilinear or the Hodge dual of a gauge boson Chern-Simons term (though for the Chern Simons term gauge invariance would require β = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because this term is covariant without use of the metric it does not acquire factors of V2 or Mp/M when going to 4D Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The dual effective theory for a is then found by adding (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9) to the kinetic term (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='4), imposing the Bianchi identity dG = Ω/M 2 and integrating out hµνλ, modifying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='5) to Ldual = −√−g � M 2 pe2φ 2V2 2 Dµa Dµa + 1 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' a ϵµνβρΩµνβρ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='10) where Dµa := ∂µa + c M 2p e−(β+2)φ V2 2 Jµ = ∂µa + c f 2a e−βφ Jµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='11) As before we use the kinetic term to identify fa = (Mp/V2⋆) eφ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Using M 2 p = M 2V2⋆ with V2⋆ = (MRKK)2 for a Kaluza-Klein length scale RKK, this implies fa ∼ MV−1/2 2⋆ eφ⋆ ∼ (1/RKK) eφ⋆, and so fa ∼ mKK ∼ 1/RKK when eφ⋆ is not that much smaller than order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The physical coupling that comes from comparing the kinetic and ∂µa Jµ term to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='8) is gagg ∼ gaff = 1 Faff ∼ cV2⋆e−(β+1)φ⋆ Mp ∼ c e−βφ⋆ fa for couplings to J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='12) In the special case where Jµ is the Hodge dual of a gauge-field Chern Simons term, gauge invariance also requires we take β = 0, and once this is done the coupling in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='12) agrees (up to numerical factors) with the physical coupling to Ω implied by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This coupling becomes strong when E ∼ fa, which we’ve seen is of order the Kaluza-Klein scale in the special case of two extra dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' T-type axion The lowest-dimension generally covariant and gauge invariant interaction that couples HMNP to matter localized on a space-filling 3-brane and that is linear in the components Hµmn has the form Sint = � Σ e−2φ ⋆H ∧ J ∋ M 2 p M 2 � d4x√−g e−2φ V2 2 gµν∂µb(x)Jν(x) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) where ⋆H denotes the 6D Hodge dual and we choose the φ coupling to be the same as the kinetic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The kinetic and interactions terms combine to give the effective action (in 4D Einstein frame) Seff = � d4x √−g M 2 p τ 2 � (∂b)2 + ∂µbJµ M 2 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='14) where τ := V2 eφ as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Inspection of the kinetic term identifies the decay constant as fb ≃ Mp/τ⋆ with τ⋆ ∝ V2⋆ denoting the present value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Because M 2 p ≃ M 2V2⋆ we see that fb ∼ MV−1/2 2⋆ ∼ R−1 KK is of order the KK scale in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Canonically normalizing by rescaling b = Mp b/τ⋆ —- then produces a lagrangian of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='8) with F ∼ M 2τ⋆ Mp ∼ Mp ≫ fb ∼ Mp τ⋆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='15) As is typical for KK modes the field b ∈ Bmn couples with gravitational strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Notice that the ratio F/fb ∝ V2⋆ can be enormous, since V2⋆ can be as large as 1028 in the extreme case of two large – 21 – eV-scale dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' In this case gauge invariance precludes choosing J to be the dual of the Chern Simons form of a brane localized gauge sector, even in the absence of any φ-dependence in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It is not that surprising to have a breakdown of 4D EFT methods at the KK scale, but the above discussion shows there is a difference between what happens at this scale for T- and S-type axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For T-type axions the coupling to matter is order 1/Mp and this remains true above the KK scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The breakdown of the 4D EFT is about the appearance of a multitude of new KK modes, all of which couple with gravitational strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' But the S-type axion’s coupling to matter is proportional to E/fa and so actually grows to become order unity at the KK scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' what does this order unity coupling match to in the UV theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It matches to a dimensionless extra-dimensional coupling in the UV theory: either to the coupling c appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='9) or to the coupling gcs of BMN to the Chern-Simons term SMNP that is implied by the field strength G = dB +gcsS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Although gcs is order 1/M 2 when BMN is dimensionless (as above), it is dimensionless once BMN is canonically normalized in six dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Since gauge invariance prevents the coupling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='13) from containing a coupling between T-type axions and a gauge sector localized on a 3-brane, one can ask whether such couplings are more generally forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The answer to this is ‘no’ if we allow ourselves to consider gauge sectors localized on higher- dimensional branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For instance for a six-dimensional 5-brane Σ6 they can arise from an interaction of the form Sint,g = cM 2 � Σ6 B ∧ F ∧ F ∝ c � d4x√−g b ϵµνλρFµνFλρ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='16) where the explicit factor of V2 coming from the integration over the additional two dimensions cancels the normalization of the harmonic form ωmn ∝ V−1 2 εmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For a canonically normalized scalar this would imply gaγγ ∼ 1/fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The upshot is this: the model-dependent T-type axions can couple surprisingly weakly to non- gauge matter compared to the scale set by their decay constant: 1/F ∼ 1/Mp ≪ 1/fb ∼ RKK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' By contrast, the model-independent S-type axion couples to matter with strength 1/F ∼ 1/fa ∼ 1/fb and the same is true of T-type couplings to gauge fields on higher-dimensional branes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' From the point of view of the underlying string coupling eφ the duality that maps bµν to a is also a strong/weak coupling duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 5 Conclusions Axions (or ALPs) are often motivated by appealing to string theory, which seems to provide them with abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' But string theory also provides strong concrete evidence for the assertion that exact global symmetries cannot survive contact with quantum gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' the observation that underlies the UV quality problem for attempts to solve the strong-CP problem using a global PQ symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We here reconsider some of the implications that follow from the observation that axions arise as antisymmetric tensor fields in higher dimensions and that Standard Model fields usually live in localised objects like D-branes within the extra dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Axions arise in two general types in this way: the model independent S-type axion originating from a two-form field in 4-dimensions after compactification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' and the model dependent T-type axion such as arises as a Kaluza-Klein mode for an extra-dimensional tensor field (of which we focus for simplicity on two-form potentials in two extra dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='17 These allow for a rich structure of axion phenomenology and each type of axion can be adapted to realize the PQ solution to the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It has been known for a while that UV effects 17Axions may come from other forms such as three or four forms in ten dimensions depending of which type of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 22 – can affect the original PQ proposal by generating effective interactions that violate the global PQ symmetry and modify the prediction for the axion mass: the axion quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We revisit how this problem arises for the two types of axion using the UV tools at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We find that for T-type axions the quality problem resembles the form originally studied, since the UV theory directly provides a pseudoscalar field once compactified to four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Our framework differs from early versions of the quality problem that imagine the PQ symmetry to be linearly realized by a complex scalar at energies E > f, but generally agree with estimates based on the contributions due to wormholes or gravitational instantons below a compactification scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' To the extent that these contributions are exponentially suppressed their constraints are mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The S-type case is more interesting since both the strong CP and axion quality problems must first be reformulated in terms of the two-form field and its field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The PQ mechanism involves giving a mass to the axion and so on the dual side involves the ‘eating’ of a 3-form potential along the lines proposed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The required 3-form potential is generated by the QCD sector itself as a non-propagating topological field in the EFT below the QCD scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' As applied to QCD our re-analysis broadly agrees with that of [14] in concluding that the quality problem gets recast as an issue that arises when there are multiple 3-form fields present in the low-energy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This might be imagined to be a problem for string theory, for which 3-form potentials are as ubiquitous as axions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Prompted by recent discussions of this problem [15, 16] we formulate the two properties which new 3-form fields must have if they are to threaten the PQ solution to the strong CP problem, arguing why string-generated 3-form fields are not generically a problem, largely because these fields need not couple to QCD (in string theory it depends on how bulk fields couple to brane fields and usually only one couples to QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The gravitational Chern-Simons term does couple to QCD but whether or not it sparks a new strong CP problem depends on whether or not gravity is strongly coupled in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The discussions of [14, 16] assume that it is, but we argue that if it is not (such as if the UV completion is a weakly coupled string vacuum) then the estimates for the size of the problem are again exponentially suppressed and so would not pose a quality problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Finally we also explore other UV implications for axion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' We found that depending on the brane configuration hosting the Standard Model, extra dimensions can dramatically suppress physical couplings between the axion and Standard Model sector relative to the axion decay constant appearing in the axion kinetic term, especially if the volume of the extra dimensions is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This is possible for T-type axions but in the the examples examined does so only for non-gauge couplings (making this observation more pertinent for ALPs, whose properties would tend to be ‘fermiophobic’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' For S-type fields both kinds of couplings have similar size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content='18 For this case though, we argue that the duality relating the 2-form to the axion field swaps weak and strong couplings, and suggests a semiclassical description of 2-form response need not correspond to the usual semiclassical description of a scalar axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' This again motivates better exploring the 2-form side of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' It is an old argument that UV information can have important implications for low-energy natu- ralness questions such as the strong CP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The observation that this could be informative in situations where the questions are solved using features like global symmetries that apparently should not be present at very high energies has sparked a revival of studies of generalised and non-invertible symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Many of these ideas resonate well with string-motivated constructions, such as those we explore here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' 18For supersymmetric realizations this can be seen because both the K¨ahler potential and gauge kinetic function depend directly on the S field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' – 23 – Acknowledgements We thank Philippe Brax and Junwu Huang for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' CB’s research was partially supported by funds from the Natural Sciences and Engineering Research Council (NSERC) of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfqvix/content/2301.00549v1.pdf'} +page_content=' The work of FQ has been partially supported by STFC consolidated grants ST/P000681/1, ST/T000694/1.' metadata={'source': 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Filip Mari´c*, ** Ivan Markovi´c ∗ Ivan Petrovi´c ∗ +∗ University of Zagreb, Faculty of Electrical Engineering and +Computing, Laboratory for Autonomous Systems and Mobile Robotics +(e-mail: name.surname@fer.hr). +∗∗ University of Toronto, Space and Terrestrial Autonomous Robotic +Systems Laboratory (e-mail: name.surname@mail.utoronto.ca) +Abstract: Autonomous manipulation systems operating in domains where human intervention +is difficult or impossible (e.g., underwater, extraterrestrial or hazardous environments) require +a high degree of robustness to sensing and communication failures. Crucially, motion planning +and control algorithms require a stream of accurate joint angle data provided by joint encoders, +the failure of which may result in an unrecoverable loss of functionality. In this paper, we +present a novel method for retrieving the joint angles of a robot manipulator using only a +single RGB image of its current configuration, opening up an avenue for recovering system +functionality when conventional proprioceptive sensing is unavailable. Our approach, based +on a distance-geometric representation of the configuration space, exploits the knowledge of +a robot’s kinematic model with the goal of training a shallow neural network that performs +a 2D-to-3D regression of distances associated with detected structural keypoints. It is shown +that the resulting Euclidean distance matrix uniquely corresponds to the observed configuration, +where joint angles can be recovered via multidimensional scaling and a simple inverse kinematics +procedure. We evaluate the performance of our approach on real RGB images of a Franka +Emika Panda manipulator, showing that the proposed method is efficient and exhibits solid +generalization ability. Furthermore, we show that our method can be easily combined with a +dense refinement technique to obtain superior results. +Keywords: Manipulation; Mechatronic Systems; Robotics; Joint angle estimation +1. INTRODUCTION +Autonomous manipulation systems are ideal for perform- +ing various tasks in environments where human presence is +limited, such as underwater or orbital laboratories, as well +as hazardous (e.g., radioactive, toxic) environments. In +addition to effective planning and control algorithms, these +systems require a high degree of robustness to sensing +and communication failures, as a timely intervention by +humans may be impossible. We propose a method for +recovering the joint angles (i.e., the configuration) of an +articulated robotic manipulator using only a single RGB +image, providing an alternative source of proprioceptive +data that can be used when data from joint encoders is +unavailable. This is a challenging task for multiple reasons: +fundamentally different robot configurations may result +in similar images and certain configurations may have +diminished observability due to physical occlusion by other +parts of the robot. However, even a rough estimate of the +joint configuration may enable the use of simple control +methods to steer the robot to an approximate desired +state, enabling the planning and execution of critical re- +covery protocols (Ortenzi et al. (2018)). +⋆ This work has been supported by the European Regional Devel- +opment Fund under the grant KK.01.1.1.01.0009 (DATACROSS). +The problem of recovering a robot configuration from +spatial constraints such as gripper pose is known as inverse +kinematics and features a variety of well known solutions +(Lynch and Park (2017)). However, previous work has +also explored instances of this problem where spatial con- +straints may result from a variety of sensing modalities, +such as depth or RGB images. Widmaier et al. (2016) +use synthetic depth images to train semantic classifiers for +direct joint angle regression in order to estimate the robot +arm pose. Bohg et al. (2014) also use depth images to train +a random forest classifier for pixel-wise part classification, +while using joint encoder readings to initialize an incre- +mental update scheme. Liu et al. (2020) use a model-free +approach and estimate the joint angles in an unsupervised +fashion by using pretrained deep models for optical flow +and monocular depth estimation. Conversely, our method +aims to recover joint configuration using only a single RGB +image, which necessitates first finding the appropriate set +of spatial constraints using 2D-to-3D regression. +The constraints resulting from regressed 3D keypoints may +be under-determined and therefore correspond to multi- +ple configurations. Instead, we use a distance-geometric +model that integrates structural data (e.g., link lengths) +to remove ambiguity. Distance geometry is highly relevant +for applications such as molecular conformation, sensor +arXiv:2301.02051v1 [cs.RO] 5 Jan 2023 + +network localization (SNL) and statics (Liberti et al. +(2014)). For instance, SNL is commonly framed as an Eu- +clidean distance matrix (EDM) completion problem (Ding +et al. (2008))(Dokmanic et al. (2015)) and tackled through +semidefinite programming (SDP) (Biswas et al. (2006)), +(Alfakih et al. (1999)). Mari´c et al. (2022) formalize the +equivalence of distance-based inverse kinematics and the +distance geometry problem for a large class of articulated +robots. Furthermore, Moreno-Noguer (2017) tackles the +problem of 3D human pose estimation from a single RGB +image and demonstrates that representing human poses +with EDMs instead of Cartesian coordinates results in +more precise and less ambiguous pose estimates. Our work +is partly inspired by these observations, and we show that +the problem of recovering robot’s joint angles is in general +highly related to distance geometry. +To the best of our knowledge, the only similar approaches +to ours are that of Zuo et al. (2019) and Labb´e et al. +(2021), in the sense that only a single RGB image is +used as an input for joint angle estimation. Zuo et al. +(2019) train a joint keypoint detector to recognize a +predefined, specifically chosen set of 17 keypoints in the +image displaying a 4-DoF toy robot. These keypoints are +then fed to a nonlinear non-convex 2D-to-3D optimization +algorithm in order to recover the 6D pose together with +joint angles. Although our distance-geometric method is +also keypoint-based, it only requires a number of keypoints +equal to the robot’s DoF, placed in a robot-invariant +manner. Furthermore, the optimization proposed in Zuo +et al. (2019) is complex, which diminishes its potential to +improve from scaling the model and data size. On the other +hand, Labb´e et al. (2021) use a dense, rendering-based +deep iterative matching approach (Li et al. (2018)) to +jointly learn the 6D camera-to-robot pose and joint angle +updates. Although primarily concerned with estimating +the 6D pose, Labb´e et al. (2021) also demonstrate that +joint angles, if unknown, can be accurately reconstructed, +at least with a sufficient number of iterations. We tested +a hybrid approach, i.e combining our method with the +deep refinement proposed in Labb´e et al. (2021). Overall, +the dense approach seems to generally outperform sparse +approaches in terms of accuracy due to incorporating +global information, at the cost of high computation time. +In this paper, we propose a novel method for recovering +the joint angles of an articulated robotic manipulator using +only a single RGB image, based on a distance-geometric +representation of the configuration space and the knowl- +edge of a robot’s kinematic model. Instead of training a +single large model to directly predict the solution, our +method divides the problem into a set of smaller sub- +problems in a theoretically justified manner. First, state- +of-the-art keypoint detectors (Lee et al. (2020)) are used +to detect joint keypoints in the image corresponding to +the robot’s joints, which are insufficient for configuration +recovery on their own. Then, our method takes the full +set of inter-point distances and uses a learned 2D-to- +3D regression to produce an EDM corresponding to the +associated 3D keypoints. Following the approach in (Mari´c +et al. (2022)) this EDM is extended to include distances +between auxiliary points determined by the robot’s struc- +ture, removing ambiguity related to joint angle recovery. +Given a complete EDM, joint angles can be computed +using parameter-free transformations; classical multidi- +mensional scaling (MDS) and kinematic transformations. +The former maps an EDM to a geometrically centered set +of 3D points, while the latter calculates the joint angles +based on these points, forming a fully-differentiable IK +layer that supports batching and can be ran on a GPU. +In addition to generating the complete EDM, ground- +truth joint angles are used to compute the loss in the +configuration space. Finally, our method is evaluated on +a large set of real images displaying a 7-DoF robot arm in +various configurations. We opt for a shallow architecture +for all our experiments since the primary goal of this work +is developing a framework, while a more thorough test of +its potential is left for future work. As the results show, our +method exhibits solid generalization ability, while being +simple and computationally efficient. +2. METHODOLOGICAL BACKGROUND +Euclidean distance geometry is an important tool in sev- +eral applications whose aim is to reconstruct a complete +set of distances (or points that realize them) in Euclidean +space, given an incomplete set of distances as an input. +In addition to a small subset of distance geometry which +relates to EDMs, this section describes a kinematics proce- +dure responsible for generating a set of 3D points sufficient +for recovering the robot’s configuration as well as distance +constraints emerging from kinematics. +2.1 Euclidean distance matrices +Let P ∈ Rn×d denote a matrix representing a set of +n points in a d-dimensional Euclidean space. Then, the +pairwise distances du,v between points can be determined +via the Euclidean norm: +du,v = ∥pu − pv∥. +(1) +For the sake of notation simplicity, EDMs and individual +distances are assumed to be squared in the remaining of +the paper. Expansion of (1) reveals that EDM is a function +of the Gram matrix G = PP⊤: +edm(G) = diag(G)1⊤ + 1 diag(G)⊤ − 2G, +(2) +with diag(G) representing the diagonal entries of G in the +form of a column vector and 1 stands for column vector +filled with ones. +Equation (2) establishes a one-way connection between an +EDM and a Gram matrix. Consider an inverse problem, +i.e. recovering the set of points that generated the distance +matrix D. Let D be a squared EDM. Then, a Gram matrix +that satisfies (2) can be recovered via +G = −1 +2JDJ, +(3) +where +J = I − 1 +N 11T +(4) +denotes a geometric centering matrix. Moreover, G is a +real symmetric matrix, hence it can be factored into a +canonical form via eigenvalue decomposition: + +G = UΛUT +(5) +where Λ += +diag (λ0, λ1, . . . , λn−1) contains the non- +negative eigenvalues λi and U is an orthonormal matrix. +Now, assuming the eigenvalues are sorted in the descend- +ing order, the point set ˆP ∈ Rn×d can finally be recovered +by taking: +ˆP⊤ = +� +diag +�� +λ0, +� +λ1, . . . , +� +λd−1 +� +, 0d×N−d +� +U⊤. (6) +Computing the collection of points ˆP from a distance +matrix D using (3), (4), (5) and (6) is also known as +classical multidimensional scaling (cMDS). Note how in +(6), all but the d largest eigenvalues are discarded. Assum- +ing that G is generated by a d-dimensional set of points, +all but the d largest eigenvalues will be zeros. If this is +not the case, we can assume the presence of noise which +is handled by the truncation (Dokmanic et al. (2015)). +Additionally, plugging the estimated ˆP in (2) yields an +EDM that equals D. However, distances are preserved +under rigid transformations, thus ˆP and the original P are +not generally equal. Absolute position and orientation of +the point set can be recovered through Procrustes analysis +(Sch¨onemann (1966)), assuming that a set of at least d +fixed points (i.e. anchors) is known beforehand. Then, +a rigid transformation that aligns the anchors in P to +their corresponding points in ˆP can be found. Finally, the +original set of points P can be recovered by applying the +obtained rigid transformation to all the points in ˆP. +qi +j +i +i +j +j +q +i +qi +j +q +j +q +i,j +Fig. 1. Visualization of neighbouring revolute joints con- +nected with rigid links. Dashed lines represent the +rotation axis of each joint, rotated by a an angle θ. +Auxiliary points q are placed along these axis. Points +p correspond to the position of coordinate frames +defined in the center of their respective joints (placed +below for better visibility). +2.2 Distance-based kinematics +Consider an n-DoF robotic manipulator comprised of +single-axis revolute joints, forming a kinematic chain. +The procedure introduced in Mari´c et al. (2022) defines +a sparse set of points whose positions are sufficient for +recovering the full set of joint angles determining the +robot’s configuration. As shown in Figure 1, a set of points +pi centered at the joint coordinate frames are introduced, +which we associate with the keypoints detected in the +RGB image of the robot by our network. Then, ”virtual” +auxiliary points qi are placed at a unit distance along the +joints’ rotation axes ˜z using joints’ orientation Ri +qi = pi + Ri˜z, +(7) +adding information on the relative orientation of neigh- +bouring joints required for joint angle recovery. Finally, +the model is completed with the addition of three points, +x, y and z, corresponding to the root coordinate frame, +defined by distances +∥p0 − x∥ = ∥p0 − y∥ = ∥p0 − z∥ = 1, +∥x − y∥ = ∥x − z∥ = ∥y − z∥ = +√ +2. +(8) +The proof in Mari´c et al. (2022) shows that the distances +between these points are sufficient for recovering the full +set of joint angles for a large variety of manipulator +structures. +Our training data is generated from ground truth joint +angle vectors Θ ∈ C corresponding to the robot observed +in the image I, where C ⊆ Rn represents the configu- +ration space. This allows us to simply generate the full +distance-geometric robot description. Each data point is +constructed by sequentially taking the position pi and +orientation Ri of a parent joint i, as well as its joint angle +θi, giving the position and orientation of the child joint j +as +Rj = RiRz (θi) Ri,j, +pj = pi + RiRz (θi) pi,j. +(9) +The relative positions and orientations pi,j and Ri,j of +neighbouring joints, are completely defined by the robot’s +structure. Thus, the distances between the set of the four +neighbouring points can also be defined from structural +parameters as +∥pi − qi∥ = ∥pj − qj∥ = 1 +∥pi − pj∥ = ∥pi,j∥ +∥pi − qj∥ = ∥pi,j + Ri,j˜z∥ +∥qi − pj∥ = ∥pi,j − ˜z∥ +∥qi − qj∥ = ∥pi,j − ˜z + Ri,j˜z∥ , +(10) +indicating that they are invariant to changes in joint +angles. Conversely, the distances corresponding to the +remaining non-neighbouring pairs exist as a function of +the configuration Θ. Using the constructive procedure in +(7) and (9), together with distance constraints stated in +(8) and (10), we construct a complete EDM D ∈ Rˆn׈n +that uniquely (up to a rigid transformation) represents +a set of 3D points associated with the configuration Θ +observed in the RGB image. Concretely, for an n-DoF +articulated robot comprised of single-axis revolute joints +- one auxiliary point per joint to determine the axis of +rotation, and two additional points are required to fully +define the base coordinate frame. Finally, a single data +sample used for training the model can thus be formalized +as D := (Θ, D). +3. METHOD +This section is dedicated to describing the system dis- +played in Figure 2 as a pipeline through which the pro- +posed method is implemented. First, given the input RGB +image and 2D joint keypoint detections, one needs to + +cMDS +IK +2D EDM +3D EDM +EDM Regression +Fig. 2. System overview. In the image space, EDM is constructed from detected 2D joint keypoints and fed to a 2D- +to-3D EDM regression network (black), which outputs the complete EDM in the 3D space. The cMDS layer maps +the complete EDM to a set of geometrically centered points, from which an IK layer computes the joint angles. +Estimated joint angles are depicted as a 2D projection of the robot rendered in respective configuration. +bridge the gap between 2D and 3D information. Theoreti- +cally, infinitely many sets of ˆn points which are equal up to +a rigid transformation, and generated as described in Sec- +tion 2.2, can map to a single feasible robot configuration +Θ. As inter-point distances remain identical irrespective of +such point set transformation, we associate a configuration +Θ with a single unique EDM. Therefore, instead of di- +rectly predicting the points, we predict the corresponding +distance matrix. We frame the distance matrix regression +problem as learning a mapping ζ : Rn×n ++ +�→ Rˆn׈n ++ +, where +R+ = {x ∈ R : x ≥ 0}, n equals the number of joints, and +ˆn = 2n + 2. Note that we don’t use auxiliary points in the +image space, only keypoints directly corresponding to the +robot’s joints. The mapping is implemented as a shallow +feedforward neural network that takes an EDM computed +from 2D joint keypoints as input and outputs a complete +EDM which is treated as if it was generated by a set of 3D +points comprised of joint correspondences and auxiliary +points. We define a distance-based loss as: +Ld = ∥ ˆD − D∥F , +(11) +where ∥ · ∥F denotes Frobenius norm, while ˆD and D +are the predicted and ground-truth EDMs, respectively. +Due to the fact that EDMs are symmetric matrices, +the actual implementation works with upper-triangular +elements from which the full EDM is computed afterwards. +This amounts to using n(n − 1)/2 elements instead of +n2 with no loss of information. The ground-truth EDMs +are computed from ground-truth configurations Θ via a +function f : Θn �→ Dˆn׈n based on equations described in +Section 2.2. Figure 3 demonstrates that EDM regression +and joint angle recovery are highly related, by depicting +a mean absolute joint angle error as a function of mean +absolute EDM error, using the Kinect dataset (unseen +during the learning process). Note that the model used +for this figure is trained for EDM regression, i.e. using +the loss function (11), while the joint angles are only +computed during inference via cMDS and IK layers. We +use our library 1 for most of the distance-geometry and +kinematics-related computation. +After the complete EDM D ∈ Rˆn׈n is estimated, the +cMDS layer (a set of fixed, differentiable transformations +described in Section 2) is used to obtain the geometrically +centered set of points which generate the respective EDM. +This mapping can be formally defined as Ω : Rˆn׈n ++ +�→ +Pˆn×d. The set of points is then fed to an IK layer which +computes the joint angles Θ. Note that what we refer to +1 https://github.com/utiasSTARS/graphIK +as an IK layer is not an inverse kinematics solver; it is a +sequence of differentiable kinematics transformations that, +given the estimated set of points, compute the joint angles +by computing the respective coordinate frame positions +and orientations together with the axis of rotation for +each joint (Mari´c et al. (2022)). The default configuration +(corresponding to zero joint angles) of the robot is known +from its model (an Unified Robotics Description Format +file), hence the joint angles can be computed. This allows +us to define a loss in the configuration space between the +predicted and ground-truth configuration: +Lc = | ˆΘ − Θ|, +(12) +with |·| denoting the L1 norm. We use a linear combination +of the two losses as a final loss to train the model: +L = Lc + λLd, +(13) +with λ set to 0.5. Note that Lc causes the gradients to +be propagated through IK and cMDS layers, while Ld +is applied directly at the output of the EDM regression +network and serves to provide the model with additional +information. +The EDM regression network is comprised of two Linear → +BatchNorm → ReLU → Dropout layers, and an output +Linear → ReLU layer. The layers consist of 512−512−120 +neurons (∼ 300k parameters) respectively, and the output +size is determined by ˆn(ˆn − 1)/2 (upper triangular EDM), +where ˆn = 2n + 2 and n = 7 for the Franka Emika Panda +robot. Among other necessary conditons, EDM elements +must be positive, which is enforced by the last ReLU +activation. We use Adam (Kingma and Ba (2014)) for +optimization, with initial learning rate α = 1e − 3, linear +warmup (Ma and Yarats (2021)) over 2k iterations, and +a batch size of 64 EDMs. The training is carried out for +100 epochs and learning rate is reduced by a factor of 2 +after 50 epochs. The model is initialized as proposed by +(He et al. (2015)). +4. EXPERIMENTS +All our experiments were conducted on images of a 7-DoF +Franka Emika Panda robot observed in various configu- +rations, using three different datasets. We report mean +absolute error as a joint angle error metric. Training and +evaluation were carried out on a single NVIDIA RTX +A5000. Using the setup described in Section 3, a model +requires ∼3 hours to train. Regarding the running time, +when using a batch size of 64, our system runs at 1.6ms +per data sample, and takes up 1.5GB of GPU memory for +inference. In reality, it would be limited by the running +time of the chosen keypoint detector. + +Fig. 3. Joint angles vs EDMs, using mean absolute error +(MAE). The model is trained for EDM regression, +while cMDS and IK layers are only used at test- +time to compute the joint angles. The errors are +highly correlated; Pearson correlation coefficient is +0.94. Kinect dataset (unseen during training) is used +for this plot. +4.1 Dataset +For our experiments, we use the DREAM dataset intro- +duced by Lee et al. (2020) in their recently proposed +state-of-the-art method for single-view camera-to-robot +pose estimation. The dataset is comprised of real and +synthetic parts. The synthetic part is photorealistic and +generated with domain randomization. We focus on the +real part of the dataset, which is split into 4 different +Panda-3CAM datasets - Realsense, Azure, Kinect, and +Orb which contain 5944, 6394, 4966 and 32315 samples, +respectively. Each of these datasets is comprised of RGB +images of the 7-DoF Franka Emika Panda robotic ma- +nipulator, captured using different cameras with different +intrinsic parameters. The robot is observed in various +configurations, including images with joint occlusions and +even out-of-view joints. Besides RGB images, the datasets +contain 2D joint keypoint annotations together with their +3D correspondences and ground-truth robot configuration. +The camera-to-robot pose is different and fixed for each +dataset, except for Orb which is captured from 27 different +viewpoints. For all our experiments we used 8x subsampled +version of the Orb dataset for training, which we refer to as +Orb in the rest of the section. We automatically adjusted +all the 2D joint keypoint annotations in all the datasets so +that they match the Panda’s coordinate frame definitions +exactly. +4.2 Results +The results of applying our method on the Panda-3CAM +datasets are displayed in Table 1. For evaluating on Kinect +and Azure datasets the model is trained on Realsense +and Orb, while for Realsense evaluation we trained it on +the Kinect and Orb datasets. The results show that our +method gives solid joint angle approximations on unseen +data, while being simple and computationally efficient. +The top 50% predictions mostly correspond to images +which display configurations relatively close to coplanar +with respect to the image plane, for a given robot-camera +pose. However, the datasets also contain images with +Table 1. Results on the Panda-3CAM datasets. +The mean absolute joint angle error and stan- +dard deviation are reported on all images and +on the top 50% images with respect to the +ground truth. +Dataset +num. images +all [◦] +top 50% [◦] +Realsense +5944 +12.61 ± 2.18 +5.62 ± 0.8 +Kinect +4966 +10.61 ± 2 +3.44 ± 0.4 +Azure +6394 +14.33 ± 2.64 +7.33 ± 0.92 +joint occlusions and highly non-coplanar configurations +with respect to the image plane, making the task more +difficult for a sparse, keypoint-based method such as ours. +By manual inspection, we detected that Azure dataset +contains more such images compared to the other two +datasets, which reflects on the results. The more thorough +error analysis is left for future work. +Fig. 4. Input image (left) and rendered joint angle pre- +dictions - our method (middle), our method with +refinement (right) +4.3 Deep refinement +The proposed distance-geometric method exhibits solid +generalization in addition to being lightweight, thus it can +be easily used in conjunction with a refinement procedure +without introducing noticeable computational complexity. +To this end, we use the RoboPose model, introduced by +Labb´e et al. (2021) and trained using a deep iterative +matching procedure (Li et al. (2018)) on the DREAM +dataset. This procedure can be briefly described as follows. +First, the joint angles are initialized randomly within +joint angle limits and used to render an RGB image +of the 3D robot model in this configuration. Then, the +rendered and input RGB images are cropped (to suppress +background information) and fed to a ResNet34 backbone. +The backbone outputs the relative 6D pose together with a +joint angle residual which are used to update the input and +the process is repeated iteratively. We use their publicly +available pretrained model which was trained on 100k +images using 44 GPUs. +We combine our method with RoboPose by using it to +initialize the deep refinement procedure. The results are +shown in Table 2. Clearly, the combined approach outper- +forms both our method (results in Table 1) and RoboPose +applied independently. This is because our model provides +a good initial guess, thereby making the refinement task +much easier in contrast to using a feasible random config- +uration as an initial guess. An exemplary robot configura- +tion estimated by our method and the combined approach +is displayed in Figure 4 in the form of a rendered image. +Furthermore, the goal is to obtain an accurate estimation +using as few iterations as possible, hence the results were + +generated using 3 iterations, where each iteration requires +a rendering operation and a forward pass of a deep CNN +backbone. On the contrary, RoboPose, when applied inde- +pendently, requires at least 10 iterations to achieve similar +accuracy. Note that we have not trained the model from +scratch in order to adjust it to our initialization - the model +is pretrained with random initialization. We would expect +retraining the model to introduce further improvements +as the joint angle residual is much smaller when using our +method compared to a feasible random configuration. This +investigation is left for future work. +Table 2. Results on the Panda-3CAM datasets before +and after combining the deep refinement model with +our method. The mean absolute joint angle error is +reported on all images. +Dataset +#images +RoboPose [◦] +Ours + RoboPose [◦] +Realsense +5944 +12.21 +5.85 +Kinect +4966 +13.68 +6.13 +Azure +6394 +9.3 +5.44 +5. CONCLUSION +In this paper, we have proposed a novel distance-geometric +framework for recovering the joint angles determining the +configuration of the robot from a single RGB image. +Our method is computationally efficient and exhibits solid +generalization ability when tested on a large set of images +displaying a state-of-the-art 7-DoF robot. We also show +that, due to its computational efficiency, it can be easily +used in conjuction with a dense refinement approach +to obtain superior results. We believe that a modular +approach is promising in the long-term, i.e. tackling the +larger problem through a set of smaller, simpler problems. +If one can detect the joint keypoints and recover the +respective EDM in the 3D space accurately, the joint +angles can accurately be recovered since the leap from +EDMs to joint angles is done via deterministic, parameter- +free transformations. This is valuable since interpreting, +analyzing, and generally understanding deep models is +hard due to their nature; thus, it is perhaps ”easier” to +develop and understand models that aim to solve smaller +pieces of the problem. However, due to the sensitivity +of sparse methods, global information should also be +incorporated, but in a computationally efficient way. 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In 2019 IEEE/CVF Conference +on Computer Vision and Pattern Recognition (CVPR), +4209–4218. doi:10.1109/CVPR.2019.00434. + diff --git a/cdA0T4oBgHgl3EQfGv9r/content/tmp_files/load_file.txt b/cdA0T4oBgHgl3EQfGv9r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e2b9fa29a1b8f6b61a3ec3eab7fb1d100da63354 --- /dev/null +++ b/cdA0T4oBgHgl3EQfGv9r/content/tmp_files/load_file.txt @@ -0,0 +1,488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf,len=487 +page_content='A Distance-Geometric Method for Recovering Robot Joint Angles From an RGB Image ⋆ Ivan Bili´c ∗ Filip Mari´c*, ** Ivan Markovi´c ∗ Ivan Petrovi´c ∗ ∗ University of Zagreb, Faculty of Electrical Engineering and Computing, Laboratory for Autonomous Systems and Mobile Robotics (e-mail: name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='surname@fer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' ∗∗ University of Toronto, Space and Terrestrial Autonomous Robotic Systems Laboratory (e-mail: name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='surname@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='ca) Abstract: Autonomous manipulation systems operating in domains where human intervention is difficult or impossible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=', underwater, extraterrestrial or hazardous environments) require a high degree of robustness to sensing and communication failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Crucially, motion planning and control algorithms require a stream of accurate joint angle data provided by joint encoders, the failure of which may result in an unrecoverable loss of functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In this paper, we present a novel method for retrieving the joint angles of a robot manipulator using only a single RGB image of its current configuration, opening up an avenue for recovering system functionality when conventional proprioceptive sensing is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Our approach, based on a distance-geometric representation of the configuration space, exploits the knowledge of a robot’s kinematic model with the goal of training a shallow neural network that performs a 2D-to-3D regression of distances associated with detected structural keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' It is shown that the resulting Euclidean distance matrix uniquely corresponds to the observed configuration, where joint angles can be recovered via multidimensional scaling and a simple inverse kinematics procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We evaluate the performance of our approach on real RGB images of a Franka Emika Panda manipulator, showing that the proposed method is efficient and exhibits solid generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Furthermore, we show that our method can be easily combined with a dense refinement technique to obtain superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Keywords: Manipulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Mechatronic Systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Robotics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Joint angle estimation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' INTRODUCTION Autonomous manipulation systems are ideal for perform- ing various tasks in environments where human presence is limited, such as underwater or orbital laboratories, as well as hazardous (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=', radioactive, toxic) environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In addition to effective planning and control algorithms, these systems require a high degree of robustness to sensing and communication failures, as a timely intervention by humans may be impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We propose a method for recovering the joint angles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=', the configuration) of an articulated robotic manipulator using only a single RGB image, providing an alternative source of proprioceptive data that can be used when data from joint encoders is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This is a challenging task for multiple reasons: fundamentally different robot configurations may result in similar images and certain configurations may have diminished observability due to physical occlusion by other parts of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' However, even a rough estimate of the joint configuration may enable the use of simple control methods to steer the robot to an approximate desired state, enabling the planning and execution of critical re- covery protocols (Ortenzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' ⋆ This work has been supported by the European Regional Devel- opment Fund under the grant KK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='0009 (DATACROSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The problem of recovering a robot configuration from spatial constraints such as gripper pose is known as inverse kinematics and features a variety of well known solutions (Lynch and Park (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' However, previous work has also explored instances of this problem where spatial con- straints may result from a variety of sensing modalities, such as depth or RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Widmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2016) use synthetic depth images to train semantic classifiers for direct joint angle regression in order to estimate the robot arm pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Bohg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2014) also use depth images to train a random forest classifier for pixel-wise part classification, while using joint encoder readings to initialize an incre- mental update scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2020) use a model-free approach and estimate the joint angles in an unsupervised fashion by using pretrained deep models for optical flow and monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Conversely, our method aims to recover joint configuration using only a single RGB image, which necessitates first finding the appropriate set of spatial constraints using 2D-to-3D regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The constraints resulting from regressed 3D keypoints may be under-determined and therefore correspond to multi- ple configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Instead, we use a distance-geometric model that integrates structural data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=', link lengths) to remove ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Distance geometry is highly relevant for applications such as molecular conformation, sensor arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='02051v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='RO] 5 Jan 2023 network localization (SNL) and statics (Liberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' For instance, SNL is commonly framed as an Eu- clidean distance matrix (EDM) completion problem (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2008))(Dokmanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2015)) and tackled through semidefinite programming (SDP) (Biswas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2006)), (Alfakih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Mari´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2022) formalize the equivalence of distance-based inverse kinematics and the distance geometry problem for a large class of articulated robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Furthermore, Moreno-Noguer (2017) tackles the problem of 3D human pose estimation from a single RGB image and demonstrates that representing human poses with EDMs instead of Cartesian coordinates results in more precise and less ambiguous pose estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Our work is partly inspired by these observations, and we show that the problem of recovering robot’s joint angles is in general highly related to distance geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' To the best of our knowledge, the only similar approaches to ours are that of Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2019) and Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2021), in the sense that only a single RGB image is used as an input for joint angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2019) train a joint keypoint detector to recognize a predefined, specifically chosen set of 17 keypoints in the image displaying a 4-DoF toy robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' These keypoints are then fed to a nonlinear non-convex 2D-to-3D optimization algorithm in order to recover the 6D pose together with joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Although our distance-geometric method is also keypoint-based, it only requires a number of keypoints equal to the robot’s DoF, placed in a robot-invariant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Furthermore, the optimization proposed in Zuo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2019) is complex, which diminishes its potential to improve from scaling the model and data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' On the other hand, Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2021) use a dense, rendering-based deep iterative matching approach (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2018)) to jointly learn the 6D camera-to-robot pose and joint angle updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Although primarily concerned with estimating the 6D pose, Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2021) also demonstrate that joint angles, if unknown, can be accurately reconstructed, at least with a sufficient number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We tested a hybrid approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e combining our method with the deep refinement proposed in Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Overall, the dense approach seems to generally outperform sparse approaches in terms of accuracy due to incorporating global information, at the cost of high computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In this paper, we propose a novel method for recovering the joint angles of an articulated robotic manipulator using only a single RGB image, based on a distance-geometric representation of the configuration space and the knowl- edge of a robot’s kinematic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Instead of training a single large model to directly predict the solution, our method divides the problem into a set of smaller sub- problems in a theoretically justified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' First, state- of-the-art keypoint detectors (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2020)) are used to detect joint keypoints in the image corresponding to the robot’s joints, which are insufficient for configuration recovery on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, our method takes the full set of inter-point distances and uses a learned 2D-to- 3D regression to produce an EDM corresponding to the associated 3D keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Following the approach in (Mari´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2022)) this EDM is extended to include distances between auxiliary points determined by the robot’s struc- ture, removing ambiguity related to joint angle recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Given a complete EDM, joint angles can be computed using parameter-free transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' classical multidi- mensional scaling (MDS) and kinematic transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The former maps an EDM to a geometrically centered set of 3D points, while the latter calculates the joint angles based on these points, forming a fully-differentiable IK layer that supports batching and can be ran on a GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In addition to generating the complete EDM, ground- truth joint angles are used to compute the loss in the configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Finally, our method is evaluated on a large set of real images displaying a 7-DoF robot arm in various configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We opt for a shallow architecture for all our experiments since the primary goal of this work is developing a framework, while a more thorough test of its potential is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' As the results show, our method exhibits solid generalization ability, while being simple and computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' METHODOLOGICAL BACKGROUND Euclidean distance geometry is an important tool in sev- eral applications whose aim is to reconstruct a complete set of distances (or points that realize them) in Euclidean space, given an incomplete set of distances as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In addition to a small subset of distance geometry which relates to EDMs, this section describes a kinematics proce- dure responsible for generating a set of 3D points sufficient for recovering the robot’s configuration as well as distance constraints emerging from kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='1 Euclidean distance matrices Let P ∈ Rn×d denote a matrix representing a set of n points in a d-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, the pairwise distances du,v between points can be determined via the Euclidean norm: du,v = ∥pu − pv∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (1) For the sake of notation simplicity, EDMs and individual distances are assumed to be squared in the remaining of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Expansion of (1) reveals that EDM is a function of the Gram matrix G = PP⊤: edm(G) = diag(G)1⊤ + 1 diag(G)⊤ − 2G, (2) with diag(G) representing the diagonal entries of G in the form of a column vector and 1 stands for column vector filled with ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Equation (2) establishes a one-way connection between an EDM and a Gram matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Consider an inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' recovering the set of points that generated the distance matrix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Let D be a squared EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, a Gram matrix that satisfies (2) can be recovered via G = −1 2JDJ, (3) where J = I − 1 N 11T (4) denotes a geometric centering matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Moreover, G is a real symmetric matrix, hence it can be factored into a canonical form via eigenvalue decomposition: G = UΛUT (5) where Λ = diag (λ0, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' , λn−1) contains the non- negative eigenvalues λi and U is an orthonormal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Now, assuming the eigenvalues are sorted in the descend- ing order, the point set ˆP ∈ Rn×d can finally be recovered by taking: ˆP⊤ = � diag �� λ0, � λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' , � λd−1 � , 0d×N−d � U⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (6) Computing the collection of points ˆP from a distance matrix D using (3), (4), (5) and (6) is also known as classical multidimensional scaling (cMDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note how in (6), all but the d largest eigenvalues are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Assum- ing that G is generated by a d-dimensional set of points, all but the d largest eigenvalues will be zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' If this is not the case, we can assume the presence of noise which is handled by the truncation (Dokmanic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Additionally, plugging the estimated ˆP in (2) yields an EDM that equals D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' However, distances are preserved under rigid transformations, thus ˆP and the original P are not generally equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Absolute position and orientation of the point set can be recovered through Procrustes analysis (Sch¨onemann (1966)), assuming that a set of at least d fixed points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' anchors) is known beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, a rigid transformation that aligns the anchors in P to their corresponding points in ˆP can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Finally, the original set of points P can be recovered by applying the obtained rigid transformation to all the points in ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' qi j i i j j q i qi j q j q i,j Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Visualization of neighbouring revolute joints con- nected with rigid links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Dashed lines represent the rotation axis of each joint, rotated by a an angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Auxiliary points q are placed along these axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Points p correspond to the position of coordinate frames defined in the center of their respective joints (placed below for better visibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='2 Distance-based kinematics Consider an n-DoF robotic manipulator comprised of single-axis revolute joints, forming a kinematic chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The procedure introduced in Mari´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2022) defines a sparse set of points whose positions are sufficient for recovering the full set of joint angles determining the robot’s configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' As shown in Figure 1, a set of points pi centered at the joint coordinate frames are introduced, which we associate with the keypoints detected in the RGB image of the robot by our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, ”virtual” auxiliary points qi are placed at a unit distance along the joints’ rotation axes ˜z using joints’ orientation Ri qi = pi + Ri˜z, (7) adding information on the relative orientation of neigh- bouring joints required for joint angle recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Finally, the model is completed with the addition of three points, x, y and z, corresponding to the root coordinate frame, defined by distances ∥p0 − x∥ = ∥p0 − y∥ = ∥p0 − z∥ = 1, ∥x − y∥ = ∥x − z∥ = ∥y − z∥ = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (8) The proof in Mari´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2022) shows that the distances between these points are sufficient for recovering the full set of joint angles for a large variety of manipulator structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Our training data is generated from ground truth joint angle vectors Θ ∈ C corresponding to the robot observed in the image I, where C ⊆ Rn represents the configu- ration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This allows us to simply generate the full distance-geometric robot description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Each data point is constructed by sequentially taking the position pi and orientation Ri of a parent joint i, as well as its joint angle θi, giving the position and orientation of the child joint j as Rj = RiRz (θi) Ri,j, pj = pi + RiRz (θi) pi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (9) The relative positions and orientations pi,j and Ri,j of neighbouring joints, are completely defined by the robot’s structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Thus, the distances between the set of the four neighbouring points can also be defined from structural parameters as ∥pi − qi∥ = ∥pj − qj∥ = 1 ∥pi − pj∥ = ∥pi,j∥ ∥pi − qj∥ = ∥pi,j + Ri,j˜z∥ ∥qi − pj∥ = ∥pi,j − ˜z∥ ∥qi − qj∥ = ∥pi,j − ˜z + Ri,j˜z∥ , (10) indicating that they are invariant to changes in joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Conversely, the distances corresponding to the remaining non-neighbouring pairs exist as a function of the configuration Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Using the constructive procedure in (7) and (9), together with distance constraints stated in (8) and (10), we construct a complete EDM D ∈ Rˆn׈n that uniquely (up to a rigid transformation) represents a set of 3D points associated with the configuration Θ observed in the RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Concretely, for an n-DoF articulated robot comprised of single-axis revolute joints one auxiliary point per joint to determine the axis of rotation, and two additional points are required to fully define the base coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Finally, a single data sample used for training the model can thus be formalized as D := (Θ, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' METHOD This section is dedicated to describing the system dis- played in Figure 2 as a pipeline through which the pro- posed method is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' First, given the input RGB image and 2D joint keypoint detections, one needs to cMDS IK 2D EDM 3D EDM EDM Regression Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' System overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In the image space, EDM is constructed from detected 2D joint keypoints and fed to a 2D- to-3D EDM regression network (black), which outputs the complete EDM in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The cMDS layer maps the complete EDM to a set of geometrically centered points, from which an IK layer computes the joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Estimated joint angles are depicted as a 2D projection of the robot rendered in respective configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' bridge the gap between 2D and 3D information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Theoreti- cally, infinitely many sets of ˆn points which are equal up to a rigid transformation, and generated as described in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='2, can map to a single feasible robot configuration Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' As inter-point distances remain identical irrespective of such point set transformation, we associate a configuration Θ with a single unique EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Therefore, instead of di- rectly predicting the points, we predict the corresponding distance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We frame the distance matrix regression problem as learning a mapping ζ : Rn×n + �→ Rˆn׈n + , where R+ = {x ∈ R : x ≥ 0}, n equals the number of joints, and ˆn = 2n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note that we don’t use auxiliary points in the image space, only keypoints directly corresponding to the robot’s joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The mapping is implemented as a shallow feedforward neural network that takes an EDM computed from 2D joint keypoints as input and outputs a complete EDM which is treated as if it was generated by a set of 3D points comprised of joint correspondences and auxiliary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We define a distance-based loss as: Ld = ∥ ˆD − D∥F , (11) where ∥ · ∥F denotes Frobenius norm, while ˆD and D are the predicted and ground-truth EDMs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Due to the fact that EDMs are symmetric matrices, the actual implementation works with upper-triangular elements from which the full EDM is computed afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This amounts to using n(n − 1)/2 elements instead of n2 with no loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The ground-truth EDMs are computed from ground-truth configurations Θ via a function f : Θn �→ Dˆn׈n based on equations described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Figure 3 demonstrates that EDM regression and joint angle recovery are highly related, by depicting a mean absolute joint angle error as a function of mean absolute EDM error, using the Kinect dataset (unseen during the learning process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note that the model used for this figure is trained for EDM regression, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' using the loss function (11), while the joint angles are only computed during inference via cMDS and IK layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We use our library 1 for most of the distance-geometry and kinematics-related computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' After the complete EDM D ∈ Rˆn׈n is estimated, the cMDS layer (a set of fixed, differentiable transformations described in Section 2) is used to obtain the geometrically centered set of points which generate the respective EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This mapping can be formally defined as Ω : Rˆn׈n + �→ Pˆn×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The set of points is then fed to an IK layer which computes the joint angles Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note that what we refer to 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='com/utiasSTARS/graphIK as an IK layer is not an inverse kinematics solver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' it is a sequence of differentiable kinematics transformations that, given the estimated set of points, compute the joint angles by computing the respective coordinate frame positions and orientations together with the axis of rotation for each joint (Mari´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The default configuration (corresponding to zero joint angles) of the robot is known from its model (an Unified Robotics Description Format file), hence the joint angles can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This allows us to define a loss in the configuration space between the predicted and ground-truth configuration: Lc = | ˆΘ − Θ|, (12) with |·| denoting the L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We use a linear combination of the two losses as a final loss to train the model: L = Lc + λLd, (13) with λ set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note that Lc causes the gradients to be propagated through IK and cMDS layers, while Ld is applied directly at the output of the EDM regression network and serves to provide the model with additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The EDM regression network is comprised of two Linear → BatchNorm → ReLU → Dropout layers, and an output Linear → ReLU layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The layers consist of 512−512−120 neurons (∼ 300k parameters) respectively, and the output size is determined by ˆn(ˆn − 1)/2 (upper triangular EDM), where ˆn = 2n + 2 and n = 7 for the Franka Emika Panda robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Among other necessary conditons, EDM elements must be positive, which is enforced by the last ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We use Adam (Kingma and Ba (2014)) for optimization, with initial learning rate α = 1e − 3, linear warmup (Ma and Yarats (2021)) over 2k iterations, and a batch size of 64 EDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The training is carried out for 100 epochs and learning rate is reduced by a factor of 2 after 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The model is initialized as proposed by (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' EXPERIMENTS All our experiments were conducted on images of a 7-DoF Franka Emika Panda robot observed in various configu- rations, using three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We report mean absolute error as a joint angle error metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Training and evaluation were carried out on a single NVIDIA RTX A5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Using the setup described in Section 3, a model requires ∼3 hours to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Regarding the running time, when using a batch size of 64, our system runs at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='6ms per data sample, and takes up 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='5GB of GPU memory for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' In reality, it would be limited by the running time of the chosen keypoint detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Joint angles vs EDMs, using mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The model is trained for EDM regression, while cMDS and IK layers are only used at test- time to compute the joint angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The errors are highly correlated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Pearson correlation coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Kinect dataset (unseen during training) is used for this plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='1 Dataset For our experiments, we use the DREAM dataset intro- duced by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2020) in their recently proposed state-of-the-art method for single-view camera-to-robot pose estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The dataset is comprised of real and synthetic parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The synthetic part is photorealistic and generated with domain randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We focus on the real part of the dataset, which is split into 4 different Panda-3CAM datasets - Realsense, Azure, Kinect, and Orb which contain 5944, 6394, 4966 and 32315 samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Each of these datasets is comprised of RGB images of the 7-DoF Franka Emika Panda robotic ma- nipulator, captured using different cameras with different intrinsic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The robot is observed in various configurations, including images with joint occlusions and even out-of-view joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Besides RGB images, the datasets contain 2D joint keypoint annotations together with their 3D correspondences and ground-truth robot configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The camera-to-robot pose is different and fixed for each dataset, except for Orb which is captured from 27 different viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' For all our experiments we used 8x subsampled version of the Orb dataset for training, which we refer to as Orb in the rest of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We automatically adjusted all the 2D joint keypoint annotations in all the datasets so that they match the Panda’s coordinate frame definitions exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='2 Results The results of applying our method on the Panda-3CAM datasets are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' For evaluating on Kinect and Azure datasets the model is trained on Realsense and Orb, while for Realsense evaluation we trained it on the Kinect and Orb datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The results show that our method gives solid joint angle approximations on unseen data, while being simple and computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The top 50% predictions mostly correspond to images which display configurations relatively close to coplanar with respect to the image plane, for a given robot-camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' However, the datasets also contain images with Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Results on the Panda-3CAM datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The mean absolute joint angle error and stan- dard deviation are reported on all images and on the top 50% images with respect to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Dataset num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' images all [◦] top 50% [◦] Realsense 5944 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='8 Kinect 4966 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='61 ± 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='4 Azure 6394 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='33 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='92 joint occlusions and highly non-coplanar configurations with respect to the image plane, making the task more difficult for a sparse, keypoint-based method such as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' By manual inspection, we detected that Azure dataset contains more such images compared to the other two datasets, which reflects on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The more thorough error analysis is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Input image (left) and rendered joint angle pre- dictions - our method (middle), our method with refinement (right) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='3 Deep refinement The proposed distance-geometric method exhibits solid generalization in addition to being lightweight, thus it can be easily used in conjunction with a refinement procedure without introducing noticeable computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' To this end, we use the RoboPose model, introduced by Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2021) and trained using a deep iterative matching procedure (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' (2018)) on the DREAM dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This procedure can be briefly described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' First, the joint angles are initialized randomly within joint angle limits and used to render an RGB image of the 3D robot model in this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Then, the rendered and input RGB images are cropped (to suppress background information) and fed to a ResNet34 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The backbone outputs the relative 6D pose together with a joint angle residual which are used to update the input and the process is repeated iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We use their publicly available pretrained model which was trained on 100k images using 44 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We combine our method with RoboPose by using it to initialize the deep refinement procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Clearly, the combined approach outper- forms both our method (results in Table 1) and RoboPose applied independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This is because our model provides a good initial guess, thereby making the refinement task much easier in contrast to using a feasible random config- uration as an initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' An exemplary robot configura- tion estimated by our method and the combined approach is displayed in Figure 4 in the form of a rendered image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Furthermore, the goal is to obtain an accurate estimation using as few iterations as possible, hence the results were generated using 3 iterations, where each iteration requires a rendering operation and a forward pass of a deep CNN backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' On the contrary, RoboPose, when applied inde- pendently, requires at least 10 iterations to achieve similar accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Note that we have not trained the model from scratch in order to adjust it to our initialization - the model is pretrained with random initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We would expect retraining the model to introduce further improvements as the joint angle residual is much smaller when using our method compared to a feasible random configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This investigation is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Results on the Panda-3CAM datasets before and after combining the deep refinement model with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' The mean absolute joint angle error is reported on all images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Dataset #images RoboPose [◦] Ours + RoboPose [◦] Realsense 5944 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='85 Kinect 4966 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='13 Azure 6394 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='44 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' CONCLUSION In this paper, we have proposed a novel distance-geometric framework for recovering the joint angles determining the configuration of the robot from a single RGB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' Our method is computationally efficient and exhibits solid generalization ability when tested on a large set of images displaying a state-of-the-art 7-DoF robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We also show that, due to its computational efficiency, it can be easily used in conjuction with a dense refinement approach to obtain superior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' We believe that a modular approach is promising in the long-term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' tackling the larger problem through a set of smaller, simpler problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' If one can detect the joint keypoints and recover the respective EDM in the 3D space accurately, the joint angles can accurately be recovered since the leap from EDMs to joint angles is done via deterministic, parameter- free transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' This is valuable since interpreting, analyzing, and generally understanding deep models is hard due to their nature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' thus, it is perhaps ”easier” to develop and understand models that aim to solve smaller pieces of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' However, due to the sensitivity of sparse methods, global information should also be incorporated, but in a computationally efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdA0T4oBgHgl3EQfGv9r/content/2301.02051v1.pdf'} +page_content=' As future work, we intend to explore the adequacy of different architectures for this task, including scaling the data 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sha256:0efb6cf7d0db86273266fb62d54c0dd340deb1cca0b0a32fc7edc22290b2c4bf +size 228823 diff --git a/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/2301.03204v1.pdf.txt b/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/2301.03204v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..692e5ac1d6a0e9bc4081ff1c94205eed4b9187b0 --- /dev/null +++ b/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/2301.03204v1.pdf.txt @@ -0,0 +1,1118 @@ +arXiv:2301.03204v1 [cs.IT] 9 Jan 2023 +1 +Secure Communication for Spatially Correlated RIS-Aided +Multiuser Massive MIMO Systems: Analysis and Optimization +Dan Yang, Jindan Xu, Wei Xu, Senior Member, IEEE, Yongming Huang, Senior Member, IEEE, and Zhaohua Lu +Abstract—This letter investigates the secure communication in +a reconfigurable intelligent surface (RIS)-aided multiuser mas- +sive multiple-input multiple-output (MIMO) system exploiting +artificial noise (AN). We first derive a closed-form expression +of the ergodic secrecy rate under spatially correlated MIMO +channels. By using this derived result, we further optimize the +power fraction of AN in closed form and the RIS phase shifts +by developing a gradient-based algorithm, which requires only +statistical channel state information (CSI). Our analysis shows +that spatial correlation at the RIS provides an additional dimen- +sion for optimizing the RIS phase shifts. Numerical simulations +validate the analytical results which show the insightful interplay +among the system parameters and the degradation of secrecy +performance due to high spatial correlation at the RIS. +Index Terms—Reconfigurable intelligent surface (RIS), ergodic +secrecy rate, spatial correlation, joint optimization. +I. INTRODUCTION +R +ECONFIGURABLE intelligent surface (RIS) has been +proposed as a promising technology for improving both +spectral and energy efficiencies for next-generation wireless +networks [1]. It consists of numerous low-cost passive reflect- +ing elements that can induce phase changes to the reflected +electromagnetic waves. As such, by properly adjusting the +phase shifts, RIS can smartly modify the channel conditions +between the base station (BS) and the users, which helps +improve the communication quality of wireless networks [2]. +Recently, there has been considerable interest in the use +of RIS to enhance the physical layer security of wireless +communication networks [3]–[6]. In [3], the transmit beam- +forming jointly with artificial noise (AN) and the RIS phase +shifts was optimized under a multiple-input multiple-output +(MIMO) wiretap channel. It was shown that the secrecy rate +performance can be strengthened with the aid of AN in RIS- +assisted systems. As for multiuser scenarios, the authors of [4] +investigated the robust secrecy design by solving a transmit +power minimization problem. Besides, in [5], the weighted +sum secrecy rate was maximized by taking both the direct link +This work was supported by the National Key Research and Develop- +ment Program 2020YFB1806600, the NSFC under grants 62022026 and +62211530108, and the ZTE Corporation and the State Key Laboratory of +Mobile Network and Mobile Multimedia Technology (Corresponding author: +Wei Xu, Zhaohua Lu) +D. Yang, and Y. Huang are with the National Mobile Communications +Research Laboratory, Southeast University, Nanjing 210096, China, and +also with Purple Mountain Laboratories, Nanjing 211111, China (email: +dyang@seu.edu.cn; huangym@seu.edu.cn). J. Xu is with the Engineering +Product Development Pillar, Singapore University of Technology and Design, +Singapore 487372 (e-mail: jindan xu@sutd.edu.sg). +W. Xu is with the National Mobile Communications Research Lab, South- +east University, Nanjing 210096, China, and also with Henan Joint Inter- +national Research Laboratory of Intelligent Networking and Data Analysis, +Zhengzhou University, Zhengzhou, 450001 China (wxu@seu.edu.cn). +Z. Lu is with ZTE Corporation, and State Key Laboratory of Mo- +bile Network and Mobile Multimedia Technology, Shenzhen, China (email: +lu.zhaohua@zte.com.cn) +and the cascaded RIS link into account. In [6], a RIS-aided +secure multiple-input single-output (MISO) communication +system was studied, where multiple colluding Eves coexist. +However, the design of RIS phase shifts of most methods +was based on instantaneous channel state information (CSI), +which can be unaffordable due to frequent phase adjustment at +RIS and channel estimation at the BS. Secondly, all the above +works were based on independent Rayleigh or Rician fading +and have not considered the impact of spatial correlation on +the secrecy performance. In fact, spatial correlation generally +exists at the RIS due to physical constraints in a rectangular +array, and it varies by adjusting the space among adjacent +RIS elements [7], [8]. Moreover, previous works on RIS-aided +systems, e.g., [5], [6], are usually restricted to single-antenna +Eves scenarios for the sake of analytical tractability. +Against the above background, the performance of spatially +correlated RIS-aided multiuser massive MIMO systems is first +studied in the presence of a multi-antenna Eve. The main +contributions of this work are listed below. +• We derive the closed-form expression for the ergodic +secrecy rate, which depends only on statistical channel +information of the users and Eve. +• We devise an alternating algorithm to maximize the +ergodic secrecy rate, where the power fraction of AN +is optimally obtained in closed form and the RIS phase +shifts are designed by a projected gradient ascent method. +• Insightful observations of the impact of spatial correlation +and the number of RIS elements on the secrecy perfor- +mance are presented. It indicates that the spatial correla- +tion enhances the ability of RIS to properly manipulate +the wireless environment. +Notation: The inverse, conjugate transpose, and trace of +matrix A are denoted by A−1, AH, tr(A), respectively. +CN(0, σ2) represents the complex Gaussian distribution with +zero mean and variance σ2. Besides, E{·} and var{·} denote +the expectation and variance of a random variable, respectively. +We use Cm×n to represent the space of all m × n matrices +with complex-valued elements. IK denotes the K-dimensional +identity matrix. +II. SYSTEM MODEL +We consider a RIS-aided multiuser massive MIMO secure +system, where K single-antenna legitimate users are served by +an M-antenna BS leveraging a RIS of N reflecting elements. +One passive Eve equipped with ME antennas is located around +users seeking to wiretap the transmitted information1. Assume +that the RIS is controlled by the BS through a perfect backhaul +link and perfect CSI of the users are available as it plays +1Note that this assumption also applies to situations where there are several +Eves collusively eavesdropping on the same secrecy data [9]. + +2 +the role of an upper bound with imperfect CSI in practice. +To evaluate the secrecy performance, channel distribution +information of Eve is assumed to be available at the BS, which +has been widely adopted and validated in literature, e.g., [9], +[10]. +We assume narrow-band quasi-static fading channels. Let +H1 ∈ CM×N, hB,k ∈ CM×1, HB,E ∈ CM×ME, hI,k ∈ +CN×1, and HI,E ∈ CN×ME, respectively, denote the channel +from the BS to RIS, BS to user k, BS to Eve, RIS to user +k, and RIS to Eve. Notably, we consider spatially correlated +rather than independent Rayleigh fading. Hence, we have +hI,k = +� +βI,kR1/2 +I,k gI,k +hB,k = +� +β2,kR1/2 +B,kgB,k, +(1) +HI,E = +� +βI,ER1/2 +I,E GI,E +HB,E = +� +β3R1/2 +B,EGB,E +(2) +where βI,k, βI,E, β3 and β2,k represent the large-scale path +losses of the corresponding channels. Elements of gI,k, gB,k, +GI,E, and GB,E are independently and identically distributed +(i.i.d.) complex Gaussian random variables with zero mean +and unit variance. In addition, RI,k and RB,k (RI,E and RB,E) +are respectively the channel correlation matrices at the RIS and +BS. Moreover, the LoS channel H1 is modeled, similar to [11], +as a full-rank channel matrix with [H1]m,n = √β1e−j2π +dm,n +λ +, +where β1 is the path loss, λ is the carrier wavelength, and +dm,n is the distance between reflecting element m of the +RIS and antenna n of the BS. Such channels are popularly +seen when deterministic scattering presents between the BS +and RIS or placing the RIS close to the BS [11]. Note that +the correlation matrices and the path losses are assumed to +be known, e.g., by the methods in [12]. In addition, denote +the phase shift matrix of the RIS by a diagonal matrix Φ = +diag(ejθ1, ..., ejθn, ..., ejθN), where θn ∈ [0, 2π) represents the +phase shift of the nth RIS reflecting element2. +In case that the instantaneous CSI of Eve is completely +unknown, AN is injected to mask the confidential information. +Before transmission, the information signal s with E{ssH} = +IK and the AN signal z ∼ CN(0M−K, IM−K) are multiplied +by data precoder W ∈ CM×K with tr(WWH) = K and +AN precoder V ∈ CM×(M−K) with tr(VVH) = M − K, +respectively. The transmit signal is expressed as +x = +� +ξP +K Ws + +� +(1 − ξ)P +M − K Vz ≜ √pWs + √qVz, +(3) +where P denotes the total transmit power and ξ ∈ [0, 1] is the +fraction of power allocated to the information (power fraction, +for short). Based on the above definitions, the transmit signal in +(3) satisfies the power constraint E{xHx} = P. For notational +simplicity, we define p ≜ +ξP +K +and q ≜ +(1−ξ)P +M−K . Then, the +received signals at user k and Eve are respectively given by +yk = √phH +k wksk+√p +� +i̸=k hH +k wisi+√qhH +k Vz+nk, (4) +yE = √pHH +E Ws + √qHH +E Vz + nE, +(5) +where nk ∼ CN(0, σ2 +k) and nE ∼ CN(0, σ2 +EIME) are the +additive white Gaussian noise (AWGN) at user k and Eve, +while hk = H1ΦhI,k + hB,k and HE = H1ΦHI,E + HB,E +represents the equivalent channel from the BS to user k and +to Eve, respectively. +2As usual, we use the amplitude-independent phase shift model for tractable +analysis. The analysis based on practical amplitude models [13] will be left +for our future work. +III. SECRECY PERFORMANCE ANALYSIS +In this section, the ergodic secrecy rate of the RIS-aided +secure system is derived in closed form. +We take advantage of channel hardening, because users do +not have any knowledge of the instantaneous CSI in practice, +but they are aware of their statistics. Therefore, the received +signal is decomposed as +yk =E{√phH +k wk}sk + +�√phH +k wk − E{√phH +k wk} +� +sk ++ √p +� +i̸=k hH +k wisi + √qhH +k Vz + nk. +(6) +By treating the interference and channel uncertainty as an +equivalent noise term, a lower bound for the achievable rate +of user k is given by +Rk = log2 +� +1 + +��E{√phH +k wk} +��2 +Ψ +� +, +(7) +where Ψ = � +i̸=k E{|√phH +k wi|2} + E{qhH +k VVHhk} + +var{√phH +k wk} + σ2 +k. For analytical tractability and low +complexity, we adopt the MRT precoding [9], and V = +[v1, ..., vi, ..., vM−K] with ∥vi∥ = 1, i = 1, ..., M − K, is +designed to lie in the null space of the user channels, i.e., +HHV = 0, where H = [h1, ..., hK]. +Considering a pessimistic case, Eve is so powerful that +it is perfectly aware of its channel and can remove all the +interference from legitimate users, i.e., strongly eavesdropping +in [10], [14]. Then, from (5), an upper bound for the capacity +of Eve is obtained as +C = E +� +log2 +� +1 + pwH +k HEX−1HH +E wk +�� +, +(8) +where X ≜ qHH +E VVHHE denotes the noise correlation +matrix at Eve. In addition, since the noise level at Eve is +unknown, it is reasonable to assume negligible thermal noise, +i.e., σ2 +E → 0, where the secure communication is guaranteed +in the worst case of a powerful Eve. To this end, the ergodic +secrecy rate is given by +Rsec = [Rk − C]+, +(9) +where [x]+ = max{0, x}. However, evaluating the expected +value in (8) analytically is cumbersome. As an alternative, a +lower bound for the ergodic secrecy rate is presented in the +following theorem. +Theorem 1: In the RIS-aided secure system with AN, the +ergodic secrecy rate of user k can be evaluated by +Rsec = [log2(1 + γk) − log2(1 + γE)]+, +(10) +with +γk = Sk/Ik, γE = SE/IE, Sk = ξP +� +tr(Rk) +�2, +(11) +Ik = ξP +� +i̸=k tr(RkRi) + σ2 +k +�K +j=1 tr(Rj), +(12) +SE = ξMME(M − K)tr +� +Rk(RE + β3RB,E) +� +, +(13) +IE = (1 − ξ)(M − K − ME)tr(RE + β3RB,E) +K +� +j=1 +tr(Rj), (14) +where Rk = β2,kRB,k + βI,kH1ΦRI,kΦHHH +1 +and RE = +βI,EH1ΦRI,EΦHHH +1 . +Proof: See Appendix A. +■ +Remark 1: It is observed from Theorem 1 that the ergodic +secrecy rate depends only on the statistical CSI of the users and +Eve, phase shifts Φ, and power fraction ξ, motivating further +optimization concerning Φ and ξ at the BS. + +3 +Corollary 1: For uncorrelated Rayleigh fading, i.e., RI,k = +RI,E = IN and RB,k = RB,E = IM, we obtain γk and +γE as (15) and (16) at the bottom of the next page. From +(15), we observe that the inter-user interference always exists +even with an infinite number of BS antennas M. This is +because the cascaded channels through the RIS for the multiple +users are not asymptotically orthogonal due to the common +component H1. In addition, the RIS’s ability to modify the +wireless medium is significantly impeded since the secrecy +rate becomes independent of the RIS phase shifts Φ but only +dependent on the size of RIS. +Corollary 2: When N ≫ M, we have H1HH +1 → β1NIM. +By substituting ∥H1HH +1 ∥2 +2 = β2 +1N 2M into (15) and (16), the +secrecy rate is given by (17) at the bottom of this page. We +evince that the achievable rate of user k increases logarithmi- +cally with the number of BS antennas and the capacity of Eve +hardly changes with M. This implies that a promising secrecy +performance gain is achieved for large N. +Corollary 3: Without the existence of RIS, i.e., βI,k = 0 +and βI,E = 0, the ergodic secrecy rate in (10) reduces to +Rsec = +� +log2 +� +1 + +ξβ2 +2,kPM 2/ �K +j=1 β2,j +ξPδ � +i̸=k β2,itr(RB,kRB,i) + σ2 +k +� +− log2 +� +1 + +ξME(M − K)tr +� +RB,kRB,E +� +(1 − ξ)M(M − K − ME) �K +j=1 β2,j +��+ +, +(18) +where δ = β2,k/ �K +j=1 β2,j. Specifically, when the spatial +correlation at the BS disappears, the derived Rsec in (18) +retrieves the result in [14, Theorem 1] as a special case. +IV. PROPOSED DESIGN FOR SECRECY RATE +MAXIMIZATION +In this section, we study the joint optimization of the +power fraction ξ and phase shifts Φ to maximize the ergodic +secrecy rate in (10). Mathematically, the optimization problem +is formulated as +(P1) max +ξ,Φ +Rsec(ξ, Φ) +(19) +s.t. +ξ ∈ [0, 1]; |φi| = 1, i = 1, ..., N, +where φi = exp(jθi). It is challenging to jointly optimize +Rsec(ξ, Φ) as it is a non-convex function of ξ and Φ. To +address this, the alternating optimization (AO) technique is +applied to optimize ξ and Φ by executing refinement processes +with efficient closed-form calculations at the BS. +First, we consider the optimization of ξ by fixing Φ. The +following lemma provides a closed-form solution to the fixed- +point equation for solving (P1). +Lemma 1: For given Φ, the optimal solution of ξ is +ξ∗ = −b + +√ +b2 − 4ac +2a +, +(20) +where a = B1(A1A2 + A1A3 + A2 +2) − A1A3, b = 2A3A1, +and c = B1A2 +3 −A1A3 are constants with respect to statistical +channel spatial correlation matrices. +Proof: By taking the first derivative of Rsec in (10), it yields +R′ +sec = ∂Rsec +∂ξ += +A1A3 +ln 2(A3 + A2ξ)[A3 + (A1 + A2)ξ] +− +B1 +ln 2(ξ − 1)[1 + (B1 − 1)ξ], +(21) +where +B1 +≜ +MME(M−K)ζ2tr +� +Rk(RE+β3RB,E) +� +K(M−K−ME)tr(RE+β3RB,E) �K +j=1 tr(Rj), +A1 +≜ +Ptr(Rk)2, +A2 +≜ +P � +i̸=k tr(RkRi), +and +A3 +≜ +σ2 +k +�K +j=1 tr(Rj). Since ξ +∈ +[0, 1], after some +algebraic manipulations, it is easily checked that R′′ +sec < 0, +which implies that R′ +sec is a strictly decreasing function on ξ. +Moreover, we have R′ +sec > 0 for small ξ, while R′ +sec < 0 for +large ξ. Hence, there exists an optimal choice of ξ achieving +the unique maximum of secrecy rate. Therefore, considering +the concavity of Rsec with respect to ξ, the optimal power +fraction in (20) is obtained by solving R′ +sec = 0. +■ +Then, we optimize the RIS phase matrix Φ for fixed ξ, +which is less tractable due to the unit-modulus constraints. Due +to the complicated form of Rsec in (10), we apply the projected +gradient ascent method to obtain a locally optimal solution, +eventually converging to a stationary point [11]. Specifically +at the lth step, denote by vl = [φl +1, ..., φl +n, ..., φl +N]T the induced +phases and by qk the adopted ascent direction, where [ql]n = +∂Rsec +∂φ∗ +n +with respect to φn = ejθn is obtained in the following +Lemma 2. The subsequent (l + 1)th iteration step is updated +according to +˜vl+1 = vl + µkql and vl+1 = exp +� +j arg +�˜vl+1�� +, +(22) +where µk is the step size computed at each step. +Lemma 2: The gradient of the ergodic secrecy rate, Rsec, +with respect to φn is computed as +∂Rsec +∂φ∗n += +1 +ln 2 +� +∂γk +∂φ∗n +1 + γk +− +∂γE +∂φ∗n +1 + γE +� +, +(23) +where ∂γk +∂φ∗n and ∂γE +∂φ∗n are given in (28) and (29), respectively. +Proof: See Appendix B. +■ +Now by incorporating Lemma 1 and the gradient ascent +method, concrete steps of the proposed algorithm are sum- +marized in Algorithm 1. +Proposition 1: The proposed algorithm always converges to +a stationary point of (P1). +Proof: This is directly checked by the following +Rsec +� +ξ(t), Φ(t) +� (a) +≥ Rsec +� +ξ(t−1), Φ(t) +� (b) +≥ Rsec +� +ξ(t−1), Φ(t−1) +� +, +(24) +where (a) holds since the optimization of ξ is convex for +given Φ, and (b) holds because the gradient search is along a +monotonically increasing direction of Rsec [15]. +■ +The algorithm comes with low computational complexity +because it consists of simple matrix operations. In particular, +the complexity of Algorithm 1, depending on the computations +involved in updating the power faction in (20) and the gradient +in (23), is O(MN 2 + NM 2), which is lower compared with +that of [6] under practical settings. +γk = +ξP +� +β2 +I,kβ2 +1M 2N 2 + βI,kβ2,kβ1M 2N + β2 +2,kM 2� +ξP � +i̸=k +� +β2,kβ2,iM + (β2,kβI,i + βI,kβ2,i) β1MN + βI,kβI,i∥H1HH +1 ∥2 +2 +� ++ σ2 +k +�K +j=1 +� +βI,jβ1MN + β2,jM +� +(15) +γE = ξME(M − K) +� +βI,kβI,E∥H1HH +1 ∥2 +2 + (βI,kβ3 + β2,kβI,E) β1MN + β2,kβ3M +� +(1 − ξ)(M − K − ME) (β3 + βI,Eβ1N) �K +j=1 +� +βI,jβ1MN + β2,jM +� +(16) +Rsec = +� +log2 +� +1 + MβI,kβ2 +1 +� +i̸=k βI,i +� +− log2 +� +1 + +ξME(M − K)βI,k +(1 − ξ)(M − K − ME)β2 +1 +�K +j=1 βI,j +��+ +(17) + +4 +Algorithm 1 Proposed algorithm for solving P1 +1: Initialize: v0 = exp(jπ/2)1N, Φ0 = diag(v0), R0 +sec = +f(ξ, Φ0) given by (9), ξ ∈ [0, 1], t = 0, and ǫ > 0. +2: Repeat t ← t + 1 +3: Find ξ(t) with fixed Φ(t−1) as per (20); +4: for l = 0, 1, 2, ..., do +5: +Find [ql]n = +∂Rsec +∂φ∗n , n = 1, ..., N, as per (23) and µ by +backtrack line search [15]; +6: +˜vl+1 = vl + µql; +vl+1 = exp(j arg(˜vl+1)); +7: +Φl+1 = diag(vl+1); +Rl+1 +sec = f(ξ(t), Φl+1); +8: +Until |Rsec(ξ(t), Φl+1) − Rsec(ξ(t), Φl)| < ǫ; +9: +Φ(t) = Φl+1; +10: Until |Rsec(ξ(t), Φ(t)) − Rsec(ξ(t−1), Φ(t−1))| < ǫ. +V. NUMERICAL RESULTS +In this section, numerical simulations are provided to vali- +date the effectiveness of the proposed methods. The distance- +dependent large-scale path loss coefficient is β = C0( d +D0 )−ζ, +where C0 = −20 dB is the path loss at the reference distance +D0 = 1 m, d represents the individual link distance, and ζ +denotes the path loss exponent. The pass-loss exponents for +the RIS-aided links are set as 2 and 2.2 while the pass-loss +exponent for the direct links is set as 3. The distance between +the BS and RIS is set to be 20 m, and all the users and Eve +are assumed to be located in a circular regime, whose center +is 50 m away from the RIS and 60 m away from the BS, and +the radius is 3 m. The spatial correlation matrices at the BS +are generated according to [14] as [R(ρ)]i,j = ρ|i−j|, while +the spatial correlation matrices at the RIS are given as in [7]. +The RIS element spacing is given by dH = dV = λ/4. Also, +the signal-to-noise ratio (SNR) is defined as 10 log 10(P/σ2 +k). +Unless otherwise specified, we also set σ2 +k = σ2 +E = −60 dBm, +SNR = 5 dB, ρ = 0.4, N = 256, M = 128, and K = 8. +Fig. 1 illustrates that the derived analytical results and +numerical results match well for varying number of Eve’s +antennas. We observe that a higher number of Eve’s antennas +degrades the secrecy rates as expected. For comparison, we +also depict the results with ZF precoding. It is shown that +MRT outperforms ZF at low SNRs while for high SNR values +ZF attains a higher secrecy rate since ZF offers interference- +free communication to users in the high SNR regime. In the +case of imperfect CSI, the estimated channel is modeled as +zk = +√ +1 − τ2ˆzk + τek by representing hk = R +1 +2 +k zk, where +ˆzk is an imperfect observation of zk, ek is the Gaussian noise, +and 0 < τ < 1 characterizes the CSI imperfection. We observe +that the secrecy performance loss is marginal with estimation +error τ = 0.1 in the tested cases. +Fig. 2 depicts the secrecy rate versus the power fraction for +ME = 4, where the optimal value for ξ in (20) is marked by +black stars. It is shown that ξ∗ is decreasing in the number of +BS antennas M, i.e., more power should be allocated to AN. +This is because the correlation between hk and HE becomes +strong with growing M due to the increasing dimension of +H1, resulting in potentially more information leakage to Eve. +On the other hand, ξ∗ is increasing in the RIS size N, since +the effective degree of freedom of the channels from RIS to +users increases with N. In this case, it can be useful to allocate +less power to AN for improving the secrecy performance. +Fig. 3 presents the secrecy performance of the proposed +scheme versus N for ME = 2 by using equal power fraction +[16] and random phase shifts [7] as benchmarks, i.e., ξ = 0.5 +and θn ∼ U[0, 2π]. For small N, the optimal power fraction +with random phase shifts achieves higher secrecy rates than the +equal power fraction with random and optimal phase shifts. +This is because a small RIS provides limited signal energy +boosting for the system where the system tends to be a power- +limited scenario and this power fraction optimization plays a +dominating role. In addition, we notice that spatial correlation +at the RIS should be taken into account to benefit from the +phase shifts design in the case of statistical CSI. Also, it is +shown that the secrecy rates decrease when the inter-element +spacing reduces from λ/4 to λ/8. This is due to an increase in +the spatial correlation, which reduces the spatial diversity. We +observe a small performance gap by comparing the proposed +algorithm with an approximate of the global optimum, which is +achieved by running the algorithm twenty times with different +initializations and then choosing the best. +Fig. 4 shows the ergodic secrecy rate versus the number of +iterations for various numbers of RIS elements with ME = 2. +We observe that the algorithm converges fast in all the tested +cases, where the algorithm converges within 10 iterations. +VI. CONCLUSION +We considered the secure communication in RIS-aided +multiuser massive MIMO systems. A closed-form expression +of the ergodic secrecy rate was derived. Then, based on the +expression, we optimized RIS phase shifts and AN power +fraction. We showed that a large number of RIS elements and +low spatial correlation at the RIS are preferred to achieve high +secrecy rates. Future works include extending to Rician and +even millimeter-wave channels. +APPENDIX A +For the sake of exposition, we denote the cascade channel +of user k by hk = +� +βI,kH1ΦR1/2 +I,k gI,k + +� +β2,kR1/2 +B,kgB,k. +Since gI,k and gB,k are independent random vectors, we have +that hk follows the complex Gaussian distribution, i.e., hk ∼ +CN(0, Rk), where Rk = β2,kRB,k + βI,kH1ΦRI,kΦHHH +1 . +1) Compute Rk: Consider MRT satisfying tr(WWH) = K, +which leads to W = +� +K +�K +j=1 tr(Rj)H. First, we directly obtain +��E{hH +k wk} +��2 = +K +�K +j=1 tr(Rj) [tr (Rk)]2 and E +���hH +k wi +��2� += +K +�K +j=1 tr(Rj)tr (RkRi). Then, the variance is calculated as +1 +M 2 var +� +hH +k wk +� += +K +�K +j=1 tr(Rj) +E +����� +1 +M hH +k hk − 1 +M tr(Rk) +���� +2� +M→∞ +−−−−→ 0, +(25) +where (25) is obtained according to [17, Lemma 4]. For the +term of E +� +hH +k VVHhk +� +, it is obviously zero due to the null- +space AN method. +2) Compute C: To begin with, we rewrite X in (8) as +X = +qX1 + qX2, where X1 +≜ HH +B,EVVHHB,E and +X2 ≜ (H1ΦHI,E)HVVH(H1ΦHI,E) are uncorrelated due +to the definition in (2). Eigendecompose RB,E = UΛUH to +decorrelate the channel matrix HB,E as Z = HB,EΛ−1/2UH, +where Λ = diag(λ1, ..., λN) contains the eigenvalues of R and +the columns of U are the corresponding eigenvectors. Since + +5 +-10 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Fig. 1. Ergodic secrecy rate versus +SNR +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.5 +1 +1.5 +Fig. 2. Ergodic secrecy rate versus +power fraction +3264128 +256 +512 +1024 +0 +1 +2 +3 +4 +Fig. 3. Ergodic secrecy rate versus +the number of RIS elements +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +1.5 +2 +2.5 +3 +Fig. 4. Ergodic secrecy rate versus +the number of iterations +U is unitary, the statistics of ZU are identical to those of Z. +Hence, the distribution of X1 is the same as +�N +i=1 +�N +j=1 λ1/2 +i +λ1/2 +j +zivivH +j zH +j , +(26) +where zi is the ith row of Z and vi is the ith column of V. +Considering that zi and vi are independent, it is known from +[18] that �N +n=1 λnznvnvH +n zH +n follows a Wishart distribution, +i.e., �N +n=1 λnWME(M − K, 1 +M IME). The distribution of X2 +is obtained analogously by rewriting H1ΦHI,E = R1/2 +E GI,E +with RE = βI,EH1ΦRI,EΦHHH +1 . Then, by applying the +Jensen’s inequality, the capacity of Eve is bounded as +C ≤ log2 +� +1 + pE +� +wH +k HEX−1HH +E wk +�� +(a) += log2 +� +1 + +ξM(M − K)E +� +wH +k HEHH +E wk +� +K(1 − ξ)(M − K − ME)tr(RE + β3RB,E) +� +(b) += log2 +� +1 + ξζ2MME(M − K)tr +� +Rk(RE + β3RB,E) +� +K(1 − ξ)(M − K − ME)tr(RE + β3RB,E) +� +, +(27) +where (a) uses the property that A−1 +a.s. +−−→ 1/(n − m)Im +for a Wishart matrix A ∼ Wm(n, Im) with n > m [18, +Sec 2.1.6] and �N +n=1 λn = tr(RE), and (b) results form +E +� +wH +k HEHH +E wk +� += ζ2MEtr +� +Rk(RE + β3RB,E) +� +where +ζ2 = K +�� �K +j=1 tr(Rj) +� +. +APPENDIX B +Using the standard quotient rule of derivatives, we have +∂γk +∂φ∗n += 1 +I2 +k +� +Ik ∂Sk +∂φ∗n +− Sk ∂Ik +∂φ∗n +� +, +(28) +∂γE +∂φ∗n += 1 +I2 +E +� +IE ∂SE +∂φ∗n +− SE ∂IE +∂φ∗n +� +. +(29) +For simplicity, we use the notation (·)′ to represent the partial +derivative with respect to φ∗ +n. Specifically, the term S′ +k is +given by S′ +k = 2ξPtr(Rk)tr(R′ +k), which requires a further +derivation of tr(R′ +k). Since all terms in Rk depend on φ∗ +n, we +have +tr +� +R′ +k +� += tr +� +β2,k ∂RB,k +∂φ∗n ++ βI,k +∂ +� +H1ΦRI,kΦHHH +1 +� +∂φ∗n +� +(a) +=βI,k +� +i,j +� +H1ΦRI,k +� +j,n[HH +1 ]T +i,n = βI,k +� +HH +1 H1ΦRI,k +� +n,n, +(30) +where (a) is obtained by using Lemma 1 in [8]. To +this +end, +the +partial +derivatives +of +Sk, +Ik, +SE, +and +IE with respect to φ∗ +n in (28) and (29) are expressed +as +follows: +S′ +k += +2ξPβI,ktr(Rk) +� +HH +1 H1ΦRI,k +� +n,n, +I′ +k += +ξP � +i̸=k +� +HH +1 +� +βI,kRkH1ΦRI,k ++ +βI,iRiH1ΦRI,i +�� +n,n ++ +σ2 +k +�k +j=1 +� +HH +1 H1ΦRI,j +� +n,n, +S′ +E = ξMME(M − K) +� +HH +1 +� +βI,k +� +RE + β3RB,E +� +H1ΦRI,k + +βI,ERkH1ΦRI,E +�� +n,n, and I′ +E += +(1 − ξ)(M − K − +ME) +� +βI,E +�k +j=1 tr(Rj) +� +HH +1 H1ΦRI,E +� +n,n ++ +tr(RE ++ +β3RB,E) �k +j=1 +� +HH +1 H1ΦRI,j +� +n,n +� +. +REFERENCES +[1] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: +Intelligent reflecting surface aided wireless network,” IEEE Commun. +Mag., vol. 58, no. 1, pp. 106–112, Jan. 2020. +[2] W. Shi et al., “Intelligent reflection enabling technologies for integrated +and green internet-of-everything beyond 5G: Communication, sensing, +and security,” IEEE Wireless Commun., early access, May 9, 2022. +[3] S. Hong et al., “Artificial-noise-aided secure MIMO wireless commu- +nications via intelligent reflecting surface,” IEEE Trans. Commun., vol. +68, no. 12, pp. 7851–7866, Dec. 2020. +[4] X. Yu et al., “Robust and secure wireless communications via intelligent +reflecting surfaces,” IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. +2637–2652, Nov. 2020. +[5] H. Niu et al., “Weighted sum secrecy rate maximization using intelligent +reflecting surface,” IEEE Trans. Commun., vol. 69, no. 9, pp. 6170–6184, +Sep. 2021. +[6] Y. 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Boston, MA, USA: Now, 2004. + diff --git a/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/load_file.txt b/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5489a1f7267451aa0ce98d72516dd25fe8080829 --- /dev/null +++ b/eNE1T4oBgHgl3EQfeQRR/content/tmp_files/load_file.txt @@ -0,0 +1,494 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf,len=493 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='03204v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='IT] 9 Jan 2023 1 Secure Communication for Spatially Correlated RIS-Aided Multiuser Massive MIMO Systems: Analysis and Optimization Dan Yang, Jindan Xu, Wei Xu, Senior Member, IEEE, Yongming Huang, Senior Member, IEEE, and Zhaohua Lu Abstract—This letter investigates the secure communication in a reconfigurable intelligent surface (RIS)-aided multiuser mas- sive multiple-input multiple-output (MIMO) system exploiting artificial noise (AN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We first derive a closed-form expression of the ergodic secrecy rate under spatially correlated MIMO channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' By using this derived result, we further optimize the power fraction of AN in closed form and the RIS phase shifts by developing a gradient-based algorithm, which requires only statistical channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Our analysis shows that spatial correlation at the RIS provides an additional dimen- sion for optimizing the RIS phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Numerical simulations validate the analytical results which show the insightful interplay among the system parameters and the degradation of secrecy performance due to high spatial correlation at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Index Terms—Reconfigurable intelligent surface (RIS), ergodic secrecy rate, spatial correlation, joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' INTRODUCTION R ECONFIGURABLE intelligent surface (RIS) has been proposed as a promising technology for improving both spectral and energy efficiencies for next-generation wireless networks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It consists of numerous low-cost passive reflect- ing elements that can induce phase changes to the reflected electromagnetic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' As such, by properly adjusting the phase shifts, RIS can smartly modify the channel conditions between the base station (BS) and the users, which helps improve the communication quality of wireless networks [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Recently, there has been considerable interest in the use of RIS to enhance the physical layer security of wireless communication networks [3]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In [3], the transmit beam- forming jointly with artificial noise (AN) and the RIS phase shifts was optimized under a multiple-input multiple-output (MIMO) wiretap channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It was shown that the secrecy rate performance can be strengthened with the aid of AN in RIS- assisted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' As for multiuser scenarios, the authors of [4] investigated the robust secrecy design by solving a transmit power minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Besides, in [5], the weighted sum secrecy rate was maximized by taking both the direct link This work was supported by the National Key Research and Develop- ment Program 2020YFB1806600, the NSFC under grants 62022026 and 62211530108, and the ZTE Corporation and the State Key Laboratory of Mobile Network and Mobile Multimedia Technology (Corresponding author: Wei Xu, Zhaohua Lu) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Yang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Huang are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China, and also with Purple Mountain Laboratories, Nanjing 211111, China (email: dyang@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' huangym@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Xu is with the Engineering Product Development Pillar, Singapore University of Technology and Design, Singapore 487372 (e-mail: jindan xu@sutd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='sg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Xu is with the National Mobile Communications Research Lab, South- east University, Nanjing 210096, China, and also with Henan Joint Inter- national Research Laboratory of Intelligent Networking and Data Analysis, Zhengzhou University, Zhengzhou, 450001 China (wxu@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Lu is with ZTE Corporation, and State Key Laboratory of Mo- bile Network and Mobile Multimedia Technology, Shenzhen, China (email: lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='zhaohua@zte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='cn) and the cascaded RIS link into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In [6], a RIS-aided secure multiple-input single-output (MISO) communication system was studied, where multiple colluding Eves coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' However, the design of RIS phase shifts of most methods was based on instantaneous channel state information (CSI), which can be unaffordable due to frequent phase adjustment at RIS and channel estimation at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Secondly, all the above works were based on independent Rayleigh or Rician fading and have not considered the impact of spatial correlation on the secrecy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In fact, spatial correlation generally exists at the RIS due to physical constraints in a rectangular array, and it varies by adjusting the space among adjacent RIS elements [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Moreover, previous works on RIS-aided systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', [5], [6], are usually restricted to single-antenna Eves scenarios for the sake of analytical tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Against the above background, the performance of spatially correlated RIS-aided multiuser massive MIMO systems is first studied in the presence of a multi-antenna Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The main contributions of this work are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We derive the closed-form expression for the ergodic secrecy rate, which depends only on statistical channel information of the users and Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We devise an alternating algorithm to maximize the ergodic secrecy rate, where the power fraction of AN is optimally obtained in closed form and the RIS phase shifts are designed by a projected gradient ascent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Insightful observations of the impact of spatial correlation and the number of RIS elements on the secrecy perfor- mance are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It indicates that the spatial correla- tion enhances the ability of RIS to properly manipulate the wireless environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Notation: The inverse, conjugate transpose, and trace of matrix A are denoted by A−1, AH, tr(A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' CN(0, σ2) represents the complex Gaussian distribution with zero mean and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Besides, E{·} and var{·} denote the expectation and variance of a random variable, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We use Cm×n to represent the space of all m × n matrices with complex-valued elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' IK denotes the K-dimensional identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' SYSTEM MODEL We consider a RIS-aided multiuser massive MIMO secure system, where K single-antenna legitimate users are served by an M-antenna BS leveraging a RIS of N reflecting elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' One passive Eve equipped with ME antennas is located around users seeking to wiretap the transmitted information1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Assume that the RIS is controlled by the BS through a perfect backhaul link and perfect CSI of the users are available as it plays 1Note that this assumption also applies to situations where there are several Eves collusively eavesdropping on the same secrecy data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2 the role of an upper bound with imperfect CSI in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' To evaluate the secrecy performance, channel distribution information of Eve is assumed to be available at the BS, which has been widely adopted and validated in literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We assume narrow-band quasi-static fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Let H1 ∈ CM×N, hB,k ∈ CM×1, HB,E ∈ CM×ME, hI,k ∈ CN×1, and HI,E ∈ CN×ME, respectively, denote the channel from the BS to RIS, BS to user k, BS to Eve, RIS to user k, and RIS to Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Notably, we consider spatially correlated rather than independent Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Hence, we have hI,k = � βI,kR1/2 I,k gI,k hB,k = � β2,kR1/2 B,kgB,k, (1) HI,E = � βI,ER1/2 I,E GI,E HB,E = � β3R1/2 B,EGB,E (2) where βI,k, βI,E, β3 and β2,k represent the large-scale path losses of the corresponding channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Elements of gI,k, gB,k, GI,E, and GB,E are independently and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=') complex Gaussian random variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In addition, RI,k and RB,k (RI,E and RB,E) are respectively the channel correlation matrices at the RIS and BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Moreover, the LoS channel H1 is modeled, similar to [11], as a full-rank channel matrix with [H1]m,n = √β1e−j2π dm,n λ , where β1 is the path loss, λ is the carrier wavelength, and dm,n is the distance between reflecting element m of the RIS and antenna n of the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Such channels are popularly seen when deterministic scattering presents between the BS and RIS or placing the RIS close to the BS [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Note that the correlation matrices and the path losses are assumed to be known, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', by the methods in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In addition, denote the phase shift matrix of the RIS by a diagonal matrix Φ = diag(ejθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', ejθn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', ejθN), where θn ∈ [0, 2π) represents the phase shift of the nth RIS reflecting element2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In case that the instantaneous CSI of Eve is completely unknown, AN is injected to mask the confidential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Before transmission, the information signal s with E{ssH} = IK and the AN signal z ∼ CN(0M−K, IM−K) are multiplied by data precoder W ∈ CM×K with tr(WWH) = K and AN precoder V ∈ CM×(M−K) with tr(VVH) = M − K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The transmit signal is expressed as x = � ξP K Ws + � (1 − ξ)P M − K Vz ≜ √pWs + √qVz, (3) where P denotes the total transmit power and ξ ∈ [0, 1] is the fraction of power allocated to the information (power fraction, for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Based on the above definitions, the transmit signal in (3) satisfies the power constraint E{xHx} = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' For notational simplicity, we define p ≜ ξP K and q ≜ (1−ξ)P M−K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Then, the received signals at user k and Eve are respectively given by yk = √phH k wksk+√p � i̸=k hH k wisi+√qhH k Vz+nk, (4) yE = √pHH E Ws + √qHH E Vz + nE, (5) where nk ∼ CN(0, σ2 k) and nE ∼ CN(0, σ2 EIME) are the additive white Gaussian noise (AWGN) at user k and Eve, while hk = H1ΦhI,k + hB,k and HE = H1ΦHI,E + HB,E represents the equivalent channel from the BS to user k and to Eve, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2As usual, we use the amplitude-independent phase shift model for tractable analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The analysis based on practical amplitude models [13] will be left for our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' SECRECY PERFORMANCE ANALYSIS In this section, the ergodic secrecy rate of the RIS-aided secure system is derived in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We take advantage of channel hardening, because users do not have any knowledge of the instantaneous CSI in practice, but they are aware of their statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Therefore, the received signal is decomposed as yk =E{√phH k wk}sk + �√phH k wk − E{√phH k wk} � sk + √p � i̸=k hH k wisi + √qhH k Vz + nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' (6) By treating the interference and channel uncertainty as an equivalent noise term, a lower bound for the achievable rate of user k is given by Rk = log2 � 1 + ��E{√phH k wk} ��2 Ψ � , (7) where Ψ = � i̸=k E{|√phH k wi|2} + E{qhH k VVHhk} + var{√phH k wk} + σ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' For analytical tractability and low complexity, we adopt the MRT precoding [9], and V = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', vM−K] with ∥vi∥ = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', M − K, is designed to lie in the null space of the user channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', HHV = 0, where H = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', hK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Considering a pessimistic case, Eve is so powerful that it is perfectly aware of its channel and can remove all the interference from legitimate users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', strongly eavesdropping in [10], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Then, from (5), an upper bound for the capacity of Eve is obtained as C = E � log2 � 1 + pwH k HEX−1HH E wk �� , (8) where X ≜ qHH E VVHHE denotes the noise correlation matrix at Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In addition, since the noise level at Eve is unknown, it is reasonable to assume negligible thermal noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', σ2 E → 0, where the secure communication is guaranteed in the worst case of a powerful Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' To this end, the ergodic secrecy rate is given by Rsec = [Rk − C]+, (9) where [x]+ = max{0, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' However, evaluating the expected value in (8) analytically is cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' As an alternative, a lower bound for the ergodic secrecy rate is presented in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Theorem 1: In the RIS-aided secure system with AN, the ergodic secrecy rate of user k can be evaluated by Rsec = [log2(1 + γk) − log2(1 + γE)]+, (10) with γk = Sk/Ik, γE = SE/IE, Sk = ξP � tr(Rk) �2, (11) Ik = ξP � i̸=k tr(RkRi) + σ2 k �K j=1 tr(Rj), (12) SE = ξMME(M − K)tr � Rk(RE + β3RB,E) � , (13) IE = (1 − ξ)(M − K − ME)tr(RE + β3RB,E) K � j=1 tr(Rj), (14) where Rk = β2,kRB,k + βI,kH1ΦRI,kΦHHH 1 and RE = βI,EH1ΦRI,EΦHHH 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ■ Remark 1: It is observed from Theorem 1 that the ergodic secrecy rate depends only on the statistical CSI of the users and Eve, phase shifts Φ, and power fraction ξ, motivating further optimization concerning Φ and ξ at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 3 Corollary 1: For uncorrelated Rayleigh fading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', RI,k = RI,E = IN and RB,k = RB,E = IM, we obtain γk and γE as (15) and (16) at the bottom of the next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' From (15), we observe that the inter-user interference always exists even with an infinite number of BS antennas M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' This is because the cascaded channels through the RIS for the multiple users are not asymptotically orthogonal due to the common component H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In addition, the RIS’s ability to modify the wireless medium is significantly impeded since the secrecy rate becomes independent of the RIS phase shifts Φ but only dependent on the size of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Corollary 2: When N ≫ M, we have H1HH 1 → β1NIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' By substituting ∥H1HH 1 ∥2 2 = β2 1N 2M into (15) and (16), the secrecy rate is given by (17) at the bottom of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We evince that the achievable rate of user k increases logarithmi- cally with the number of BS antennas and the capacity of Eve hardly changes with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' This implies that a promising secrecy performance gain is achieved for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Corollary 3: Without the existence of RIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', βI,k = 0 and βI,E = 0, the ergodic secrecy rate in (10) reduces to Rsec = � log2 � 1 + ξβ2 2,kPM 2/ �K j=1 β2,j ξPδ � i̸=k β2,itr(RB,kRB,i) + σ2 k � − log2 � 1 + ξME(M − K)tr � RB,kRB,E � (1 − ξ)M(M − K − ME) �K j=1 β2,j ��+ , (18) where δ = β2,k/ �K j=1 β2,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Specifically, when the spatial correlation at the BS disappears, the derived Rsec in (18) retrieves the result in [14, Theorem 1] as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' PROPOSED DESIGN FOR SECRECY RATE MAXIMIZATION In this section, we study the joint optimization of the power fraction ξ and phase shifts Φ to maximize the ergodic secrecy rate in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Mathematically, the optimization problem is formulated as (P1) max ξ,Φ Rsec(ξ, Φ) (19) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ξ ∈ [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' |φi| = 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', N, where φi = exp(jθi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It is challenging to jointly optimize Rsec(ξ, Φ) as it is a non-convex function of ξ and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' To address this, the alternating optimization (AO) technique is applied to optimize ξ and Φ by executing refinement processes with efficient closed-form calculations at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' First, we consider the optimization of ξ by fixing Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The following lemma provides a closed-form solution to the fixed- point equation for solving (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Lemma 1: For given Φ, the optimal solution of ξ is ξ∗ = −b + √ b2 − 4ac 2a , (20) where a = B1(A1A2 + A1A3 + A2 2) − A1A3, b = 2A3A1, and c = B1A2 3 −A1A3 are constants with respect to statistical channel spatial correlation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Proof: By taking the first derivative of Rsec in (10), it yields R′ sec = ∂Rsec ∂ξ = A1A3 ln 2(A3 + A2ξ)[A3 + (A1 + A2)ξ] − B1 ln 2(ξ − 1)[1 + (B1 − 1)ξ], (21) where B1 ≜ MME(M−K)ζ2tr � Rk(RE+β3RB,E) � K(M−K−ME)tr(RE+β3RB,E) �K j=1 tr(Rj), A1 ≜ Ptr(Rk)2, A2 ≜ P � i̸=k tr(RkRi), and A3 ≜ σ2 k �K j=1 tr(Rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Since ξ ∈ [0, 1], after some algebraic manipulations, it is easily checked that R′′ sec < 0, which implies that R′ sec is a strictly decreasing function on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Moreover, we have R′ sec > 0 for small ξ, while R′ sec < 0 for large ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Hence, there exists an optimal choice of ξ achieving the unique maximum of secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Therefore, considering the concavity of Rsec with respect to ξ, the optimal power fraction in (20) is obtained by solving R′ sec = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ■ Then, we optimize the RIS phase matrix Φ for fixed ξ, which is less tractable due to the unit-modulus constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Due to the complicated form of Rsec in (10), we apply the projected gradient ascent method to obtain a locally optimal solution, eventually converging to a stationary point [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Specifically at the lth step, denote by vl = [φl 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', φl n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', φl N]T the induced phases and by qk the adopted ascent direction, where [ql]n = ∂Rsec ∂φ∗ n with respect to φn = ejθn is obtained in the following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The subsequent (l + 1)th iteration step is updated according to ˜vl+1 = vl + µkql and vl+1 = exp � j arg �˜vl+1�� , (22) where µk is the step size computed at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Lemma 2: The gradient of the ergodic secrecy rate, Rsec, with respect to φn is computed as ∂Rsec ∂φ∗n = 1 ln 2 � ∂γk ∂φ∗n 1 + γk − ∂γE ∂φ∗n 1 + γE � , (23) where ∂γk ∂φ∗n and ∂γE ∂φ∗n are given in (28) and (29), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ■ Now by incorporating Lemma 1 and the gradient ascent method, concrete steps of the proposed algorithm are sum- marized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Proposition 1: The proposed algorithm always converges to a stationary point of (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Proof: This is directly checked by the following Rsec � ξ(t), Φ(t) � (a) ≥ Rsec � ξ(t−1), Φ(t) � (b) ≥ Rsec � ξ(t−1), Φ(t−1) � , (24) where (a) holds since the optimization of ξ is convex for given Φ, and (b) holds because the gradient search is along a monotonically increasing direction of Rsec [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ■ The algorithm comes with low computational complexity because it consists of simple matrix operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In particular, the complexity of Algorithm 1, depending on the computations involved in updating the power faction in (20) and the gradient in (23), is O(MN 2 + NM 2), which is lower compared with that of [6] under practical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' γk = ξP � β2 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ2 1M 2N 2 + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ1M 2N + β2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kM 2� ξP � i̸=k � β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='iM + (β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i) β1MN + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i∥H1HH 1 ∥2 2 � + σ2 k �K j=1 � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='jβ1MN + β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='jM � (15) γE = ξME(M − K) � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E∥H1HH 1 ∥2 2 + (βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ3 + β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E) β1MN + β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ3M � (1 − ξ)(M − K − ME) (β3 + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='Eβ1N) �K j=1 � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='jβ1MN + β2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='jM � (16) Rsec = � log2 � 1 + MβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kβ2 1 � i̸=k βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i � − log2 � 1 + ξME(M − K)βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='k (1 − ξ)(M − K − ME)β2 1 �K j=1 βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='j ��+ (17) 4 Algorithm 1 Proposed algorithm for solving P1 1: Initialize: v0 = exp(jπ/2)1N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Φ0 = diag(v0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' R0 sec = f(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Φ0) given by (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' ξ ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2: Repeat t ← t + 1 3: Find ξ(t) with fixed Φ(t−1) as per (20);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 4: for l = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', do 5: Find [ql]n = ∂Rsec ∂φ∗n , n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', N, as per (23) and µ by backtrack line search [15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 6: ˜vl+1 = vl + µql;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' vl+1 = exp(j arg(˜vl+1));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 7: Φl+1 = diag(vl+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Rl+1 sec = f(ξ(t), Φl+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 8: Until |Rsec(ξ(t), Φl+1) − Rsec(ξ(t), Φl)| < ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 9: Φ(t) = Φl+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 10: Until |Rsec(ξ(t), Φ(t)) − Rsec(ξ(t−1), Φ(t−1))| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, numerical simulations are provided to vali- date the effectiveness of the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The distance- dependent large-scale path loss coefficient is β = C0( d D0 )−ζ, where C0 = −20 dB is the path loss at the reference distance D0 = 1 m, d represents the individual link distance, and ζ denotes the path loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The pass-loss exponents for the RIS-aided links are set as 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='2 while the pass-loss exponent for the direct links is set as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The distance between the BS and RIS is set to be 20 m, and all the users and Eve are assumed to be located in a circular regime, whose center is 50 m away from the RIS and 60 m away from the BS, and the radius is 3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The spatial correlation matrices at the BS are generated according to [14] as [R(ρ)]i,j = ρ|i−j|, while the spatial correlation matrices at the RIS are given as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The RIS element spacing is given by dH = dV = λ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Also, the signal-to-noise ratio (SNR) is defined as 10 log 10(P/σ2 k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Unless otherwise specified, we also set σ2 k = σ2 E = −60 dBm, SNR = 5 dB, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='4, N = 256, M = 128, and K = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 1 illustrates that the derived analytical results and numerical results match well for varying number of Eve’s antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We observe that a higher number of Eve’s antennas degrades the secrecy rates as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' For comparison, we also depict the results with ZF precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It is shown that MRT outperforms ZF at low SNRs while for high SNR values ZF attains a higher secrecy rate since ZF offers interference- free communication to users in the high SNR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In the case of imperfect CSI, the estimated channel is modeled as zk = √ 1 − τ2ˆzk + τek by representing hk = R 1 2 k zk, where ˆzk is an imperfect observation of zk, ek is the Gaussian noise, and 0 < τ < 1 characterizes the CSI imperfection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We observe that the secrecy performance loss is marginal with estimation error τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='1 in the tested cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2 depicts the secrecy rate versus the power fraction for ME = 4, where the optimal value for ξ in (20) is marked by black stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' It is shown that ξ∗ is decreasing in the number of BS antennas M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', more power should be allocated to AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' This is because the correlation between hk and HE becomes strong with growing M due to the increasing dimension of H1, resulting in potentially more information leakage to Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' On the other hand, ξ∗ is increasing in the RIS size N, since the effective degree of freedom of the channels from RIS to users increases with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In this case, it can be useful to allocate less power to AN for improving the secrecy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 3 presents the secrecy performance of the proposed scheme versus N for ME = 2 by using equal power fraction [16] and random phase shifts [7] as benchmarks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 and θn ∼ U[0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' For small N, the optimal power fraction with random phase shifts achieves higher secrecy rates than the equal power fraction with random and optimal phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' This is because a small RIS provides limited signal energy boosting for the system where the system tends to be a power- limited scenario and this power fraction optimization plays a dominating role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' In addition, we notice that spatial correlation at the RIS should be taken into account to benefit from the phase shifts design in the case of statistical CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Also, it is shown that the secrecy rates decrease when the inter-element spacing reduces from λ/4 to λ/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' This is due to an increase in the spatial correlation, which reduces the spatial diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We observe a small performance gap by comparing the proposed algorithm with an approximate of the global optimum, which is achieved by running the algorithm twenty times with different initializations and then choosing the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 4 shows the ergodic secrecy rate versus the number of iterations for various numbers of RIS elements with ME = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We observe that the algorithm converges fast in all the tested cases, where the algorithm converges within 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' CONCLUSION We considered the secure communication in RIS-aided multiuser massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' A closed-form expression of the ergodic secrecy rate was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Then, based on the expression, we optimized RIS phase shifts and AN power fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' We showed that a large number of RIS elements and low spatial correlation at the RIS are preferred to achieve high secrecy rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Future works include extending to Rician and even millimeter-wave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' APPENDIX A For the sake of exposition, we denote the cascade channel of user k by hk = � βI,kH1ΦR1/2 I,k gI,k + � β2,kR1/2 B,kgB,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Since gI,k and gB,k are independent random vectors, we have that hk follows the complex Gaussian distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', hk ∼ CN(0, Rk), where Rk = β2,kRB,k + βI,kH1ΦRI,kΦHHH 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 1) Compute Rk: Consider MRT satisfying tr(WWH) = K, which leads to W = � K �K j=1 tr(Rj)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' First, we directly obtain ��E{hH k wk} ��2 = K �K j=1 tr(Rj) [tr (Rk)]2 and E ���hH k wi ��2� = K �K j=1 tr(Rj)tr (RkRi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Then, the variance is calculated as 1 M 2 var � hH k wk � = K �K j=1 tr(Rj) E ����� 1 M hH k hk − 1 M tr(Rk) ���� 2� M→∞ −−−−→ 0, (25) where (25) is obtained according to [17, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' For the term of E � hH k VVHhk � , it is obviously zero due to the null- space AN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2) Compute C: To begin with, we rewrite X in (8) as X = qX1 + qX2, where X1 ≜ HH B,EVVHHB,E and X2 ≜ (H1ΦHI,E)HVVH(H1ΦHI,E) are uncorrelated due to the definition in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Eigendecompose RB,E = UΛUH to decorrelate the channel matrix HB,E as Z = HB,EΛ−1/2UH, where Λ = diag(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', λN) contains the eigenvalues of R and the columns of U are the corresponding eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Since 5 10 8 6 4 2 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Ergodic secrecy rate versus SNR 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Ergodic secrecy rate versus power fraction 3264128 256 512 1024 0 1 2 3 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Ergodic secrecy rate versus the number of RIS elements 1 2 3 4 5 6 7 8 9 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='5 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Ergodic secrecy rate versus the number of iterations U is unitary, the statistics of ZU are identical to those of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Hence, the distribution of X1 is the same as �N i=1 �N j=1 λ1/2 i λ1/2 j zivivH j zH j , (26) where zi is the ith row of Z and vi is the ith column of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Considering that zi and vi are independent, it is known from [18] that �N n=1 λnznvnvH n zH n follows a Wishart distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=', �N n=1 λnWME(M − K, 1 M IME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' The distribution of X2 is obtained analogously by rewriting H1ΦHI,E = R1/2 E GI,E with RE = βI,EH1ΦRI,EΦHHH 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Then, by applying the Jensen’s inequality, the capacity of Eve is bounded as C ≤ log2 � 1 + pE � wH k HEX−1HH E wk �� (a) = log2 � 1 + ξM(M − K)E � wH k HEHH E wk � K(1 − ξ)(M − K − ME)tr(RE + β3RB,E) � (b) = log2 � 1 + ξζ2MME(M − K)tr � Rk(RE + β3RB,E) � K(1 − ξ)(M − K − ME)tr(RE + β3RB,E) � , (27) where (a) uses the property that A−1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' −−→ 1/(n − m)Im for a Wishart matrix A ∼ Wm(n, Im) with n > m [18, Sec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='6] and �N n=1 λn = tr(RE), and (b) results form E � wH k HEHH E wk � = ζ2MEtr � Rk(RE + β3RB,E) � where ζ2 = K �� �K j=1 tr(Rj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' APPENDIX B Using the standard quotient rule of derivatives, we have ∂γk ∂φ∗n = 1 I2 k � Ik ∂Sk ∂φ∗n − Sk ∂Ik ∂φ∗n � , (28) ∂γE ∂φ∗n = 1 I2 E � IE ∂SE ∂φ∗n − SE ∂IE ∂φ∗n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' (29) For simplicity, we use the notation (·)′ to represent the partial derivative with respect to φ∗ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Specifically, the term S′ k is given by S′ k = 2ξPtr(Rk)tr(R′ k), which requires a further derivation of tr(R′ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Since all terms in Rk depend on φ∗ n, we have tr � R′ k � = tr � β2,k ∂RB,k ∂φ∗n + βI,k ∂ � H1ΦRI,kΦHHH 1 � ∂φ∗n � (a) =βI,k � i,j � H1ΦRI,k � j,n[HH 1 ]T i,n = βI,k � HH 1 H1ΦRI,k � n,n, (30) where (a) is obtained by using Lemma 1 in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' the partial derivatives of Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Ik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' and IE with respect to φ∗ n in (28) and (29) are expressed as follows: S′ k = 2ξPβI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='ktr(Rk) � HH 1 H1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='k � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' I′ k = ξP � i̸=k � HH 1 � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='kRkH1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='k + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='iRiH1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='i �� n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n + σ2 k �k j=1 � HH 1 H1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='j � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' S′ E = ξMME(M − K) � HH 1 � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='k � RE + β3RB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E � H1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='k + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='ERkH1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E �� n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' and I′ E = (1 − ξ)(M − K − ME) � βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E �k j=1 tr(Rj) � HH 1 H1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n + tr(RE + β3RB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='E) �k j=1 � HH 1 H1ΦRI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='j � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content='n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' REFERENCES [1] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Wu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNE1T4oBgHgl3EQfeQRR/content/2301.03204v1.pdf'} +page_content=' Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless 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Research Asia +Shanghai, China +dongsheng.li@microsoft.com +Ming Gao∗ +East China Normal University +Shanghai, China +mgao@dase.ecnu.edu.cn +ABSTRACT +Generative graph self-supervised learning (SSL) aims to learn node +representations by reconstructing the input graph data. However, +most existing methods focus on unsupervised learning tasks only +and very few work has shown its superiority over the state-of- +the-art graph contrastive learning (GCL) models, especially on the +classification task. While a very recent model has been proposed +to bridge the gap, its performance on unsupervised learning tasks +is still unknown. In this paper, to comprehensively enhance the +performance of generative graph SSL against other GCL models on +both unsupervised and supervised learning tasks, we propose the +SeeGera model, which is based on the family of self-supervised vari- +ational graph auto-encoder (VGAE). Specifically, SeeGera adopts +the semi-implicit variational inference framework, a hierarchical +variational framework, and mainly focuses on feature reconstruc- +tion and structure/feature masking. On the one hand, SeeGera +co-embeds both nodes and features in the encoder and reconstructs +both links and features in the decoder. Since feature embeddings +contain rich semantic information on features, they can be com- +bined with node embeddings to provide fine-grained knowledge +for feature reconstruction. On the other hand, SeeGera adds an +additional layer for structure/feature masking to the hierarchical +variational framework, which boosts the model generalizability. We +conduct extensive experiments comparing SeeGera with 9 other +state-of-the-art competitors. Our results show that SeeGera can +compare favorably against other state-of-the-art GCL methods in a +variety of unsupervised and supervised learning tasks. +KEYWORDS +Graph neural networks, graph self-supervised learning, variational +graph auto-encoder +∗Corresponding author +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2023 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/10.1145/nnnnnnn.nnnnnnn +ACM Reference Format: +Xiang Li, Tiandi Ye, Caihua Shan, Dongsheng Li, and Ming Gao. 2023. Self- +supervised Semi-implicit Graph Variational Auto-encoders with Masking. +In Proceedings of ACM Conference (Conference’17). ACM, New York, NY, +USA, 12 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn +1 +INTRODUCTION +Self-supervised learning (SSL) [2, 6, 9, 36] has attracted significant +attention recently. By extracting and employing supervisions from +data itself, SSL can heavily reduce the dependence of neural network +models on the labeled data, which is costly to obtain. To facilitate +graph-based learning, SSL has been applied on graph-structured +data. For example, it can learn representations for nodes (e.g., web +pages in search engines), and detect the anomalies on webs (e.g., +malicious users) [16]. Recently, graph contrastive learning (GCL), as +one of the main SSL types, has experienced a surge [10, 26, 29, 35]. +The core idea of GCL is to first construct positive and negative pairs +for nodes, and then maximize the similarity between positive pairs +while minimizing that between negative ones1. +Despite the success, existing GCL methods suffer from two main +problems. On the one hand, negative samples are needed in most +contrastive objectives, which generally construct one positive and +𝐾 negative samples for each node. However, these models are eas- +ily affected by the value of 𝐾. When 𝐾 is small, the model cannot +learn sufficient discriminative information, which degrades the +model effectiveness; otherwise, there could lead to a large number +of false-negative samples and slow convergence. Generally, 𝐾 is +set empirically and there lack theoretical supports. On the other +hand, for the rest of methods based on positive pairs only, they +are easily trapped into a degenerate solution [40], where all the +output embeddings of nodes collapse to a constant. To tackle the +issue, additional strategies are necessary, such as asymmetric dual +encoders with momentum updates and exponential moving aver- +age [21, 27]. Recently, some studies [14] have showed that although +these training strategies can alleviate collapse to some extent, they +may still cause collapse in partial dimensions of the representation, +which leads to worse performance. +To address the shortcomings of GCL methods, generative graph +SSL methods can be used instead. In particular, self-supervised +graph auto-encoders (GAEs) [12], whose objective is to reconstruct +1Note that some GCL methods require positive pairs only and they only maximize the +similarity between positive pairs. +1 +arXiv:2301.12458v1 [cs.LG] 29 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Xiang Li, et al. +the input graph data, have been widely studied. Existing methods +mainly differ in their adopted reconstruction components, such +as the adjacency matrix reconstruction [19], the node feature re- +construction [20] and a combination of both graph structure and +node feature reconstruction [23]. However, most of these methods +focus on the unsupervised learning tasks like link prediction and +node clustering, and very few work has shown its superiority over +the state-of-the-art GCL methods, especially on the classification +task. While a masked GAE model GraphMAE [7] is very recently +proposed to bridge the gap, its performance on the unsupervised +learning tasks is still unexplored. Since the goal of SSL is to learn +versatile representations, a further study on self-supervised GAE +model that can achieve comprehensive superiority on both unsu- +pervised and supervised learning tasks is needed. Further, although +GraphMAE is an auto-encoding method, it is based on GAE and +is essentially not a generative model. This also calls our attention +back to the study of generative graph SSL model, such as variational +graph auto-encoder (VGAE) [12]. +Different from GAE, VGAE consists of an inference model and a +generative model. Specifically, the inference model encodes obser- +vations (links and features) into latent variables (node embeddings) +while the generative model decodes from these latent variables +to reconstruct links. However, as pointed out in [7], node feature +reconstruction is beneficial for learning high-quality representa- +tions. Therefore, the lack of feature reconstruction could degrade +the model effectiveness. To solve the issue, most existing methods +adopt MLP [8, 9] and GNNs [7, 20] as their decoders for feature +reconstruction. However, they utilize node-level embeddings only +and ignore feature-level embeddings that contain rich semantic +information on node features and can be used to help feature recon- +struction. Recently, CAN [18] is proposed to co-embed both nodes +and features, and use the inner product of their embeddings as the +decoder to recover node features. Despite the success, it has three +main problems. First, the linear decoder is generally less powerful +than MLP and GNNs, which restricts the model’s capability in re- +constructing node features. Second, it assumes the independence +between node and feature embeddings in the variational inference +stage, but practically these two types of embeddings are highly +correlated. Third, it lacks structure/feature masking in the learning +process, which has been shown to degrade the model’s performance +on the classification task [7]. +In this paper, we study generative graph SSL and our goal is to en- +hance the family of self-supervised VGAE on graph representation +learning in a variety of downstream tasks. Recently, semi-implicit +variational inference (SIVI) [32], which is a hierarchical variational +framework, has been applied to VGAE to model a wide range of un- +derlying true posteriors with multi-modality, skewness and heavy +tails [4]. We thus adopt the framework to remove the explicit Gauss- +ian restriction on the variational distribution and mainly focus +on the component of feature reconstruction and structure/feature +masking. We propose a Self-supervised semi-implicit Graph varia- +tional auto-encoder with masking, namely, SeeGera. Specifically, +the model co-embeds both nodes and features in the encoder and +jointly reconstructs links and features in the decoder. Note that the +feature embeddings can provide fine-grained information that is +supplementary to the node embeddings when reconstructing node +features. Specifically, for each node, we take its feature values as +weights and compute the weighted average of feature embeddings +w.r.t. the node. The weighted embedding characterizes the affini- +ties between the node and all the features. After that, we combine +the weighted embedding with the node embedding, and feed the +fused embedding into GNNs to reconstruct the node’s features. +Further, to generate node and feature embeddings in the encoder, +we first assume the independence between them and propose the +base SeeGera model. Then we upgrade the model by capturing +the correlations between node and feature embeddings. Finally, we +add an additional layer to the hierarchical variational framework +to integrate SeeGera with the masking mechanism and boost the +model performance. In summary, our main contributions are listed: +• We propose a generative graph SSL model SeeGera. To our knowl- +edge, this is the first generative graph SSL method that is compre- +hensively compared with the SOTA GCL models in terms of both +unsupervised and supervised learning tasks, and shows superiority. +• We present a novel feature reconstruction method that lever- +ages both node and feature embeddings to provide fine-grained +information for reconstructing features. We further introduce the +structure/feature masking mechanism by adding an additional layer +to the hierarchical variational framework. +• We conduct extensive experiments to evaluate the performance +of SeeGera on two unsupervised learning tasks: link prediction +and attribute inference, and one supervised learning task: node +classification. Experimental results show that SeeGera can signifi- +cantly outperform other competitors on both link prediction and +attribute inference tasks, and perform comparably with them in +node classification. This effectively verifies the power of generative +graph SSL in graph representation learning. +2 +RELATED WORK +In this section, we summarize the related work on both graph self- +supervised learning and generative graph self-supervised learning, +respectively. +2.1 +Graph self-supervised learning +Graph self-supervised learning [7, 26, 31, 35] aims to employ super- +visions extracted from graph-structured data without the need for +annotated data. Existing methods can be mainly divided into four +types: (1) generative models [12], whose objective is to reconstruct +the input graph data. (2) auxiliary-property-based methods [36], +which first obtain graph-related properties and then take them as +supervisions, such as the pseudo labels of unlabeled nodes; (3) con- +trastive models [29], which construct positive and negative pairs +for contrast. (4) hybrid approaches [38], which combine the objec- +tives of the first three types in a multi-task learning fashion. For a +comprehensive survey on graph self-supervised learning, see [15]. +Recently, graph contrastive learning has been widely studied. +According to whether negative samples are used in the learning pro- +cess, existing methods include negative-sample-based and negative- +sample-free ones. For the former, DGI [29] and InfoGraph [26] em- +ploy corruptions to construct negative pairs. GRACE [41], GCA [42] +and GraphCL [35] take samples in a mini-batch as a dictionary +whose size is constrained by the batch size and consider other sam- +ples in the same mini-batch as negatives of a sample, while GCC [21] +maintains a dynamic dictionary with larger size as in MoCo [6]. +2 + +Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking +Conference’17, July 2017, Washington, DC, USA +For the latter, BGRL [27] and CCA-SSG [37] are two representative +models that are based on asymmetric encoding architectures. How- +ever, they require special training strategies to avoid the collapse +of learned node embeddings to a constant, such as momentum up- +date [6], exponential moving average [27] and stop gradient [27]. +Further, existing GCL methods heavily rely on graph augmentation +strategies to construct different graph views for contrast, including +feature-oriented (e.g., masking [35] and shuffling [29]), proximity- +oriented (e.g., perturbation [35]), and graph-sampling-based (e.g., +random-walk [5]) augmentations. +2.2 +Generative graph self-supervised learning +Generative graph self-supervised learning aims to take the input +graph as self-supervision and recover the input data. It mainly con- +sists of two families of models: graph autoregressive models and +graph autoencoders (GAEs). Autoregressive models [33, 34] decom- +pose joint probability distributions as a product of conditionals. The +representative graph autoregressive model is GPT-GNN [9], which +takes attributed graph generation as its objective. However, since +autoregressive models require an explicit ordering to generate, they +might not work well on graphs that do not exhibit inherent orders. +Different from graph autoregressive models, GAEs do not require +any decoding ordering and they aim to reconstruct part of the input +graph data. According to the reconstructed components, existing +self-supervised GAE methods include those that reconstruct links +only (e.g., ARVGA [19], GAE [12], VGAE [12]), features only (e.g., +GraphMAE [7], GALA [20], MGAE [30], EP [3]), and a combination +of both links and features (e.g., GATE [23], CAN [18], DGE [39]). +However, most of these methods focus on the link prediction and +node clustering tasks, and few of them compares favorably against +the state-of-the-art GCL methods, especially in the classification +task. While GraphMAE is recently proposed to bridge the gap, its +performance on unsupervised learning tasks remains unexplored. +Further, it is based on GAE and is essentially not a generative model. +Different from GAE, variational graph auto-encoder (VGAE) is a +generative model that recovers links only in the decoder. While +there exist some self-supervised VGAE models that reconstruct fea- +tures [23, 39], most of them only leverage node-level embeddings +but ignore feature-level embeddings that contain fine-grained infor- +mation for node features and can help boost feature reconstruction. +In this paper, we reconsider generative graph self-supervised learn- +ing and show that self-supervised VGAE can outperform or perform +comparably against other SOTA GCL models in a variety of tasks, +such as link prediction, attribute inference and node classification. +3 +PRELIMINARY +3.1 +Notations +Let G = (V, E) denote a graph, where V = {𝑥𝑖}𝑛 +𝑖=1 is a set of nodes +and E ⊆ V × V is a set of edges. Let A be the adjacency matrix of +𝐺, such that A𝑖𝑗 represents the weight of edge 𝑒𝑖𝑗 between objects +𝑥𝑖 and 𝑥𝑗. For simplicity, we set A𝑖𝑗 = 1 if 𝑒𝑖𝑗 ∈ E; 0, otherwise. +Further, since nodes in a graph are usually associated with features, +we denote F = {𝑓𝑟 }𝑙 +𝑟=1 as a set of node features and X ∈ R𝑛×𝑙 as +the node feature matrix, where the 𝑖-th row X𝑖 is the feature vector +of node 𝑥𝑖. For the node representation matrix, let it be ZV ∈ R𝑛×𝑑, +where 𝑑 is the output embedding dimension satisfying 𝑑 ≪ |V|. +Note that the 𝑖-th row ZV +𝑖 +represents the embedding of node 𝑥𝑖. +Similarly, ZF ∈ R𝑙×𝑑 denotes the feature representation matrix, +whose 𝑟-th row ZF𝑟 is the embedding of node feature 𝑓𝑟. In this +paper, we learn both node and feature representations, and use +node representations in various downstream tasks. +3.2 +SIVI and SIG-VAE +Given observations Y and latent variable Z, the vanilla variational +inference (VI) derives an evidence lower bound +ELBO = −EZ∼𝑞(Z|𝜓) [log𝑞(Z|𝜓) − log𝑝(Y, Z)] , +(1) +where 𝜓 is variational parameter, 𝑞(Z|𝜓) is variational distribution +and 𝑝(Y, Z) is joint distribution. However, VI restricts an exponen- +tial family assumption to the posterior. To address the problem, +semi-implicit variational inference (SIVI) [32] considers variational +parameters as random variables drawn from a mixing distribution. +Specifically, the semi-implicit variational distribution for Z is de- +fined in a hierarchical manner, which follows Z ∼ 𝑞(Z|𝜓) and +𝜓 ∼ 𝑞𝜙 (𝜓). Here, 𝜙 is the parameter of the mixing distribution +𝑞𝜙 (𝜓). Further, 𝜓 can be marginalized out to derive a distribution +family H indexed by 𝜙 for Z: +H = +� +ℎ(Z) : ℎ(Z) = +∫ +𝜓 +𝑞(Z|𝜓)𝑞𝜙 (𝜓)d𝜓 +� +. +(2) +Note that 𝑞(Z|𝜓) is required to be explicit, but the mixing distri- +bution 𝑞𝜙 (𝜓) is allowed to be implicit. Moreover, the marginal +distribution ℎ(Z) ∈ H is often implicit unless 𝑞𝜙 (𝜓) is conjugate +to 𝑞(Z|𝜓). These are the reasons why the method is referred to as +“semi-implicit” VI. To maintain simple optimization, 𝑞(Z|𝜓) is re- +quired to be either reparameterizable [11] or allow the ELBO under +𝑞(Z|𝜓) to be analytic. For 𝑞𝜙 (𝜓), it needs to be reparameterizable. +Generally, SIVI draws from 𝑞𝜙 (𝜓) by injecting random noise 𝜖 into +node features and transforming the features via neural networks. +Recently, Hasanzadeh et al. [4] apply SIVI to VGAE and propose +the semi-implicit graph variational auto-encoder (SIG-VAE) model. +Specifically, it sets 𝑞(Z|𝜓) to be Gaussian distribution and uses +GNNs to characterize the mixing distribution𝑞𝜙 (𝜓). While SIG-VAE +uses the hierarchical variational framework to capture complex non- +Gaussian posteriors, it still has the problem of ignorance of feature +reconstruction and structure/feature masking. Therefore, based on +the framework of SIG-VAE, we next explore how to enhance self- +supervised VGAE for unsupervised graph representation learning. +4 +ALGORITHM +In this section, we present our model SeeGera. Different from SIG- +VAE that uses node embeddings only, SeeGera further generates +feature embeddings to capture the rich semantic information on +node features, which can be used to enhance feature reconstruction. +Specifically, we consider two cases in the encoder when generating +node and feature embeddings: (1) they are independent; (2) they +are correlated. After that, in the decoder part, we utilize GNNs +to reconstruct node features based on both node and feature em- +beddings. Finally, we show how structure/feature masking can be +integrated with the hierarchical variational framework and gives +the optimization techniques. The overall framework of SeeGera is +summarized in Figure 1. +3 + +Conference’17, July 2017, Washington, DC, USA +Xiang Li, et al. +4 +0 +3 +2 +1 +GCN +MLP +GCN +Encoding +Decoding +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +MASKING +NOISE +INJECTION +SAMPLING +Ber(0.5) +[Inner Product] +[Combine] +[Weighted Average] +[Independent] +[Correlated] +Figure 1: The overall framework of SeeGera. +4.1 +Variational lower bound +In VI, given a graph G with an adjacency matrix A and a feature +matrix X, we approximate the true posterior 𝑝(ZV, ZF|A, X) with +a variational distribution 𝑞(ZV, ZF|𝜓1,𝜓2), where 𝜓1 and 𝜓2 are +variational parameters. To capture more complex posteriors that go +beyond the exponential family, we adopt the hierarchical variational +framework in SIVI and assume +ZV ∼ 𝑞1(ZV |𝜓1), 𝜓1 ∼ 𝑞𝜙1 (𝜓1), ZF ∼ 𝑞2(ZF |𝜓2), 𝜓2 ∼ 𝑞𝜙2 (𝜓2), +(3) +where𝜙1 and𝜙2 are parameters of mixing distributions. We marginal- +ize 𝜓1 and 𝜓2 out and derive +ZV ∼ ℎ𝜙1 (ZV) = +∫ +𝜓1 +𝑞1(ZV |𝜓1)𝑞𝜙1 (𝜓1)d𝜓1, +ZF ∼ ℎ𝜙2 (ZF) = +∫ +𝜓2 +𝑞2(ZF |𝜓2)𝑞𝜙2 (𝜓2)d𝜓2. +(4) +We maximize the log-likelihood of observations A and X, and use +Jensen’s inequality to get +log𝑝(A, X) ≥ Eℎ𝜙 (ZV,ZF) +� +log 𝑝(A, X, ZV, ZF) +ℎ𝜙 (ZV, ZF) +� += L, +(5) +where L is ELBO and +ℎ𝜙 (ZV, ZF) = +∫ +𝜓1 +∫ +𝜓2 +𝑞(ZV, ZF|𝜓1,𝜓2)𝑞𝜙 (𝜓1,𝜓2)d𝜓1d𝜓2 +(6) +is the marginal distribution over ZV and ZF. Since ℎ𝜙 is often +intractable, the Monte Carlo estimation of ELBO could be prohibited. +To address the problem, we first take the mean-field assumption: +𝑞(ZV, ZF|𝜓1,𝜓2) = 𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2), +𝑞𝜙 (𝜓1,𝜓2) = 𝑞𝜙1 (𝜓1)𝑞𝜙2 (𝜓2), +(7) +and substitute Eq. 7 into Eq. 6 to get: +ℎ𝜙 (ZV, ZF) = ℎ𝜙1 (ZV)ℎ𝜙2 (ZF). +(8) +From Eq. 8, we see that ZV and ZF are independent. Then we +derive a lower bound for the ELBO based on Eq. 8: +L = Eℎ𝜙1 (ZV)Eℎ𝜙2 (ZF) +� +log 𝑝(A, X, ZV, ZF) +ℎ𝜙1 (ZV)ℎ𝜙2 (ZF) +� +≥ E𝜓1∼𝑞𝜙1 (𝜓)EZV∼𝑞1(ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓)EZF∼𝑞2(ZF |𝜓2) +� +log +𝑝(A, X, ZV, ZF) +𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2) +� += L1 +(9) +Details on the derivation of Equation 9 are deferred to Appendix E. +In L1, 𝑞1 and 𝑞2 are required to be explicit and have analytic den- +sity function, while 𝑞𝜙1 and 𝑞𝜙2 could be implicit but have to be +convenient to be sampled from. Directly optimizing L1 by Monte +Carlo Estimation is much easier. +However, in practice, nodes and their features are highly cor- +related. On the one hand, node embeddings are generated based +on features. On the other hand, the semantic information of fea- +tures are directly reflected by nodes. Therefore, the independence +between ZV and ZF in Equation 8 is inappropriate. To tackle the +issue, we modify Eq. 7 into: +𝑞(ZV, ZF|𝜓1,𝜓2) = 𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2), +𝑞𝜙 (𝜓1,𝜓2) = 𝑞𝜙2 (𝜓2|𝜓1)𝑞𝜙1 (𝜓1), +(10) +which explicitly characterizes the dependence between variational +parameters𝜓1 and𝜓2. In this way,ℎ𝜙 (ZV, ZF) ≠ ℎ𝜙1 (ZV)ℎ𝜙2 (ZF), +which shows that ZV and ZF are correlated. Then we can derive +another lower bound for the ELBO in Equation 5: +L ≥ E𝜓1∼𝑞𝜙1 (𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2 |𝜓1)E(ZV,ZF)∼𝑞(ZV,ZF |𝜓1,𝜓2) +� +log 𝑝(A, X, ZV, ZF) +𝑞(ZV, ZF|𝜓1,𝜓2) +� += L2 +(11) +4 + +Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking +Conference’17, July 2017, Washington, DC, USA +4.2 +Encoder +In the encoder, we generate ZV and ZF from observations A and X. +We next show how to generate ZV and ZF according to whether +they are independent or not. +[ZV and ZF are independent]. To generate ZV, we assume +that 𝑞1(ZV |𝜓1) = �𝑛 +𝑖=1 𝑞1(ZV +𝑖 |𝜇V +𝑖 , ΣV +𝑖 ), where 𝑞1(ZV +𝑖 |𝜇V +𝑖 , ΣV +𝑖 ) += N (𝜇V +𝑖 , ΣV +𝑖 ) and N is multivariate Gaussian distribution with +mean 𝜇V +𝑖 +and diagonal co-variance matrix ΣV +𝑖 . Since 𝜇V +𝑖 +and ΣV +𝑖 +are random variables, we draw them by injecting noise ˜𝜖 into a +GNN model: +˜X = CONCAT(X, ˜𝜖), ˜𝜖 ∼ ˜𝑞(𝜖), [𝜇V +𝑖 , ΣV +𝑖 ] = GNN1(A, ˜X), +(12) +where CONCAT(·) is the concatenation function and GNN1(·) is a +GNN model. Note that ˜𝜖 is random noise sampled from distribution +˜𝑞(𝜖), whose row size should be the same as X. The injected noise ˜𝜖 +enables the uncertainty propagation between neighboring nodes in +the GNN layer, which drives the outputs of the GNN to be random +variables rather than deterministic values. Similarly, for ZF, we +assume 𝑞2(ZF|𝜓2) = �𝑙 +𝑟=1 𝑞2(ZF𝑟 |𝜇 F +𝑟 , ΣF𝑟 ) with 𝑞2(ZF𝑟 |𝜇 F +𝑟 , ΣF𝑟 ) = +N (𝜇 F +𝑟 , ΣF𝑟 ). To infer 𝜇 F +𝑟 and ΣF𝑟 , we use a MLP model: +ˆX = CONCAT(X𝑇, ˆ𝜖), ˆ𝜖 ∼ ˆ𝑞(𝜖), [𝜇 F +𝑟 , ΣF +𝑟 ] = MLP1( ˆX). +(13) +Note that X𝑇 ∈ R𝑙×𝑛 and the 𝑟-th row X𝑇𝑟 ∈ R𝑛 can be considered +as the feature vector of feature 𝑓𝑟 . The random noise ˆ𝜖 drawn from +ˆ𝑞(𝜖) injects uncertainty to the matrix X𝑇 , which models 𝜇 F +𝑟 and +ΣF𝑟 as random variables. +[ZV and ZF are correlated]. We also assume 𝑞1 and 𝑞2 follow +Gaussian distribution and use the same method as in Equation 12 to +generate [𝜇V +𝑖 , ΣV +𝑖 ]. However, to capture the dependence between +𝜓1 and 𝜓2 2 in Equation 10, we compute [𝜇 F +𝑟 , ΣF𝑟 ] for feature 𝑓𝑟 +based on node embeddings. Specifically, since the rich semantic +information of each feature is directly reflected by values of nodes +in the feature, we take the feature vector X𝑇𝑟 ∈ R𝑛 as the weight +vector over all the nodes, and compute: +[𝜇 F +𝑟 , ΣF +𝑟 ] = MLP2 +� �𝑛 +𝑖=1 X𝑇 +𝑟𝑖 [𝜇V +𝑖 , ΣV +𝑖 ] +�𝑛 +𝑖=1 X𝑇 +𝑟𝑖 +� +. +(14) +In this way, [𝜇 F +𝑟 , ΣF𝑟 ] is derived from node embeddings. Since 𝜇V +𝑖 +and ΣV +𝑖 +are random variables, 𝜇 F +𝑟 and ΣF𝑟 will also be random +variables. +4.3 +Decoder +In the decoder, we aim to reconstruct both edges and features in +the given graph. The generative process is described as follows. +First, for each node 𝑥𝑖 and each feature 𝑓𝑟 , we draw (ZV +𝑖 , ZF𝑟 ) ∼ +ℎ𝜙 (ZV +𝑖 , ZF𝑟 )3. Second, for each edge A𝑖𝑗 in the adjacency matrix +A, draw A𝑖𝑗 ∼ Ber(𝑝A +𝑖𝑗). Here, Ber(·) denotes Bernoulli distribu- +tion and 𝑝A +𝑖𝑗 is the probability for the existence of edge A𝑖𝑗. We +implement 𝑝A +𝑖𝑗 simply by inner product as: 𝑝A +𝑖𝑗 = 𝜎((ZV +𝑖 )𝑇 ZV +𝑗 ), +where 𝜎 is the sigmoid function. Third, for each attribute X𝑖𝑟 in the +2Here, we denote 𝜓1 = [𝜇V, ΣV ] and 𝜓2 = [𝜇F, ΣF ], respectively. +3When ZV +𝑖 +and ZF +𝑟 are independent, we draw ZV +𝑖 +∼ ℎ𝜙1 (ZV +𝑖 ) and ZF +𝑟 ∼ ℎ𝜙2 (ZF +𝑟 ). +attribute matrix X, draw X𝑖𝑟 ∼ N (𝜇X +𝑖𝑟, ΣX +𝑖𝑟 |ZV +𝑖 , ZF𝑟 ). Here, 𝜇X +𝑖𝑟, ΣX +𝑖𝑟 +are functions of ZV and ZF. +We next introduce how to compute 𝜇X +𝑖𝑟 and ΣX +𝑖𝑟. Since ZF𝑟 con- +tains rich semantic information on feature 𝑓𝑟, the affinity between +𝑥𝑖 and 𝑓𝑟 can provide fine-grained knowledge for feature reconstruc- +tion. Given a node 𝑥𝑖, to capture its affinities with all the features, +the attention mechanism [28] can be applied on node and feature +embeddings. However, this will increase the time complexity of the +model. For simplicity, we directly use the feature vector X𝑖 ∈ R𝑙 of +𝑥𝑖 as the weight vector and calculate the weighted average over all +the feature embeddings: +¯ZV +𝑖 += +�𝑙 +𝑟=1 X𝑖𝑟ZF𝑟 +�𝑙 +𝑟=1 X𝑖𝑟 +. +(15) +Compared with ZV +𝑖 , ¯ZV +𝑖 +contains more details on how each feature +can be reconstructed. After that, we combine ZV +𝑖 +and ¯ZV +𝑖 +to get: +ZV +𝑖 += COMBINE(ZV +𝑖 , ¯ZV +𝑖 ). +(16) +In our experiments, we set the COMBINE function to be CONCAT. +Finally, the updated ZV +𝑖 +is taken as input and fed into a GNN model +to learn parameters w.r.t. node 𝑥𝑖: +[𝜇X +𝑖 , ΣX +𝑖 ] = GNN2(A, ZV +𝑖 ), +(17) +where GNN2(·) is a GNN model. +4.4 +Masking +To further improve the model generalizability, we introduce the +masking mechanism in SeeGera by adding an additional layer to +the hierarchical variational framework. Specifically, we transform +Equation 6 into: +ℎ𝜙 (ZV, ZF) = +∫ +𝜓1 +∫ +𝜓2 +∫ +˜𝐺 +𝑞(ZV, ZF|𝜓1,𝜓2)· +𝑞𝜙 (𝜓1,𝜓2| ˜𝐺)𝑝( ˜𝐺|A, X)d𝜓1d𝜓2d ˜𝐺 +(18) +From the above equation, we see that in addition to 𝜓1 and 𝜓2, the +integration is performed over a new variable ˜𝐺 and a probability +function 𝑝( ˜𝐺|A, X). Here, ˜𝐺 denotes a new graph and 𝑝 is the graph +augmentation probability function. The equation can lead to a new +variational lower bound for the ELBO, but it is more difficult to +optimize compared with L1 and L2. To tackle the issue, we can +first perform graph augmentation and generate a perturbed graph +˜𝐺. After that, based on ˜𝐺, node and feature embeddings are learned +based on L1 or L2. We repeat the above process until convergence. +Although graph augmentation can include more operations than +masking, we mainly focus on structure/feature masking in this +paper, because masking is beneficial for node classification [7]. +4.5 +Optimization +In Section 4.1, we have derived two lower bounds L1 and L2 for +the ELBO, according to whether ZV and ZF are independent. For +notation brevity, we use L to overload both L1 and L2. However, +directly optimizing L could lead to the degeneracy problem [32] +that 𝑞𝜙1 (𝜓1), 𝑞𝜙2 (𝜓2) and 𝑞𝜙 (𝜓1,𝜓2) might converge to a point +mass density, which degenerates SIVI to the vanilla VI. To address +5 + +Conference’17, July 2017, Washington, DC, USA +Xiang Li, et al. +the problem, we can regularize L by 𝐵𝐾: +𝐵𝐾 = E(𝜓1,𝜓2),{( ˜𝜓𝑘 +1 , ˜𝜓𝑘 +2 )}𝐾 +𝑘=1∼𝑞𝜙 (𝜓1,𝜓2)DKL +� +𝑞(ZV, ZF |𝜓1,𝜓2) || ˜ℎ𝐾 (ZV, ZF) +� +˜ℎ𝐾 (ZV, ZF) = +𝑞(ZV, ZF|𝜓1,𝜓2) + �𝐾 +𝑘=1 𝑞(ZV, ZF| ˜𝜓𝑘 +1 , ˜𝜓𝑘 +2 ) +𝐾 + 1 +. +Note that 𝐵𝐾 satisfies (1) 𝐵𝐾 ≥ 0; (2) 𝐵𝐾 = 0 if and only if 𝐾 = 0 or +𝑞𝜙 degenerates to a point mass density. According to [32], L𝐾 = +L + 𝐵𝐾 is an asymptotically exact surrogate ELBO that satisfies +L0 = L and lim𝐾→∞ L𝐾 = L. Maximizing L𝐾 with 𝐾 ≥ 1 +derives positive 𝐵𝐾 and could drive 𝑞𝜙 away from degeneracy. +Moreover, importance reweighting [1] can be further introduced to +tighten L𝐾 by drawing 𝐽 samples {(ZV)𝑗, (ZF)𝑗,𝜓 𝑗 +1,𝜓 𝑗 +2 }𝐽 +𝑗=1 from +𝑞(ZV, ZF,𝜓1,𝜓2). The objective can be formulated as +L𝐽 +𝐾 =E{(ZV ) 𝑗,(ZF ) 𝑗,𝜓 𝑗 +1 ,𝜓 𝑗 +2 }𝐽 +𝑗=1∼𝑞(ZV,ZF |𝜓1,𝜓2)𝑞𝜙 (𝜓1,𝜓2) +E{ ˜𝜓𝑘 +1 , ˜𝜓𝑘 +2 }𝐾 +𝑘=1∼𝑞𝜙 (𝜓1,𝜓2) log 1 +𝐽 +𝐽∑︁ +𝑗=1 +𝑝 (A, X, (ZV)𝑗, (ZF)𝑗) +Ω𝑗 +, +(19) +where +Ω𝑗 = +1 +𝐾 + 1 +� +𝑞((ZV)𝑗, (ZF)𝑗 |𝜓 𝑗 +1 ,𝜓 𝑗 +2 ) + +𝐾 +∑︁ +𝑘=1 +𝑞((ZV)𝑗, (ZF)𝑗 | ˜𝜓𝑘 +1 , ˜𝜓𝑘 +2 ) +� +. +Then we take L𝐽 +𝐾 as the surrogate ELBO and use stochastic gradient +ascent to optimize it. Note that L1 and L2 treats differently for +𝑞𝜙 (𝜓1,𝜓2). Finally, we summarize the pseudocodes of SeeGera in +Algorithm 1 (see Appendix B). +[Complexity analysis] The major time complexity in the en- +coder comes from GNN and MLP. Suppose we use GCN as the GNN +model. Since the adjacency matrix is generally sparse, let 𝑑𝐴 be the +average number of non-zero entries in each row of the adjacency +matrix. Let 𝑙 be the number of features and 𝑑 be the embedding +dimension. Further, we denote ˜𝑑 and ˆ𝑑 as the dimensions of injected +noise to the GCN and MLP, respectively. Then, the time complexi- +ties for GCN and MLP are 𝑂(𝑛𝑑𝐴(𝑙 + ˜𝑑)+𝑛(𝑙 + ˜𝑑)𝑑) and 𝑂(𝑙(𝑛+ ˆ𝑑)𝑑), +respectively. In the decoder, suppose we still adopt GCN as the GNN +model. Then the time complexities for reconstructing links and fea- +tures are 𝑂(𝑛2𝑑) and 𝑂(𝑛𝑑𝐴𝑑 + 𝑛𝑑𝑙), respectively. As suggested +by [12], we can down-sample the number of nonexistent edges in +the graph to reduce the time complexity for recovering links. +5 +EXPERIMENT +In this section we comprehensively evaluate the quality of node +embeddings learned by SeeGera. We mainly study four research +questions: +(RQ1) How does SeeGera perform in the link prediction task? +(RQ2) Can SeeGera effectively predict node attributes? +(RQ3) While SeeGera is an unsupervised learning method, can +it perform well when generalized to the node classification task? +(RQ4) How does structure/feature masking influence the perfor- +mance of SeeGera? +5.1 +Datasets and Baselines +To answer the above four questions, we conduct extensive experi- +ments on seven public datasets: Cora, Citeseer, Pubmed, Coauthor CS, +Coauthor Physics, Amazon Computer and Amazon Photo. Detailed +descriptions and statistics on these datasets are provided in Ap- +pendix A. We also compare SeeGera with 9 other SOTA baselines, +which can be categorized into two groups: +•[Generative graph SSL methods]. This group of methods are +based on GAE/VGAE and aim to reconstruct links and/or features, +including SIG-VAE [4], CAN [18], GATE [23] and GraphMAE [7]. +Note that GraphMAE is the SOTA generative graph SSL model. +•[Graph contrastive learning methods]. Models in this type +construct positive (and negative) pairs for contrast to learn node rep- +resentations, including DGI [29], MVGRL [5], GRACE [41], GCA [42] +and CCA-SSG [37]. +5.2 +Implementation Details +We implemented SeeGera by PyTorch. The model is initialized +by Glorot initialization and trained by Adam. We run the model +for 3500 epochs on all the datasets. In particular, for the task of +node classification, we further run the logistic regression classifier +for 500 epochs. We adopt GCNs in both encoder and decoder for +all the datasets except CS. For CS, we use MLPs to replace GNNs +instead. We choose Bernoulli noise Ber(0.5) for both ˜𝜖 ∼ ˜𝑞(𝜖) and +ˆ𝜖 ∼ ˆ𝑞(𝜖), and set the noise dimension to 5 in all tasks. We fine-tune +the model hyper-parameters by grid search and details are given +in Appendix D. As suggested by [25], we use warm-up during the +first 300 epochs to gradually impose the prior regularization terms +DKL(𝑞1(ZV |𝜓1)||𝑝(ZV)) and DKL(𝑞2(ZF|𝜓2)||𝑝(ZF)). For both +priors 𝑝(ZV) and 𝑝(ZF), we assume that they follow standard +multivariate normal distributions. In the node classification task, +most results of baselines are publicly available and we directly re- +port these results from their original papers. For the results that are +missing, we run the released codes by their authors and fine-tune +the model hyper-parameters. In the link prediction and attribute +inference tasks, since most baselines are not studied in these tasks, +we run their released codes with fine-tuning. For fairness, we run +all the experiments on a server with a single NVIDIA A100 GPU +with 80G memory. For simplicity, we set 𝐽 = 𝐾 = 1 in Eq. 19 +for both SeeGera and SIG-VAE. One may further refer to [22] for +better value selection. All the datasets and codes are provided at +https://github.com/SeeGera/SeeGera. +5.3 +Link Prediction (RQ1) +Link prediction is a typical unsupervised learning task for graph +analysis, which aims to predict whether an edge exists between +two nodes or not. We compare SeeGera with 8 other SOTA base- +lines, including GCL models: DGI [29], MVGRL [5], GRACE [41], +GCA [42], CCA-SSG [37], and the generative graph SSL methods: +CAN [18], SIG-VAE [4], GraphMAE [7]. For our proposed method +SeeGera, we put forward three versions. Specifically, SeeGera-v1 +assumes the independence between ZV and ZF and optimizes L1, +while SeeGera-v2 captures the correlations between them and op- +timizes L2. Further, SeeGera-v3 upgrades SeeGera-v2 by adding +the masking mechanism. +To evaluate the model performance, we construct the valida- +tion/test set by randomly selecting 20%/10% edges in the original +graph as positive samples and an equal number of nonexistent edges +as negative samples. After the removal of these selected edges, we +train all the models on the resulting graph with the remaining 70% +6 + +Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking +Conference’17, July 2017, Washington, DC, USA +Table 1: Link prediction results. The error bar (±) denotes the standard deviation score of results over 10 trials. We highlight the +best score on each dataset in bold. For CAN, the released codes by the authors do not implement reconstruction for numerical +features, so we cannot run it on datasets with numerical features. OOM denotes the out-of-the-memory error. +Metrics +Method +Cora +Citeseer +Pubmed +Photo +Computer +CS +Physics +AUC +DGI +93.88 ± 1.00 +95.98 ± 0.72 +96.30 ± 0.20 +80.95 ± 0.39 +81.27 ± 0.51 +93.81 ± 0.20 +93.51 ± 0.22 +MVGRL +93.33 ± 0.68 +88.66 ± 5.27 +95.89 ± 0.22 +69.58 ± 2.04 +92.37 ± 0.78 +91.45 ± 0.67 +OOM +GRACE +82.67 ± 0.27 +87.74 ± 0.96 +94.09 ± 0.92 +81.72 ± 0.31 +82.94 ± 0.20 +85.26 ± 2.07 +83.48 ± 0.96 +GCA +81.46 ± 4.86 +84.81 ± 1.25 +94.20 ± 0.59 +70.02 ± 9.66 +89.92 ± 0.91 +84.35 ± 1.13 +85.24 ± 5.41 +CCA-SSG +93.88 ± 0.95 +94.69 ± 0.95 +96.63 ± 0.15 +73.98 ± 1.31 +75.91 ± 1.50 +96.80 ± 0.16 +96.74 ± 0.05 +CAN +93.67 ± 0.62 +94.56 ± 0.68 +− +97.00 ± 0.28 +96.03 ± 0.37 +− +− +SIG-VAE +94.10 ± 0.68 +92.88 ± 0.74 +85.89 ± 0.54 +94.98 ± 0.86 +91.14 ± 1.10 +95.26 ± 0.36 +98.76 ± 0.23 +GraphMAE +90.70 ± 0.01 +70.55 ± 0.05 +69.12 ± 0.01 +77.42 ± 0.02 +75.14 ± 0.02 +91.47 ± 0.01 +87.61 ± 0.02 +SeeGera-v1 +94.95 ± 0.72 +96.75 ± 0.54 +97.07 ± 2.20 +98.40 ± 0.08 +96.87 ± 0.29 +97.82 ± 0.11 +98.95 ± 0.06 +SeeGera-v2 +95.37 ± 0.60 +96.81 ± 0.51 +97.79 ± 0.22 +98.47 ± 0.05 +97.28 ± 0.00 +97.83 ± 0.11 +98.97 ± 0.04 +SeeGera-v3 +95.50 ± 0.71 +97.04 ± 0.47 +97.87 ± 0.20 +98.64 ± 0.05 +97.70 ± 0.19 +98.42 ± 0.13 +99.03 ± 0.05 +AP +DGI +93.60 ± 1.14 +96.18 ± 0.68 +95.65 ± 0.26 +81.01 ± 0.47 +82.05 ± 0.50 +92.79 ± 0.31 +92.10 ± 0.29 +MVGRL +92.95 ± 0.82 +89.37 ± 4.55 +95.53 ± 0.30 +63.43 ± 2.02 +91.73 ± 0.40 +89.14 ± 0.93 +OOM +GRACE +82.36 ± 0.24 +86.92 ± 1.11 +93.26 ± 1.20 +81.18 ± 0.37 +83.12 ± 0.23 +83.90 ± 2.20 +82.20 ± 1.06 +GCA +80.87 ± 4.11 +81.93 ± 1.76 +93.31 ± 0.75 +65.17 ± 10.11 +89.50 ± 0.64 +83.24 ± 1.16 +82.80 ± 4.46 +CCA-SSG +93.74 ± 1.15 +95.06 ± 0.91 +95.97 ± 0.23 +67.99 ± 1.60 +69.47 ± 1.94 +96.40 ± 0.30 +96.26 ± 0.10 +CAN +94.49 ± 0.60 +95.49 ± 0.61 +− +96.68 ± 0.30 +95.96 ± 0.38 +− +− +SIG-VAE +94.79 ± 0.71 +94.21 ± 0.53 +85.02 ± 0.49 +94.53 ± 0.93 +91.23 ± 1.04 +94.93 ± 0.37 +98.85 ± 0.12 +GraphMAE +89.52 ± 0.01 +74.50 ± 0.04 +87.92 ± 0.01 +77.18 ± 0.02 +75.80 ± 0.01 +83.58 ± 0.01 +86.44 ± 0.03 +SeeGera-v1 +95.53 ± 0.54 +97.10 ± 0.49 +97.25 ± 2.07 +98.32 ± 0.09 +96.73 ± 0.31 +98.30 ± 0.11 +99.10 ± 0.09 +SeeGera-v2 +95.90 ± 0.49 +97.17 ± 0.46 +97.89 ± 0.21 +98.37 ± 0.09 +97.15 ± 0.00 +98.33 ± 0.10 +99.13 ± 0.06 +SeeGera-v3 +95.92 ± 0.68 +97.33 ± 0.46 +97.87 ± 0.20 +98.48 ± 0.06 +97.50 ± 0.15 +98.53 ± 0.18 +99.18 ± 0.04 +Table 2: Attribute inference performance w.r.t the MSE metric. The best result in each dataset is highlighted in bold. +Method +Cora +Citeseer +Pubmed +Photo +Computer +CS +Physics +CAN +- +- +- +0.22 ± 0.00 +0.23 ± 0.01 +- +- +GATE +1.80 × 10−3 ± 2.15 × 10−4 +4.58 × 10−4 ± 8.07 × 10−5 +3.90 × 10−4 ± 1.99 × 10−5 +0.24 ± 0.01 +0.25 ± 0.01 +2.03 ± 0.25 +OOM +GraphMAE +1.57 × 10−3 ± 7.42 × 10−5 +8.68 × 10−4 ± 1.30 × 10−4 +7.29 × 10−4 ± 2.66 × 10−5 +0.48 ± 0.00 +0.48 ± 0.00 +2.70 ± 0.06 +2.97 ± 0.05 +SeeGera-v1 +1.90 × 10−3 ± 8.18 × 10−5 +4.87 × 10−4 ± 2.62 × 10−6 +4.21 × 10−4 ± 4.70 × 10−5 +0.22 ± 0.00 +0.23 ± 0.14 +2.12 ± 0.06 +2.15 ± 0.05 +SeeGera-v2 +1.89 × 10−3 ± 8.13 × 10−5 +4.86 × 10−4 ± 2.24 × 10−6 +3.68 × 10−4 ± 7.11 × 10−6 +0.21 ± 0.01 +0.23 ± 0.01 +2.08 ± 0.07 +2.14 ± 0.03 +SeeGera-v3 +1.89 × 10−3 ± 8.13 × 10−5 +4.84 × 10−4 ± 2.96 × 10−6 +3.66 × 10−4 ± 7.34 × 10−6 +0.21 ± 0.01 +0.22 ± 0.00 +1.93 ± 0.07 +2.14 ± 0.03 +Table 3: Node classification performance w.r.t. the classification accuracy. We highlight the best results in bold. +Method +Cora +Citeseer +Pubmed +Photo +Computer +CS +Physics +DGI +82.3 ± 0.6 +71.8 ± 0.7 +76.8 ± 0.6 +91.61 ± 0.22 +83.95 ± 0.47 +92.15 ± 0.63 +94.51 ± 0.52 +MVGRL +83.5 ± 0.4 +73.3 ± 0.5 +80.1 ± 0.7 +91.74 ± 0.07 +87.52 ± 0.11 +92.11 ± 0.12 +95.33 ± 0.03 +GRACE +81.9 ± 0.4 +71.2 ± 0.5 +80.6 ± 0.4 +92.15 ± 0.24 +86.25 ± 0.25 +92.93 ± 0.01 +95.26 ± 0.02 +CCA-SSG +84.0 ± 0.4 +73.1 ± 0.3 +81.0 ± 0.4 +93.14 ± 0.14 +88.74 ± 0.28 +93.31 ± 0.22 +95.38 ± 0.06 +GraphMAE +84.2 ± 0.4 +73.4 ± 0.4 +81.1 ± 0.4 +92.98 ± 0.35 +88.34 ± 0.27 +93.08 ± 0.17 +95.30 ± 0.12 +SeeGera-v1 +82.9 ± 0.4 +71.7 ± 0.6 +78.9 ± 0.9 +92.53 ± 0.41 +88.44 ± 0.24 +93.72 ± 0.29 +95.40 ± 0.10 +SeeGera-v2 +84.0 ± 0.4 +73.0 ± 0.8 +80.4 ± 0.4 +92.70 ± 0.42 +88.39 ± 0.26 +93.83 ± 0.22 +95.39 ± 0.08 +SeeGera-v3 +84.3 ± 0.4 +73.0 ± 0.8 +80.4 ± 0.4 +92.81 ± 0.45 +88.39 ± 0.26 +93.84 ± 0.11 +95.39 ± 0.08 +edges. We use two commonly used metrics, the area under the ROC +curve (AUC) and the average precision (AP), to report the model +performance. For both metrics, a larger value indicates a better +performance. We use the validation set for hyper-parameter tuning +and early stopping with a patience of 100, i.e., we stop training if +both metric scores on the validation set do not increase for 100 +consecutive epochs. Similar as in [12], the predicted probability of +an edge between nodes 𝑥𝑖 and 𝑥𝑗 is calculated by A𝑖𝑗 ∼ Ber(𝑝A +𝑖𝑗), +where 𝑝A +𝑖𝑗 = 𝜎((ZV +𝑖 )𝑇 ZV +𝑗 ) and 𝜎 is the sigmoid function. For each +method, we run experiments 10 times and report the average results +on the test set. Table 1 summarizes the results across all the datasets. +From the table, we have the following observations: +7 + +Conference’17, July 2017, Washington, DC, USA +Xiang Li, et al. +(a) Cora-AUC +(b) Cora-AP +(c) Citeseer-AUC +(d) Citeseer-AP +Figure 2: Hyper-parameter sensitivity analysis on the masking rates 𝛼1 and 𝛼2 in terms of link prediction. The darker the color, +the larger the value. +(1) The generative graph SSL methods except GraphMAE gen- +erally perform better than GCL methods. This is because these +methods learn to reconstruct links in the objective. For GraphMAE, +it only reconstructs features, which explains its poor performance. +(2) While SeeGera is based on SIG-VAE, it achieves better perfor- +mance. This demonstrates the importance of feature reconstruction +and structure/feature masking. +(3) Although CAN co-embeds both nodes and features, it still per- +forms not very well, due to the independence assumption between +node and feature embeddings. Further, it uses linear decoder for +feature reconstruction, which restricts the model’s effectiveness. +(4) SeeGera significantly outperforms other competitors across +all the datasets, which indicates the superiority of generative VGAE +model in graph representation learning. In particular, the consis- +tent outperformance of SeeGera-v2 over SeeGera-v1 verifies the +importance of capturing the correlations between node and fea- +ture embeddings. Further, the improvement of SeeGera-v3 over +SeeGera-v2 shows the necessity of structure/feature masking. +5.4 +Attribute Inference (RQ2) +Attribute inference is a task that predicts values of missing node +attributes. Similar as in link prediction, we hide a certain percent- +age of node features and train on the rest. To construct the train- +ing/validation/test set, we randomly select 70%/10%/20% node fea- +tures. The validation set is used for hyper-parameter tuning and +early stopping with a patience of 100 epoches. We take the Mean +Squared Error (MSE) as the evaluation metric. The smaller the value, +the better the performance. In this task, we compare SeeGera with +generative graph SSL methods that reconstruct features in their +decoders, including CAN [20], GATE [23] and GraphMAE [7]. For +GCL models and other generative graph SSL methods that recover +links only, they cannot be easily adapted to the task, so we do +not take them as baselines. For each method, we run experiments +10 times and report the average results in Table 2. For Cora and +Citeseer, we normalize node features for fair comparison, so CAN +cannot be applied. From the table, while CAN, GATE and Graph- +MAE can perform well on some datasets, they cannot consistently +provide excellent performance. For example, GraphMAE achieves +the best result on Cora, but it performs very poorly on Citeseer. +Further, SeeGera-v3 outperforms other competitors on 5 out of +7 datasets. This shows the effectiveness of our proposed feature +reconstruction method and also the masking mechanism. We also +notice that, in all cases, SeeGera-v2 achieves better performance +than SeeGera-v1, which again verifies the necessity of capturing +correlations between node and feature embeddings. +5.5 +Node Classification (RQ3) +To further study SeeGera, we generalize learned embeddings to +the node classification task. After node embeddings are trained on +the entire graph, we train an additional classifier. Here, we employ +Logistic Regression as the classifier. For Cora, Citeseer and Pubmed, +we use the public split for evaluation, where each class has fixed +20 nodes for training, another fixed 500 nodes and 1000 nodes for +validation and testing, respectively. For other datasets, we randomly +split the nodes into 10%/10%/80% training/validation/test sets. We +use classification accuracy as the metric to evaluate the model per- +formance. Since GCL methods have been shown to perform well in +classification tasks, we compare SeeGera with 4 state-of-the-arts, +including DGI, MVGRL, GRACE and CCA-SSA. We also take the +recently proposed generative model GraphMAE as baseline, be- +cause it bridges the gap between generative graph SSL models and +GCL methods in terms of classification tasks. Table 3 summarizes +the classification results on all the datasets. From the table, we see +that CCA-SSG, GraphMAE and SeeGera lead other competitors +and they almost tie. This shows that SeeGera achieves compa- +rable performance with the state-of-the-art methods in the node +classification task. Further, with the significant advantage in link +prediction and attribute inference tasks, we conclude that SeeGera, +a VGAE-based graph SSL method, can generate versatile node rep- +resentations that can be widely used in various downstream tasks. +5.6 +Parameter Analysis (RQ4) +We end this section with a sensitivity analysis on the key hyper- +parameters in SeeGera, i.e., the structure masking rate 𝛼1 and the +feature masking rate 𝛼2. Specifically, we explore the stability of +SeeGera w.r.t. the perturbation of 𝛼1 and 𝛼2. We conduct experi- +ments on the link prediction task by varying these parameters from +0 to 0.5, and keeping others fixed. Figure 2 illustrates the AUC and +AP scores of SeeGera-v3 under different 𝛼1 and 𝛼2 values on Cora +and Citeseer. From the figure, we see that SeeGera-v3 can give very +8 + +Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking +Conference’17, July 2017, Washington, DC, USA +stable performance over a wide range of 𝛼1 and 𝛼2 values, as shown +by the plateau in the figure. This demonstrates the insensitivity of +SeeGera w.r.t. these two hyper-parameters. +6 +CONCLUSIONS +We studied generative graph SSL in this paper and proposed SeeGera, +which enhances the family of VGAE on graph representation learn- +ing. Specifically, SeeGera adopts the hierarchical variational frame- +work in SIG-VAE and mainly focuses on feature reconstruction and +structure/feature masking. On the one hand, SeeGera co-embeds +both nodes and features in the encoder and computes their embed- +dings by assuming they are independent and correlated, respec- +tively. After that, feature embeddings that contain rich semantic +information on features are combined with node embeddings to +provide more fine-grained information for feature reconstruction in +the decoder. On the other hand, we injected the masking mechanism +into SeeGera by adding an additional layer to the hierarchical varia- +tional framework. We conducted extensive experiments to evaluate +the performance of SeeGera. The results show that SeeGera sig- +nificantly outperforms other competitors in link prediction and +attribute inference, and achieves comparable results with them in +node classification. This further verifies the power of generative +graph SSL methods in graph representation learning. +ACKNOWLEDGEMENT +This work is supported by Shanghai Pujiang Talent Program No. +21PJ1402900, Shanghai Science and Technology Committee General +Program No. 22ZR1419900 and National Natural Science Foundation +of China No. 62202172. +REFERENCES +[1] Yuri Burda, Roger Grosse, and Ruslan Salakhutdinov. 2015. Importance weighted +autoencoders. arXiv preprint arXiv:1509.00519 (2015). +[2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018. Bert: +Pre-training of deep bidirectional transformers for language understanding. arXiv +preprint arXiv:1810.04805 (2018). +[3] Alberto Garcia Duran and Mathias Niepert. 2017. Learning graph representations +with embedding propagation. Advances in neural information processing systems +30 (2017). +[4] Arman Hasanzadeh, Ehsan Hajiramezanali, Krishna Narayanan, Nick Duffield, +Mingyuan Zhou, and Xiaoning Qian. 2019. 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An empirical study of +graph contrastive learning. arXiv preprint arXiv:2109.01116 (2021). +[41] Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. 2020. +Deep graph contrastive representation learning. arXiv preprint arXiv:2006.04131 +(2020). +[42] Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. 2021. +Graph contrastive learning with adaptive augmentation. In Proceedings of the +Web Conference 2021. 2069–2080. +A +DATASETS +We use 7 public datasets which do not have license. We next briefly +introduce them as follows. +Cora, Citeseer and Pubmed [13] are three citation networks, +where nodes represent publications and edges are citation links. +Features for each node are the keywords it contains. Each dimen- +sion in the feature vector indicates the presence of a keyword in +the publication. Nodes in these datasets are associated with labels +that describe research topics of publications. +Coauther CS and Coauther Physics are co-authorship graphs +based on the Microsoft Academic Graph from the KDD Cup 2016 +challenge [24]. In these datasets, nodes are authors and edges cap- +ture the co-authorship. Further, node features represent keywords +in each author’s papers, and class labels indicate the study fields +for authors. +Amazon Computer and Amazon Photo are extracted from the +Amazon co-purchase graph [17], where nodes represent goods +and edges indicate that two goods are frequently bought together. +Node features are bag-of-words encoded product reviews and class +labels are the product categories. The statistics of these datasets +are summarized in Table 4. +Table 4: Statistics of datasets used in experiments +Datasets +#Nodes +#Edges +#Features +#Classes +Cora +2, 708 +5, 278 +1, 433 +7 +Citeseer +3, 327 +4, 676 +3, 703 +6 +Pubmed +19, 717 +88, 651 +500 +3 +Coauthor CS +18, 333 +327, 576 +6, 805 +15 +Coauthor Physics +34, 493 +991, 848 +8, 451 +5 +Amazon Computer +13, 752 +574, 418 +767 +10 +Coauthor Physics +7, 650 +287, 326 +745 +8 +B +PSEUDOCODES +This section summarizes the pseudocodes of SeeGera-v3 in Alg. 1. +Algorithm 1 SeeGera-v3 +Input: A, X, 𝑝( ˜𝐺|A, X), ˜𝑞(𝜖), ˆ𝑞(𝜖), 𝜌, neural networks 𝑇𝜙1 and 𝑇𝜙2 +Output: 𝜙1 and 𝜙2 +1: Initialize 𝜙1, 𝜙2, set L𝐽 +𝐾 = 0 +2: while not converged do +3: +Sample ˜𝐺 ∼ 𝑝( ˜𝐺|A, X) +4: +for 𝑘 = 1 to 𝐾 do +5: +Sample ˜𝜓𝑘 +1 = 𝑇𝜙1 ( ˜𝐺, ˜𝜖𝑘 +1 ), where ˜𝜖𝑘 +1 ∼ ˜𝑞(𝜖) +⊲ Eq. 12 +6: +Sample ˜𝜓𝑘 +2 = 𝑇𝜙2 ( ˜𝐺, ˜𝜓𝑘 +1 , ˆ𝜖𝑘 +2 ), where ˆ𝜖𝑘 +2 ∼ ˆ𝑞(𝜖) ⊲ Eq. 14 +7: +end for +8: +for 𝑗 = 1 to 𝐽 do +9: +Sample 𝜖 𝑗 +1 ∼ ˜𝑞(𝜖), 𝜖 𝑗 +2 ∼ ˆ𝑞(𝜖) +10: +Sample 𝜓 𝑗 +1 = [(𝜇V)𝑗, (ΣV)𝑗] = 𝑇𝜙1 ( ˜𝐺,𝜖 𝑗 +1) +⊲ Eq. 12 +11: +Sample 𝜓 𝑗 +2 = [(𝜇 F)𝑗, (ΣF)𝑗] = 𝑇𝜙2 ( ˜𝐺,𝜓 𝑗 +1,𝜖 𝑗 +2) +⊲ Eq. 14 +12: +Sample 𝜖V +𝑗 +∼ N (0, 𝐼), 𝜖 A +𝑗 +∼ N (0, 𝐼) +13: +Sample (ZV)𝑗 = (𝜇V)𝑗 + (ΣV)𝑗 ⊙ 𝜖V +𝑗 +14: +Sample (ZF)𝑗 = (𝜇 F)𝑗 + (ΣF)𝑗 ⊙ 𝜖 A +𝑗 +15: +Set 𝑡𝑚𝑝1 = − log Ω𝑗 +⊲ Eq. 19 +16: +Set 𝑡𝑚𝑝2 = log𝑝( ˜𝐺|(ZV)𝑗, (ZF)𝑗) +17: +Set 𝑡𝑚𝑝3 = log𝑝((ZV)𝑗, (ZF)𝑗) +18: +Update L𝐽 +𝐾 = L𝐽 +𝐾 + 𝑒𝑡𝑚𝑝1+𝑡𝑚𝑝2+𝑡𝑚𝑝3 +19: +end for +20: +Update L𝐽 +𝐾 = log L𝐽 +𝐾 − log 𝐽 +21: +Update 𝜙1 = 𝜙1 + 𝜌▽𝜙1L𝐽 +𝐾 +22: +Update 𝜙2 = 𝜙2 + 𝜌▽𝜙2L𝐽 +𝐾 +23: end while +24: return 𝜙1 and 𝜙2 +C +ABLATION STUDY +We conduct an ablation study to investigate the main components in +SeeGera. In particular, we have extensively compared SeeGera-v1, +SeeGera-v2 and SeeGera-v3 in our experiments. The advantage +of SeeGera-v2 over SeeGera-v1 shows the importance of captur- +ing the correlations between node and feature embeddings. Also, +the outperformance of SeeGera-v3 over SeeGera-v2 verifies the +importance of the masking mechanism. Further, to show the effec- +tiveness of our proposed feature reconstruction method, we remove +feature embeddings in the encoder and feed only node embeddings +into GCN in the decoder to reconstruct features. We call this variant +SeeGera_nf (no feature embedding). Table 5 shows the results on +attribute inference. We exclude SeeGera-v3 in the table, because it +further uses the masking mechanism while others not. From the +table, we see that both SeeGera-v1 and SeeGera-v2 outperform +SeeGera_nf. This shows the importance of using both node and +feature embeddings for feature reconstruction. +Table +5: +The +comparison +between +SeeGera +and +SeeGera_nf in the attribute inference task. +Method +Cora +Citeseer +CS +SeeGera_nf +1.91 × 10−3 ± 3.78 × 10−5 +5.15 × 10−4 ± 1.02 × 10−6 +2.14 ± 0.07 +SeeGera-v1 +1.90 × 10−3 ± 8.18 × 10−5 +4.87 × 10−4 ± 2.62 × 10−6 +2.12 ± 0.06 +SeeGera-v2 +1.89 × 10−3 ± 8.13 × 10−5 +4.86 × 10−4 ± 2.24 × 10−6 +2.08 ± 0.07 +10 + +Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking +Conference’17, July 2017, Washington, DC, USA +D +IMPLEMENTATION DETAILS +We provide the detailed hyper-parameter settings of SeeGera-v3 on +different datasets in Tables 6- 8. All hyper-parameters are selected +through small grid search, and the search space is provided as +follows: +• Number of layers in the encoder 𝐿1: {1, 2, 3} +• Number of layers in the decoder 𝐿2: {1, 2, 3} +• Learning rate of SeeGera: {1e-3, 5e-3, 1e-2} +• Dropout of SeeGera: {0, 0.1, 0.3, 0.5, 0.7, 0.9} +• Weight decay of SeeGera: {5e-5, 1e-4, 5e-4, 1e-3} +• Structure masking rate 𝛼1: {0, 0.1, 0.2, 0.3, 0.4, 0.5} +• Feature masking rate 𝛼2: {0, 0.1, 0.2, 0.3, 0.4, 0.5} +• Learning rate of logistic regression: {1e-3, 5e-3, 1e-2} +• Dropout of logistic regression: {0, 0.1, 0.3, 0.5, 0.7, 0.9} +• Weight decay of logistic regression: {5e-5, 1e-4, 5e-4, 1e-3} +Table 6: Hyper-parameter setting details of SeeGera-v3 in +link prediction. +Dataset +𝐿1 +𝐿2 +lr +dropout +wd +𝛼1 +𝛼2 +Cora +2 +3 +1e-3 +0.3 +5e-5 +0.3 +0.0 +Citeseer +1 +2 +1e-3 +0.0 +1e-4 +0.6 +0.0 +Pubmed +2 +1 +5e-3 +0.5 +0.0 +0.3 +0.1 +Photo +1 +1 +5e-3 +0.0 +0.0 +0.5 +0.0 +Computer +2 +2 +1e-3 +0.0 +0.0 +0.4 +0.0 +CS +1 +2 +1e-3 +0.0 +0.0 +0.5 +0.5 +Physics +2 +2 +1e-3 +0.0 +0.0 +0.2 +0.0 +Table 7: Hyper-parameter setting details of SeeGera-v3 in +attribute inference. +Dataset +𝐿1 +𝐿2 +lr +dropout +wd +𝛼1 +𝛼2 +Cora +1 +1 +1e-3 +0.1 +5e-5 +0.4 +0.0 +Citeseer +1 +1 +1e-3 +0.0 +0.0 +0.3 +0.0 +Pubmed +2 +2 +1e-3 +0.0 +0.0 +0.0 +0.1 +Photo +2 +3 +1e-3 +0.0 +0.0 +0.3 +0.0 +Computer +2 +3 +1e-3 +0.0 +0.0 +0.2 +0.0 +CS +1 +3 +1e-3 +0.0 +0.0 +0.1 +0.5 +Physics +2 +2 +1e-3 +0.0 +0.0 +0.0 +0.0 +E +VARIATIONAL LOWER BOUND +In this section, we show the derivation on the variational lower +bounds in detail. +L = Eℎ𝜙1 (ZV )Eℎ𝜙2 (ZF ) +� +log 𝑝 (ZV |A, X)𝑝 (ZF |X𝑇 )𝑝 (A, X) +ℎ𝜙1 (ZV)ℎ𝜙2 (ZF) +� += −DKL(ℎ𝜙1 (ZV) ||𝑝 (ZV |A, X)) − DKL(ℎ𝜙2 (ZF) ||𝑝 (ZF |X𝑇 )) + log𝑝 (A, X) +≥ −E𝜓1∼𝑞𝜙1 (𝜓1)DKL(𝑞1(ZV |𝜓1) ||𝑝 (ZV |A, X)) +− E𝜓2∼𝑞𝜙2 (𝜓2)DKL(𝑞2(ZF |𝜓2) ||𝑝 (ZF |X𝑇 )) + log𝑝 (A, X) += E𝜓1∼𝑞𝜙1 (𝜓 )EZV∼𝑞1 (ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓 )EZF∼𝑞2 (ZF |𝜓2) +� +log +𝑝 (A, X, ZV, ZF) +𝑞1(ZV |𝜓1)𝑞2(ZF |𝜓2) +� += L1, +where DKL is the KL divergence and we employ DKL(E𝜓𝑞(Z|𝜓)||𝑝(Z)) ≤ +E𝜓 DKL(𝑞(Z|𝜓)||𝑝(Z)) according to [32]. To better understand L1, +we decompose the joint distribution 𝑝(A, X, ZV, ZF) as +𝑝 (A, X, ZV, ZF) = 𝑝 (ZV)𝑝 (ZF) +� +𝑖,𝑗∈V +𝑝 (A𝑖𝑗 |ZV +𝑖 , ZV +𝑗 ) +� +𝑖∈V,𝑟∈F +𝑝 (X𝑖𝑟 |ZV +𝑖 , ZF +𝑟 ) +and expand L1 to derive: +L1 = E𝜓1∼𝑞𝜙1 (𝜓1)EZV∼𝑞1(ZV |𝜓1) + +∑︁ +𝑖,𝑗 ∈V +log𝑝(A𝑖𝑗 |ZV +𝑖 , ZV +𝑗 ) + ++ E𝜓1∼𝑞𝜙1 (𝜓1)EZV∼𝑞1(ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2)EZF∼𝑞2(ZF |𝜓2) + +∑︁ +𝑖 ∈V,𝑟 ∈F +log𝑝(X𝑖𝑟 |ZV +𝑖 , ZF +𝑟 ) + +− E𝜓1∼𝑞𝜙1 (𝜓1)DKL(𝑞1(ZV |𝜓1)||𝑝(ZV)) +− E𝜓2∼𝑞𝜙2 (𝜓2)DKL(𝑞2(ZF|𝜓2)||𝑝(ZF)). +Here, 𝑞1(ZV |𝜓1) and 𝑞2(ZF|𝜓2) are encoders that generate em- +beddings of nodes and features, respectively; 𝑝(A𝑖𝑗 |ZV +𝑖 , ZV +𝑗 ) and +𝑝(X𝑖𝑟 |ZV +𝑖 , ZF𝑟 ) are decoders that reconstruct links and features +from learned embeddings. The first two terms in the equation cor- +respond to the negative reconstruction loss for links and features, +while the last two terms are regularizers that promote the closeness +between variational distributions and prior distributions. +Similarly, we can expand L2 as: +L2 = E𝜓1∼𝑞𝜙1 (𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2 |𝜓1)E(ZV,ZF)∼𝑞(ZV,ZF |𝜓1,𝜓2) + +∑︁ +𝑖,𝑗 ∈V +log𝑝(A𝑖𝑗 |ZV +𝑖 , ZV +𝑗 ) + +∑︁ +𝑖 ∈V,𝑟 ∈F +log𝑝(X𝑖𝑟 |ZV +𝑖 , ZF +𝑟 ) + +− E(𝜓1,𝜓2)∼𝑞𝜙 (𝜓1,𝜓2)DKL(𝑞(ZV, ZF|𝜓1,𝜓2)||𝑝(ZV, ZF)). +11 + +Conference’17, July 2017, Washington, DC, USA +Xiang Li, et al. +Table 8: Hyper-parameter setting details of SeeGera-v3 in node classification. +Dataset +SeeGera +Logistic Regression +𝐿1 +𝐿2 +lr +dropout +wd +𝛼1 +𝛼2 +lr +dropout +wd +Cora +2 +2 +1e-3 +0.3 +1e-3 +0.2 +0.1 +1e-3 +0.9 +1e-3 +Citeseer +2 +2 +5e-3 +0.3 +5e-4 +0.0 +0.0 +1e-3 +0.9 +0.0 +Pubmed +2 +1 +5e-3 +0.7 +0.0 +0.0 +0.0 +1e-3 +0.3 +0.0 +Photo +1 +2 +1e-3 +0.0 +5e-4 +0.5 +0.5 +5e-3 +0.7 +5e-5 +Computer +1 +3 +5e-3 +0.0 +5e-4 +0.0 +0.0 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+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='cn Tiandi Ye East China Normal University Shanghai, China 52205903002@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='cn Caihua Shan Microsoft Research Asia Shanghai, China caihuashan@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='com Dongsheng Li Microsoft Research Asia Shanghai, China dongsheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='li@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='com Ming Gao∗ East China Normal University Shanghai, China mgao@dase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='cn ABSTRACT Generative graph self-supervised learning (SSL) aims to learn node representations by reconstructing the input graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, most existing methods focus on unsupervised learning tasks only and very few work has shown its superiority over the state-of- the-art graph contrastive learning (GCL) models, especially on the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' While a very recent model has been proposed to bridge the gap, its performance on unsupervised learning tasks is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this paper, to comprehensively enhance the performance of generative graph SSL against other GCL models on both unsupervised and supervised learning tasks, we propose the SeeGera model, which is based on the family of self-supervised vari- ational graph auto-encoder (VGAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, SeeGera adopts the semi-implicit variational inference framework, a hierarchical variational framework, and mainly focuses on feature reconstruc- tion and structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the one hand, SeeGera co-embeds both nodes and features in the encoder and reconstructs both links and features in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since feature embeddings contain rich semantic information on features, they can be com- bined with node embeddings to provide fine-grained knowledge for feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the other hand, SeeGera adds an additional layer for structure/feature masking to the hierarchical variational framework, which boosts the model generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We conduct extensive experiments comparing SeeGera with 9 other state-of-the-art competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Our results show that SeeGera can compare favorably against other state-of-the-art GCL methods in a variety of unsupervised and supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' KEYWORDS Graph neural networks, graph self-supervised learning, variational graph auto-encoder ∗Corresponding author Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Copyrights for components of this work owned by others than ACM must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='nnnnnnn ACM Reference Format: Xiang Li, Tiandi Ye, Caihua Shan, Dongsheng Li, and Ming Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Self- supervised Semi-implicit Graph Variational Auto-encoders with Masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In Proceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' ACM, New York, NY, USA, 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Self-supervised learning (SSL) [2, 6, 9, 36] has attracted significant attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' By extracting and employing supervisions from data itself, SSL can heavily reduce the dependence of neural network models on the labeled data, which is costly to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To facilitate graph-based learning, SSL has been applied on graph-structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For example, it can learn representations for nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', web pages in search engines), and detect the anomalies on webs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', malicious users) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, graph contrastive learning (GCL), as one of the main SSL types, has experienced a surge [10, 26, 29, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The core idea of GCL is to first construct positive and negative pairs for nodes, and then maximize the similarity between positive pairs while minimizing that between negative ones1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Despite the success, existing GCL methods suffer from two main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the one hand, negative samples are needed in most contrastive objectives, which generally construct one positive and 𝐾 negative samples for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, these models are eas- ily affected by the value of 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' When 𝐾 is small, the model cannot learn sufficient discriminative information, which degrades the model effectiveness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' otherwise, there could lead to a large number of false-negative samples and slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Generally, 𝐾 is set empirically and there lack theoretical supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the other hand, for the rest of methods based on positive pairs only, they are easily trapped into a degenerate solution [40], where all the output embeddings of nodes collapse to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To tackle the issue, additional strategies are necessary, such as asymmetric dual encoders with momentum updates and exponential moving aver- age [21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, some studies [14] have showed that although these training strategies can alleviate collapse to some extent, they may still cause collapse in partial dimensions of the representation, which leads to worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To address the shortcomings of GCL methods, generative graph SSL methods can be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In particular, self-supervised graph auto-encoders (GAEs) [12], whose objective is to reconstruct 1Note that some GCL methods require positive pairs only and they only maximize the similarity between positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12458v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='LG] 29 Jan 2023 Conference’17, July 2017, Washington, DC, USA Xiang Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' the input graph data, have been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Existing methods mainly differ in their adopted reconstruction components, such as the adjacency matrix reconstruction [19], the node feature re- construction [20] and a combination of both graph structure and node feature reconstruction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, most of these methods focus on the unsupervised learning tasks like link prediction and node clustering, and very few work has shown its superiority over the state-of-the-art GCL methods, especially on the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' While a masked GAE model GraphMAE [7] is very recently proposed to bridge the gap, its performance on the unsupervised learning tasks is still unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since the goal of SSL is to learn versatile representations, a further study on self-supervised GAE model that can achieve comprehensive superiority on both unsu- pervised and supervised learning tasks is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, although GraphMAE is an auto-encoding method, it is based on GAE and is essentially not a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This also calls our attention back to the study of generative graph SSL model, such as variational graph auto-encoder (VGAE) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Different from GAE, VGAE consists of an inference model and a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, the inference model encodes obser- vations (links and features) into latent variables (node embeddings) while the generative model decodes from these latent variables to reconstruct links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, as pointed out in [7], node feature reconstruction is beneficial for learning high-quality representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Therefore, the lack of feature reconstruction could degrade the model effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To solve the issue, most existing methods adopt MLP [8, 9] and GNNs [7, 20] as their decoders for feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, they utilize node-level embeddings only and ignore feature-level embeddings that contain rich semantic information on node features and can be used to help feature recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, CAN [18] is proposed to co-embed both nodes and features, and use the inner product of their embeddings as the decoder to recover node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Despite the success, it has three main problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' First, the linear decoder is generally less powerful than MLP and GNNs, which restricts the model’s capability in re- constructing node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Second, it assumes the independence between node and feature embeddings in the variational inference stage, but practically these two types of embeddings are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Third, it lacks structure/feature masking in the learning process, which has been shown to degrade the model’s performance on the classification task [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this paper, we study generative graph SSL and our goal is to en- hance the family of self-supervised VGAE on graph representation learning in a variety of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, semi-implicit variational inference (SIVI) [32], which is a hierarchical variational framework, has been applied to VGAE to model a wide range of un- derlying true posteriors with multi-modality, skewness and heavy tails [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We thus adopt the framework to remove the explicit Gauss- ian restriction on the variational distribution and mainly focus on the component of feature reconstruction and structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We propose a Self-supervised semi-implicit Graph varia- tional auto-encoder with masking, namely, SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, the model co-embeds both nodes and features in the encoder and jointly reconstructs links and features in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that the feature embeddings can provide fine-grained information that is supplementary to the node embeddings when reconstructing node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, for each node, we take its feature values as weights and compute the weighted average of feature embeddings w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The weighted embedding characterizes the affini- ties between the node and all the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After that, we combine the weighted embedding with the node embedding, and feed the fused embedding into GNNs to reconstruct the node’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, to generate node and feature embeddings in the encoder, we first assume the independence between them and propose the base SeeGera model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then we upgrade the model by capturing the correlations between node and feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Finally, we add an additional layer to the hierarchical variational framework to integrate SeeGera with the masking mechanism and boost the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In summary, our main contributions are listed: We propose a generative graph SSL model SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To our knowl- edge, this is the first generative graph SSL method that is compre- hensively compared with the SOTA GCL models in terms of both unsupervised and supervised learning tasks, and shows superiority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We present a novel feature reconstruction method that lever- ages both node and feature embeddings to provide fine-grained information for reconstructing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We further introduce the structure/feature masking mechanism by adding an additional layer to the hierarchical variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We conduct extensive experiments to evaluate the performance of SeeGera on two unsupervised learning tasks: link prediction and attribute inference, and one supervised learning task: node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Experimental results show that SeeGera can signifi- cantly outperform other competitors on both link prediction and attribute inference tasks, and perform comparably with them in node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This effectively verifies the power of generative graph SSL in graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2 RELATED WORK In this section, we summarize the related work on both graph self- supervised learning and generative graph self-supervised learning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 Graph self-supervised learning Graph self-supervised learning [7, 26, 31, 35] aims to employ super- visions extracted from graph-structured data without the need for annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Existing methods can be mainly divided into four types: (1) generative models [12], whose objective is to reconstruct the input graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (2) auxiliary-property-based methods [36], which first obtain graph-related properties and then take them as supervisions, such as the pseudo labels of unlabeled nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (3) con- trastive models [29], which construct positive and negative pairs for contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (4) hybrid approaches [38], which combine the objec- tives of the first three types in a multi-task learning fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For a comprehensive survey on graph self-supervised learning, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, graph contrastive learning has been widely studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' According to whether negative samples are used in the learning pro- cess, existing methods include negative-sample-based and negative- sample-free ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For the former, DGI [29] and InfoGraph [26] em- ploy corruptions to construct negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' GRACE [41], GCA [42] and GraphCL [35] take samples in a mini-batch as a dictionary whose size is constrained by the batch size and consider other sam- ples in the same mini-batch as negatives of a sample, while GCC [21] maintains a dynamic dictionary with larger size as in MoCo [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2 Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking Conference’17, July 2017, Washington, DC, USA For the latter, BGRL [27] and CCA-SSG [37] are two representative models that are based on asymmetric encoding architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' How- ever, they require special training strategies to avoid the collapse of learned node embeddings to a constant, such as momentum up- date [6], exponential moving average [27] and stop gradient [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, existing GCL methods heavily rely on graph augmentation strategies to construct different graph views for contrast, including feature-oriented (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', masking [35] and shuffling [29]), proximity- oriented (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', perturbation [35]), and graph-sampling-based (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', random-walk [5]) augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 Generative graph self-supervised learning Generative graph self-supervised learning aims to take the input graph as self-supervision and recover the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' It mainly con- sists of two families of models: graph autoregressive models and graph autoencoders (GAEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Autoregressive models [33, 34] decom- pose joint probability distributions as a product of conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The representative graph autoregressive model is GPT-GNN [9], which takes attributed graph generation as its objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, since autoregressive models require an explicit ordering to generate, they might not work well on graphs that do not exhibit inherent orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Different from graph autoregressive models, GAEs do not require any decoding ordering and they aim to reconstruct part of the input graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' According to the reconstructed components, existing self-supervised GAE methods include those that reconstruct links only (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', ARVGA [19], GAE [12], VGAE [12]), features only (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', GraphMAE [7], GALA [20], MGAE [30], EP [3]), and a combination of both links and features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', GATE [23], CAN [18], DGE [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, most of these methods focus on the link prediction and node clustering tasks, and few of them compares favorably against the state-of-the-art GCL methods, especially in the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' While GraphMAE is recently proposed to bridge the gap, its performance on unsupervised learning tasks remains unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, it is based on GAE and is essentially not a generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Different from GAE, variational graph auto-encoder (VGAE) is a generative model that recovers links only in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' While there exist some self-supervised VGAE models that reconstruct fea- tures [23, 39], most of them only leverage node-level embeddings but ignore feature-level embeddings that contain fine-grained infor- mation for node features and can help boost feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this paper, we reconsider generative graph self-supervised learn- ing and show that self-supervised VGAE can outperform or perform comparably against other SOTA GCL models in a variety of tasks, such as link prediction, attribute inference and node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 3 PRELIMINARY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 Notations Let G = (V, E) denote a graph, where V = {𝑥𝑖}𝑛 𝑖=1 is a set of nodes and E ⊆ V × V is a set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Let A be the adjacency matrix of 𝐺, such that A𝑖𝑗 represents the weight of edge 𝑒𝑖𝑗 between objects 𝑥𝑖 and 𝑥𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For simplicity, we set A𝑖𝑗 = 1 if 𝑒𝑖𝑗 ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, since nodes in a graph are usually associated with features, we denote F = {𝑓𝑟 }𝑙 𝑟=1 as a set of node features and X ∈ R𝑛×𝑙 as the node feature matrix, where the 𝑖-th row X𝑖 is the feature vector of node 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For the node representation matrix, let it be ZV ∈ R𝑛×𝑑, where 𝑑 is the output embedding dimension satisfying 𝑑 ≪ |V|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that the 𝑖-th row ZV 𝑖 represents the embedding of node 𝑥𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Similarly, ZF ∈ R𝑙×𝑑 denotes the feature representation matrix, whose 𝑟-th row ZF𝑟 is the embedding of node feature 𝑓𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this paper, we learn both node and feature representations, and use node representations in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 SIVI and SIG-VAE Given observations Y and latent variable Z, the vanilla variational inference (VI) derives an evidence lower bound ELBO = −EZ∼𝑞(Z|𝜓) [log𝑞(Z|𝜓) − log𝑝(Y, Z)] , (1) where 𝜓 is variational parameter, 𝑞(Z|𝜓) is variational distribution and 𝑝(Y, Z) is joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, VI restricts an exponen- tial family assumption to the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To address the problem, semi-implicit variational inference (SIVI) [32] considers variational parameters as random variables drawn from a mixing distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, the semi-implicit variational distribution for Z is de- fined in a hierarchical manner, which follows Z ∼ 𝑞(Z|𝜓) and 𝜓 ∼ 𝑞𝜙 (𝜓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, 𝜙 is the parameter of the mixing distribution 𝑞𝜙 (𝜓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, 𝜓 can be marginalized out to derive a distribution family H indexed by 𝜙 for Z: H = � ℎ(Z) : ℎ(Z) = ∫ 𝜓 𝑞(Z|𝜓)𝑞𝜙 (𝜓)d𝜓 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (2) Note that 𝑞(Z|𝜓) is required to be explicit, but the mixing distri- bution 𝑞𝜙 (𝜓) is allowed to be implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Moreover, the marginal distribution ℎ(Z) ∈ H is often implicit unless 𝑞𝜙 (𝜓) is conjugate to 𝑞(Z|𝜓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' These are the reasons why the method is referred to as “semi-implicit” VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To maintain simple optimization, 𝑞(Z|𝜓) is re- quired to be either reparameterizable [11] or allow the ELBO under 𝑞(Z|𝜓) to be analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For 𝑞𝜙 (𝜓), it needs to be reparameterizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Generally, SIVI draws from 𝑞𝜙 (𝜓) by injecting random noise 𝜖 into node features and transforming the features via neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Recently, Hasanzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [4] apply SIVI to VGAE and propose the semi-implicit graph variational auto-encoder (SIG-VAE) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, it sets 𝑞(Z|𝜓) to be Gaussian distribution and uses GNNs to characterize the mixing distribution𝑞𝜙 (𝜓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' While SIG-VAE uses the hierarchical variational framework to capture complex non- Gaussian posteriors, it still has the problem of ignorance of feature reconstruction and structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Therefore, based on the framework of SIG-VAE, we next explore how to enhance self- supervised VGAE for unsupervised graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4 ALGORITHM In this section, we present our model SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Different from SIG- VAE that uses node embeddings only, SeeGera further generates feature embeddings to capture the rich semantic information on node features, which can be used to enhance feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, we consider two cases in the encoder when generating node and feature embeddings: (1) they are independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (2) they are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After that, in the decoder part, we utilize GNNs to reconstruct node features based on both node and feature em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Finally, we show how structure/feature masking can be integrated with the hierarchical variational framework and gives the optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The overall framework of SeeGera is summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 3 Conference’17, July 2017, Washington, DC, USA Xiang Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4 0 3 2 1 GCN MLP GCN Encoding Decoding 0 1 2 3 4 0 1 2 3 4 MASKING NOISE INJECTION SAMPLING Ber(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5) [Inner Product] [Combine] [Weighted Average] [Independent] [Correlated] Figure 1: The overall framework of SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 Variational lower bound In VI, given a graph G with an adjacency matrix A and a feature matrix X, we approximate the true posterior 𝑝(ZV, ZF|A, X) with a variational distribution 𝑞(ZV, ZF|𝜓1,𝜓2), where 𝜓1 and 𝜓2 are variational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To capture more complex posteriors that go beyond the exponential family, we adopt the hierarchical variational framework in SIVI and assume ZV ∼ 𝑞1(ZV |𝜓1), 𝜓1 ∼ 𝑞𝜙1 (𝜓1), ZF ∼ 𝑞2(ZF |𝜓2), 𝜓2 ∼ 𝑞𝜙2 (𝜓2), (3) where𝜙1 and𝜙2 are parameters of mixing distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We marginal- ize 𝜓1 and 𝜓2 out and derive ZV ∼ ℎ𝜙1 (ZV) = ∫ 𝜓1 𝑞1(ZV |𝜓1)𝑞𝜙1 (𝜓1)d𝜓1, ZF ∼ ℎ𝜙2 (ZF) = ∫ 𝜓2 𝑞2(ZF |𝜓2)𝑞𝜙2 (𝜓2)d𝜓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (4) We maximize the log-likelihood of observations A and X, and use Jensen’s inequality to get log𝑝(A, X) ≥ Eℎ𝜙 (ZV,ZF) � log 𝑝(A, X, ZV, ZF) ℎ𝜙 (ZV, ZF) � = L, (5) where L is ELBO and ℎ𝜙 (ZV, ZF) = ∫ 𝜓1 ∫ 𝜓2 𝑞(ZV, ZF|𝜓1,𝜓2)𝑞𝜙 (𝜓1,𝜓2)d𝜓1d𝜓2 (6) is the marginal distribution over ZV and ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since ℎ𝜙 is often intractable, the Monte Carlo estimation of ELBO could be prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To address the problem, we first take the mean-field assumption: 𝑞(ZV, ZF|𝜓1,𝜓2) = 𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2), 𝑞𝜙 (𝜓1,𝜓2) = 𝑞𝜙1 (𝜓1)𝑞𝜙2 (𝜓2), (7) and substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 7 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 6 to get: ℎ𝜙 (ZV, ZF) = ℎ𝜙1 (ZV)ℎ𝜙2 (ZF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (8) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 8, we see that ZV and ZF are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then we derive a lower bound for the ELBO based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 8: L = Eℎ𝜙1 (ZV)Eℎ𝜙2 (ZF) � log 𝑝(A, X, ZV, ZF) ℎ𝜙1 (ZV)ℎ𝜙2 (ZF) � ≥ E𝜓1∼𝑞𝜙1 (𝜓)EZV∼𝑞1(ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓)EZF∼𝑞2(ZF |𝜓2) � log 𝑝(A, X, ZV, ZF) 𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2) � = L1 (9) Details on the derivation of Equation 9 are deferred to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In L1, 𝑞1 and 𝑞2 are required to be explicit and have analytic den- sity function, while 𝑞𝜙1 and 𝑞𝜙2 could be implicit but have to be convenient to be sampled from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Directly optimizing L1 by Monte Carlo Estimation is much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, in practice, nodes and their features are highly cor- related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the one hand, node embeddings are generated based on features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the other hand, the semantic information of fea- tures are directly reflected by nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Therefore, the independence between ZV and ZF in Equation 8 is inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To tackle the issue, we modify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 7 into: 𝑞(ZV, ZF|𝜓1,𝜓2) = 𝑞1(ZV |𝜓1)𝑞2(ZF|𝜓2), 𝑞𝜙 (𝜓1,𝜓2) = 𝑞𝜙2 (𝜓2|𝜓1)𝑞𝜙1 (𝜓1), (10) which explicitly characterizes the dependence between variational parameters𝜓1 and𝜓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this way,ℎ𝜙 (ZV, ZF) ≠ ℎ𝜙1 (ZV)ℎ𝜙2 (ZF), which shows that ZV and ZF are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then we can derive another lower bound for the ELBO in Equation 5: L ≥ E𝜓1∼𝑞𝜙1 (𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2 |𝜓1)E(ZV,ZF)∼𝑞(ZV,ZF |𝜓1,𝜓2) � log 𝑝(A, X, ZV, ZF) 𝑞(ZV, ZF|𝜓1,𝜓2) � = L2 (11) 4 Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking Conference’17, July 2017, Washington, DC, USA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 Encoder In the encoder, we generate ZV and ZF from observations A and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We next show how to generate ZV and ZF according to whether they are independent or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [ZV and ZF are independent].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To generate ZV, we assume that 𝑞1(ZV |𝜓1) = �𝑛 𝑖=1 𝑞1(ZV 𝑖 |𝜇V 𝑖 , ΣV 𝑖 ), where 𝑞1(ZV 𝑖 |𝜇V 𝑖 , ΣV 𝑖 ) = N (𝜇V 𝑖 , ΣV 𝑖 ) and N is multivariate Gaussian distribution with mean 𝜇V 𝑖 and diagonal co-variance matrix ΣV 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since 𝜇V 𝑖 and ΣV 𝑖 are random variables, we draw them by injecting noise ˜𝜖 into a GNN model: ˜X = CONCAT(X, ˜𝜖), ˜𝜖 ∼ ˜𝑞(𝜖), [𝜇V 𝑖 , ΣV 𝑖 ] = GNN1(A, ˜X), (12) where CONCAT(·) is the concatenation function and GNN1(·) is a GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that ˜𝜖 is random noise sampled from distribution ˜𝑞(𝜖), whose row size should be the same as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The injected noise ˜𝜖 enables the uncertainty propagation between neighboring nodes in the GNN layer, which drives the outputs of the GNN to be random variables rather than deterministic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Similarly, for ZF, we assume 𝑞2(ZF|𝜓2) = �𝑙 𝑟=1 𝑞2(ZF𝑟 |𝜇 F 𝑟 , ΣF𝑟 ) with 𝑞2(ZF𝑟 |𝜇 F 𝑟 , ΣF𝑟 ) = N (𝜇 F 𝑟 , ΣF𝑟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To infer 𝜇 F 𝑟 and ΣF𝑟 , we use a MLP model: ˆX = CONCAT(X𝑇, ˆ𝜖), ˆ𝜖 ∼ ˆ𝑞(𝜖), [𝜇 F 𝑟 , ΣF 𝑟 ] = MLP1( ˆX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (13) Note that X𝑇 ∈ R𝑙×𝑛 and the 𝑟-th row X𝑇𝑟 ∈ R𝑛 can be considered as the feature vector of feature 𝑓𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The random noise ˆ𝜖 drawn from ˆ𝑞(𝜖) injects uncertainty to the matrix X𝑇 , which models 𝜇 F 𝑟 and ΣF𝑟 as random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [ZV and ZF are correlated].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We also assume 𝑞1 and 𝑞2 follow Gaussian distribution and use the same method as in Equation 12 to generate [𝜇V 𝑖 , ΣV 𝑖 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, to capture the dependence between 𝜓1 and 𝜓2 2 in Equation 10, we compute [𝜇 F 𝑟 , ΣF𝑟 ] for feature 𝑓𝑟 based on node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, since the rich semantic information of each feature is directly reflected by values of nodes in the feature, we take the feature vector X𝑇𝑟 ∈ R𝑛 as the weight vector over all the nodes, and compute: [𝜇 F 𝑟 , ΣF 𝑟 ] = MLP2 � �𝑛 𝑖=1 X𝑇 𝑟𝑖 [𝜇V 𝑖 , ΣV 𝑖 ] �𝑛 𝑖=1 X𝑇 𝑟𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (14) In this way, [𝜇 F 𝑟 , ΣF𝑟 ] is derived from node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since 𝜇V 𝑖 and ΣV 𝑖 are random variables, 𝜇 F 𝑟 and ΣF𝑟 will also be random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 Decoder In the decoder, we aim to reconstruct both edges and features in the given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The generative process is described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' First, for each node 𝑥𝑖 and each feature 𝑓𝑟 , we draw (ZV 𝑖 , ZF𝑟 ) ∼ ℎ𝜙 (ZV 𝑖 , ZF𝑟 )3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Second, for each edge A𝑖𝑗 in the adjacency matrix A, draw A𝑖𝑗 ∼ Ber(𝑝A 𝑖𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, Ber(·) denotes Bernoulli distribu- tion and 𝑝A 𝑖𝑗 is the probability for the existence of edge A𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We implement 𝑝A 𝑖𝑗 simply by inner product as: 𝑝A 𝑖𝑗 = 𝜎((ZV 𝑖 )𝑇 ZV 𝑗 ), where 𝜎 is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Third, for each attribute X𝑖𝑟 in the 2Here, we denote 𝜓1 = [𝜇V, ΣV ] and 𝜓2 = [𝜇F, ΣF ], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 3When ZV 𝑖 and ZF 𝑟 are independent, we draw ZV 𝑖 ∼ ℎ𝜙1 (ZV 𝑖 ) and ZF 𝑟 ∼ ℎ𝜙2 (ZF 𝑟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' attribute matrix X, draw X𝑖𝑟 ∼ N (𝜇X 𝑖𝑟, ΣX 𝑖𝑟 |ZV 𝑖 , ZF𝑟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, 𝜇X 𝑖𝑟, ΣX 𝑖𝑟 are functions of ZV and ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We next introduce how to compute 𝜇X 𝑖𝑟 and ΣX 𝑖𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since ZF𝑟 con- tains rich semantic information on feature 𝑓𝑟, the affinity between 𝑥𝑖 and 𝑓𝑟 can provide fine-grained knowledge for feature reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Given a node 𝑥𝑖, to capture its affinities with all the features, the attention mechanism [28] can be applied on node and feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, this will increase the time complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For simplicity, we directly use the feature vector X𝑖 ∈ R𝑙 of 𝑥𝑖 as the weight vector and calculate the weighted average over all the feature embeddings: ¯ZV 𝑖 = �𝑙 𝑟=1 X𝑖𝑟ZF𝑟 �𝑙 𝑟=1 X𝑖𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (15) Compared with ZV 𝑖 , ¯ZV 𝑖 contains more details on how each feature can be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After that, we combine ZV 𝑖 and ¯ZV 𝑖 to get: ZV 𝑖 = COMBINE(ZV 𝑖 , ¯ZV 𝑖 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (16) In our experiments, we set the COMBINE function to be CONCAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Finally, the updated ZV 𝑖 is taken as input and fed into a GNN model to learn parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' node 𝑥𝑖: [𝜇X 𝑖 , ΣX 𝑖 ] = GNN2(A, ZV 𝑖 ), (17) where GNN2(·) is a GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 Masking To further improve the model generalizability, we introduce the masking mechanism in SeeGera by adding an additional layer to the hierarchical variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, we transform Equation 6 into: ℎ𝜙 (ZV, ZF) = ∫ 𝜓1 ∫ 𝜓2 ∫ ˜𝐺 𝑞(ZV, ZF|𝜓1,𝜓2)· 𝑞𝜙 (𝜓1,𝜓2| ˜𝐺)𝑝( ˜𝐺|A, X)d𝜓1d𝜓2d ˜𝐺 (18) From the above equation, we see that in addition to 𝜓1 and 𝜓2, the integration is performed over a new variable ˜𝐺 and a probability function 𝑝( ˜𝐺|A, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, ˜𝐺 denotes a new graph and 𝑝 is the graph augmentation probability function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The equation can lead to a new variational lower bound for the ELBO, but it is more difficult to optimize compared with L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To tackle the issue, we can first perform graph augmentation and generate a perturbed graph ˜𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After that, based on ˜𝐺, node and feature embeddings are learned based on L1 or L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We repeat the above process until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Although graph augmentation can include more operations than masking, we mainly focus on structure/feature masking in this paper, because masking is beneficial for node classification [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 Optimization In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1, we have derived two lower bounds L1 and L2 for the ELBO, according to whether ZV and ZF are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For notation brevity, we use L to overload both L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' However, directly optimizing L could lead to the degeneracy problem [32] that 𝑞𝜙1 (𝜓1), 𝑞𝜙2 (𝜓2) and 𝑞𝜙 (𝜓1,𝜓2) might converge to a point mass density, which degenerates SIVI to the vanilla VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To address 5 Conference’17, July 2017, Washington, DC, USA Xiang Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' the problem, we can regularize L by 𝐵𝐾: 𝐵𝐾 = E(𝜓1,𝜓2),{( ˜𝜓𝑘 1 , ˜𝜓𝑘 2 )}𝐾 𝑘=1∼𝑞𝜙 (𝜓1,𝜓2)DKL � 𝑞(ZV, ZF |𝜓1,𝜓2) || ˜ℎ𝐾 (ZV, ZF) � ˜ℎ𝐾 (ZV, ZF) = 𝑞(ZV, ZF|𝜓1,𝜓2) + �𝐾 𝑘=1 𝑞(ZV, ZF| ˜𝜓𝑘 1 , ˜𝜓𝑘 2 ) 𝐾 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that 𝐵𝐾 satisfies (1) 𝐵𝐾 ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (2) 𝐵𝐾 = 0 if and only if 𝐾 = 0 or 𝑞𝜙 degenerates to a point mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' According to [32], L𝐾 = L + 𝐵𝐾 is an asymptotically exact surrogate ELBO that satisfies L0 = L and lim𝐾→∞ L𝐾 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Maximizing L𝐾 with 𝐾 ≥ 1 derives positive 𝐵𝐾 and could drive 𝑞𝜙 away from degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Moreover, importance reweighting [1] can be further introduced to tighten L𝐾 by drawing 𝐽 samples {(ZV)𝑗, (ZF)𝑗,𝜓 𝑗 1,𝜓 𝑗 2 }𝐽 𝑗=1 from 𝑞(ZV, ZF,𝜓1,𝜓2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The objective can be formulated as L𝐽 𝐾 =E{(ZV ) 𝑗,(ZF ) 𝑗,𝜓 𝑗 1 ,𝜓 𝑗 2 }𝐽 𝑗=1∼𝑞(ZV,ZF |𝜓1,𝜓2)𝑞𝜙 (𝜓1,𝜓2) E{ ˜𝜓𝑘 1 , ˜𝜓𝑘 2 }𝐾 𝑘=1∼𝑞𝜙 (𝜓1,𝜓2) log 1 𝐽 𝐽∑︁ 𝑗=1 𝑝 (A, X, (ZV)𝑗, (ZF)𝑗) Ω𝑗 , (19) where Ω𝑗 = 1 𝐾 + 1 � 𝑞((ZV)𝑗, (ZF)𝑗 |𝜓 𝑗 1 ,𝜓 𝑗 2 ) + 𝐾 ∑︁ 𝑘=1 𝑞((ZV)𝑗, (ZF)𝑗 | ˜𝜓𝑘 1 , ˜𝜓𝑘 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then we take L𝐽 𝐾 as the surrogate ELBO and use stochastic gradient ascent to optimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that L1 and L2 treats differently for 𝑞𝜙 (𝜓1,𝜓2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Finally, we summarize the pseudocodes of SeeGera in Algorithm 1 (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [Complexity analysis] The major time complexity in the en- coder comes from GNN and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Suppose we use GCN as the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since the adjacency matrix is generally sparse, let 𝑑𝐴 be the average number of non-zero entries in each row of the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Let 𝑙 be the number of features and 𝑑 be the embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, we denote ˜𝑑 and ˆ𝑑 as the dimensions of injected noise to the GCN and MLP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then, the time complexi- ties for GCN and MLP are 𝑂(𝑛𝑑𝐴(𝑙 + ˜𝑑)+𝑛(𝑙 + ˜𝑑)𝑑) and 𝑂(𝑙(𝑛+ ˆ𝑑)𝑑), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In the decoder, suppose we still adopt GCN as the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Then the time complexities for reconstructing links and fea- tures are 𝑂(𝑛2𝑑) and 𝑂(𝑛𝑑𝐴𝑑 + 𝑛𝑑𝑙), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' As suggested by [12], we can down-sample the number of nonexistent edges in the graph to reduce the time complexity for recovering links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5 EXPERIMENT In this section we comprehensively evaluate the quality of node embeddings learned by SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We mainly study four research questions: (RQ1) How does SeeGera perform in the link prediction task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (RQ2) Can SeeGera effectively predict node attributes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (RQ3) While SeeGera is an unsupervised learning method, can it perform well when generalized to the node classification task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (RQ4) How does structure/feature masking influence the perfor- mance of SeeGera?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 Datasets and Baselines To answer the above four questions, we conduct extensive experi- ments on seven public datasets: Cora, Citeseer, Pubmed, Coauthor CS, Coauthor Physics, Amazon Computer and Amazon Photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Detailed descriptions and statistics on these datasets are provided in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We also compare SeeGera with 9 other SOTA baselines, which can be categorized into two groups: [Generative graph SSL methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This group of methods are based on GAE/VGAE and aim to reconstruct links and/or features, including SIG-VAE [4], CAN [18], GATE [23] and GraphMAE [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Note that GraphMAE is the SOTA generative graph SSL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [Graph contrastive learning methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Models in this type construct positive (and negative) pairs for contrast to learn node rep- resentations, including DGI [29], MVGRL [5], GRACE [41], GCA [42] and CCA-SSG [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 Implementation Details We implemented SeeGera by PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The model is initialized by Glorot initialization and trained by Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We run the model for 3500 epochs on all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In particular, for the task of node classification, we further run the logistic regression classifier for 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We adopt GCNs in both encoder and decoder for all the datasets except CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For CS, we use MLPs to replace GNNs instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We choose Bernoulli noise Ber(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5) for both ˜𝜖 ∼ ˜𝑞(𝜖) and ˆ𝜖 ∼ ˆ𝑞(𝜖), and set the noise dimension to 5 in all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We fine-tune the model hyper-parameters by grid search and details are given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' As suggested by [25], we use warm-up during the first 300 epochs to gradually impose the prior regularization terms DKL(𝑞1(ZV |𝜓1)||𝑝(ZV)) and DKL(𝑞2(ZF|𝜓2)||𝑝(ZF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For both priors 𝑝(ZV) and 𝑝(ZF), we assume that they follow standard multivariate normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In the node classification task, most results of baselines are publicly available and we directly re- port these results from their original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For the results that are missing, we run the released codes by their authors and fine-tune the model hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In the link prediction and attribute inference tasks, since most baselines are not studied in these tasks, we run their released codes with fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For fairness, we run all the experiments on a server with a single NVIDIA A100 GPU with 80G memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For simplicity, we set 𝐽 = 𝐾 = 1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 19 for both SeeGera and SIG-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' One may further refer to [22] for better value selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' All the datasets and codes are provided at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='com/SeeGera/SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 Link Prediction (RQ1) Link prediction is a typical unsupervised learning task for graph analysis, which aims to predict whether an edge exists between two nodes or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We compare SeeGera with 8 other SOTA base- lines, including GCL models: DGI [29], MVGRL [5], GRACE [41], GCA [42], CCA-SSG [37], and the generative graph SSL methods: CAN [18], SIG-VAE [4], GraphMAE [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For our proposed method SeeGera, we put forward three versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, SeeGera-v1 assumes the independence between ZV and ZF and optimizes L1, while SeeGera-v2 captures the correlations between them and op- timizes L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, SeeGera-v3 upgrades SeeGera-v2 by adding the masking mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To evaluate the model performance, we construct the valida- tion/test set by randomly selecting 20%/10% edges in the original graph as positive samples and an equal number of nonexistent edges as negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After the removal of these selected edges, we train all the models on the resulting graph with the remaining 70% 6 Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking Conference’17, July 2017, Washington, DC, USA Table 1: Link prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The error bar (±) denotes the standard deviation score of results over 10 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We highlight the best score on each dataset in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For CAN, the released codes by the authors do not implement reconstruction for numerical features, so we cannot run it on datasets with numerical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' OOM denotes the out-of-the-memory error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Metrics Method Cora Citeseer Pubmed Photo Computer CS Physics AUC DGI 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='88 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='72 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='20 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='39 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='51 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='20 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 MVGRL 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='66 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='27 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='58 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='04 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='78 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='67 OOM GRACE 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='27 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='96 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='92 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='31 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='20 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='96 GCA 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='46 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='86 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='81 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='59 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='66 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='91 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='35 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='13 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='41 CCA-SSG 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='95 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='95 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='98 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='31 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='91 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='50 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='16 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='05 CAN 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='62 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 − 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='28 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='37 − − SIG-VAE 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='74 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='54 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='86 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='36 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='23 GraphMAE 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='05 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 SeeGera-v1 94.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='94 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 CAN 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='60 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='61 − 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='38 − − SIG-VAE 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='71 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='53 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='49 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='93 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='23 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='04 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='37 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 GraphMAE 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='04 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 SeeGera-v1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='54 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='49 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='09 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='31 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='11 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='09 SeeGera-v2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='49 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='46 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='21 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='09 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 SeeGera-v3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='46 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='20 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='18 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='04 Table 2: Attribute inference performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t the MSE metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The best result in each dataset is highlighted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Method Cora Citeseer Pubmed Photo Computer CS Physics CAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 GATE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='80 × 10−3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 × 10−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='58 × 10−4 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 × 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='90 × 10−4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='99 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 OOM GraphMAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='57 × 10−3 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='42 × 10−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 × 10−4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 × 10−4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='29 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='66 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='05 SeeGera-v1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='90 × 10−3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='18 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='87 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='62 × 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='21 × 10−4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='70 × 10−5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='05 SeeGera-v2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 × 10−3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='13 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='86 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='68 × 10−4 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='11 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 SeeGera-v3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 × 10−3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='13 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='84 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='96 × 10−6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='66 × 10−4 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='34 × 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 Table 3: Node classification performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We highlight the best results in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Method Cora Citeseer Pubmed Photo Computer CS Physics DGI 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='47 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='63 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='52 MVGRL 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='11 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='03 GRACE 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='25 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 CCA-SSG 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='28 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 GraphMAE 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='35 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='27 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='17 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 SeeGera-v1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='41 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='29 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='10 SeeGera-v2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='42 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='22 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='08 SeeGera-v3 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='45 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='26 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='11 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='08 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We use two commonly used metrics, the area under the ROC curve (AUC) and the average precision (AP), to report the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For both metrics, a larger value indicates a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We use the validation set for hyper-parameter tuning and early stopping with a patience of 100, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', we stop training if both metric scores on the validation set do not increase for 100 consecutive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Similar as in [12], the predicted probability of an edge between nodes 𝑥𝑖 and 𝑥𝑗 is calculated by A𝑖𝑗 ∼ Ber(𝑝A 𝑖𝑗), where 𝑝A 𝑖𝑗 = 𝜎((ZV 𝑖 )𝑇 ZV 𝑗 ) and 𝜎 is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For each method, we run experiments 10 times and report the average results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 1 summarizes the results across all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' From the table, we have the following observations: 7 Conference’17, July 2017, Washington, DC, USA Xiang Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (a) Cora-AUC (b) Cora-AP (c) Citeseer-AUC (d) Citeseer-AP Figure 2: Hyper-parameter sensitivity analysis on the masking rates 𝛼1 and 𝛼2 in terms of link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The darker the color, the larger the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (1) The generative graph SSL methods except GraphMAE gen- erally perform better than GCL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This is because these methods learn to reconstruct links in the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For GraphMAE, it only reconstructs features, which explains its poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (2) While SeeGera is based on SIG-VAE, it achieves better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This demonstrates the importance of feature reconstruction and structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (3) Although CAN co-embeds both nodes and features, it still per- forms not very well, due to the independence assumption between node and feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, it uses linear decoder for feature reconstruction, which restricts the model’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' (4) SeeGera significantly outperforms other competitors across all the datasets, which indicates the superiority of generative VGAE model in graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In particular, the consis- tent outperformance of SeeGera-v2 over SeeGera-v1 verifies the importance of capturing the correlations between node and fea- ture embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, the improvement of SeeGera-v3 over SeeGera-v2 shows the necessity of structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 Attribute Inference (RQ2) Attribute inference is a task that predicts values of missing node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Similar as in link prediction, we hide a certain percent- age of node features and train on the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To construct the train- ing/validation/test set, we randomly select 70%/10%/20% node fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The validation set is used for hyper-parameter tuning and early stopping with a patience of 100 epoches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We take the Mean Squared Error (MSE) as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The smaller the value, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In this task, we compare SeeGera with generative graph SSL methods that reconstruct features in their decoders, including CAN [20], GATE [23] and GraphMAE [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For GCL models and other generative graph SSL methods that recover links only, they cannot be easily adapted to the task, so we do not take them as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For each method, we run experiments 10 times and report the average results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For Cora and Citeseer, we normalize node features for fair comparison, so CAN cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' From the table, while CAN, GATE and Graph- MAE can perform well on some datasets, they cannot consistently provide excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For example, GraphMAE achieves the best result on Cora, but it performs very poorly on Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, SeeGera-v3 outperforms other competitors on 5 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This shows the effectiveness of our proposed feature reconstruction method and also the masking mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We also notice that, in all cases, SeeGera-v2 achieves better performance than SeeGera-v1, which again verifies the necessity of capturing correlations between node and feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 Node Classification (RQ3) To further study SeeGera, we generalize learned embeddings to the node classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After node embeddings are trained on the entire graph, we train an additional classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, we employ Logistic Regression as the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For Cora, Citeseer and Pubmed, we use the public split for evaluation, where each class has fixed 20 nodes for training, another fixed 500 nodes and 1000 nodes for validation and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' For other datasets, we randomly split the nodes into 10%/10%/80% training/validation/test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We use classification accuracy as the metric to evaluate the model per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Since GCL methods have been shown to perform well in classification tasks, we compare SeeGera with 4 state-of-the-arts, including DGI, MVGRL, GRACE and CCA-SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We also take the recently proposed generative model GraphMAE as baseline, be- cause it bridges the gap between generative graph SSL models and GCL methods in terms of classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 3 summarizes the classification results on all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' From the table, we see that CCA-SSG, GraphMAE and SeeGera lead other competitors and they almost tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This shows that SeeGera achieves compa- rable performance with the state-of-the-art methods in the node classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, with the significant advantage in link prediction and attribute inference tasks, we conclude that SeeGera, a VGAE-based graph SSL method, can generate versatile node rep- resentations that can be widely used in various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 Parameter Analysis (RQ4) We end this section with a sensitivity analysis on the key hyper- parameters in SeeGera, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=', the structure masking rate 𝛼1 and the feature masking rate 𝛼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, we explore the stability of SeeGera w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' the perturbation of 𝛼1 and 𝛼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We conduct experi- ments on the link prediction task by varying these parameters from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5, and keeping others fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Figure 2 illustrates the AUC and AP scores of SeeGera-v3 under different 𝛼1 and 𝛼2 values on Cora and Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' From the figure, we see that SeeGera-v3 can give very 8 Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking Conference’17, July 2017, Washington, DC, USA stable performance over a wide range of 𝛼1 and 𝛼2 values, as shown by the plateau in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This demonstrates the insensitivity of SeeGera w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' these two hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 6 CONCLUSIONS We studied generative graph SSL in this paper and proposed SeeGera, which enhances the family of VGAE on graph representation learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Specifically, SeeGera adopts the hierarchical variational frame- work in SIG-VAE and mainly focuses on feature reconstruction and structure/feature masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the one hand, SeeGera co-embeds both nodes and features in the encoder and computes their embed- dings by assuming they are independent and correlated, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' After that, feature embeddings that contain rich semantic information on features are combined with node embeddings to provide more fine-grained information for feature reconstruction in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' On the other hand, we injected the masking mechanism into SeeGera by adding an additional layer to the hierarchical varia- tional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We conducted extensive experiments to evaluate the performance of SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The results show that SeeGera sig- nificantly outperforms other competitors in link prediction and attribute inference, and achieves comparable results with them in node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This further verifies the power of generative graph SSL methods in graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work is supported by Shanghai Pujiang Talent Program No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' An empirical study of graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='01116 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [41] Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Deep graph contrastive representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='04131 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' [42] Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Graph contrastive learning with adaptive augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In Proceedings of the Web Conference 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 2069–2080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' A DATASETS We use 7 public datasets which do not have license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We next briefly introduce them as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Cora, Citeseer and Pubmed [13] are three citation networks, where nodes represent publications and edges are citation links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Features for each node are the keywords it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Each dimen- sion in the feature vector indicates the presence of a keyword in the publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Nodes in these datasets are associated with labels that describe research topics of publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Coauther CS and Coauther Physics are co-authorship graphs based on the Microsoft Academic Graph from the KDD Cup 2016 challenge [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In these datasets, nodes are authors and edges cap- ture the co-authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, node features represent keywords in each author’s papers, and class labels indicate the study fields for authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Amazon Computer and Amazon Photo are extracted from the Amazon co-purchase graph [17], where nodes represent goods and edges indicate that two goods are frequently bought together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Node features are bag-of-words encoded product reviews and class labels are the product categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The statistics of these datasets are summarized in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 4: Statistics of datasets used in experiments Datasets #Nodes #Edges #Features #Classes Cora 2, 708 5, 278 1, 433 7 Citeseer 3, 327 4, 676 3, 703 6 Pubmed 19, 717 88, 651 500 3 Coauthor CS 18, 333 327, 576 6, 805 15 Coauthor Physics 34, 493 991, 848 8, 451 5 Amazon Computer 13, 752 574, 418 767 10 Coauthor Physics 7, 650 287, 326 745 8 B PSEUDOCODES This section summarizes the pseudocodes of SeeGera-v3 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Algorithm 1 SeeGera-v3 Input: A, X, 𝑝( ˜𝐺|A, X), ˜𝑞(𝜖), ˆ𝑞(𝜖), 𝜌, neural networks 𝑇𝜙1 and 𝑇𝜙2 Output: 𝜙1 and 𝜙2 1: Initialize 𝜙1, 𝜙2, set L𝐽 𝐾 = 0 2: while not converged do 3: Sample ˜𝐺 ∼ 𝑝( ˜𝐺|A, X) 4: for 𝑘 = 1 to 𝐾 do 5: Sample ˜𝜓𝑘 1 = 𝑇𝜙1 ( ˜𝐺, ˜𝜖𝑘 1 ), where ˜𝜖𝑘 1 ∼ ˜𝑞(𝜖) ⊲ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 12 6: Sample ˜𝜓𝑘 2 = 𝑇𝜙2 ( ˜𝐺, ˜𝜓𝑘 1 , ˆ𝜖𝑘 2 ), where ˆ𝜖𝑘 2 ∼ ˆ𝑞(𝜖) ⊲ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 14 7: end for 8: for 𝑗 = 1 to 𝐽 do 9: Sample 𝜖 𝑗 1 ∼ ˜𝑞(𝜖), 𝜖 𝑗 2 ∼ ˆ𝑞(𝜖) 10: Sample 𝜓 𝑗 1 = [(𝜇V)𝑗, (ΣV)𝑗] = 𝑇𝜙1 ( ˜𝐺,𝜖 𝑗 1) ⊲ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 12 11: Sample 𝜓 𝑗 2 = [(𝜇 F)𝑗, (ΣF)𝑗] = 𝑇𝜙2 ( ˜𝐺,𝜓 𝑗 1,𝜖 𝑗 2) ⊲ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 14 12: Sample 𝜖V 𝑗 ∼ N (0, 𝐼), 𝜖 A 𝑗 ∼ N (0, 𝐼) 13: Sample (ZV)𝑗 = (𝜇V)𝑗 + (ΣV)𝑗 ⊙ 𝜖V 𝑗 14: Sample (ZF)𝑗 = (𝜇 F)𝑗 + (ΣF)𝑗 ⊙ 𝜖 A 𝑗 15: Set 𝑡𝑚𝑝1 = − log Ω𝑗 ⊲ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 19 16: Set 𝑡𝑚𝑝2 = log𝑝( ˜𝐺|(ZV)𝑗, (ZF)𝑗) 17: Set 𝑡𝑚𝑝3 = log𝑝((ZV)𝑗, (ZF)𝑗) 18: Update L𝐽 𝐾 = L𝐽 𝐾 + 𝑒𝑡𝑚𝑝1+𝑡𝑚𝑝2+𝑡𝑚𝑝3 19: end for 20: Update L𝐽 𝐾 = log L𝐽 𝐾 − log 𝐽 21: Update 𝜙1 = 𝜙1 + 𝜌▽𝜙1L𝐽 𝐾 22: Update 𝜙2 = 𝜙2 + 𝜌▽𝜙2L𝐽 𝐾 23: end while 24: return 𝜙1 and 𝜙2 C ABLATION STUDY We conduct an ablation study to investigate the main components in SeeGera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' In particular, we have extensively compared SeeGera-v1, SeeGera-v2 and SeeGera-v3 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The advantage of SeeGera-v2 over SeeGera-v1 shows the importance of captur- ing the correlations between node and feature embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Also, the outperformance of SeeGera-v3 over SeeGera-v2 verifies the importance of the masking mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Further, to show the effec- tiveness of our proposed feature reconstruction method, we remove feature embeddings in the encoder and feed only node embeddings into GCN in the decoder to reconstruct features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We call this variant SeeGera_nf (no feature embedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 5 shows the results on attribute inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' We exclude SeeGera-v3 in the table, because it further uses the masking mechanism while others not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' From the table, we see that both SeeGera-v1 and SeeGera-v2 outperform SeeGera_nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' This shows the importance of using both node and feature embeddings for feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 5: The comparison between SeeGera and SeeGera_nf in the attribute inference task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Method Cora Citeseer CS SeeGera_nf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='91 × 10−3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='78 × 10−5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='15 × 10−4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='02 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 SeeGera-v1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='90 × 10−3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='18 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='87 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='62 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='06 SeeGera-v2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='89 × 10−3 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='13 × 10−5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='86 × 10−4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='24 × 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='07 10 Self-supervised Semi-implicit Graph Variational Auto-encoders with Masking Conference’17, July 2017, Washington, DC, USA D IMPLEMENTATION DETAILS We provide the detailed hyper-parameter settings of SeeGera-v3 on different datasets in Tables 6- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' All hyper-parameters are selected through small grid search, and the search space is provided as follows: Number of layers in the encoder 𝐿1: {1, 2, 3} Number of layers in the decoder 𝐿2: {1, 2, 3} Learning rate of SeeGera: {1e-3, 5e-3, 1e-2} Dropout of SeeGera: {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9} Weight decay of SeeGera: {5e-5, 1e-4, 5e-4, 1e-3} Structure masking rate 𝛼1: {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5} Feature masking rate 𝛼2: {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5} Learning rate of logistic regression: {1e-3, 5e-3, 1e-2} Dropout of logistic regression: {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='9} Weight decay of logistic regression: {5e-5, 1e-4, 5e-4, 1e-3} Table 6: Hyper-parameter setting details of SeeGera-v3 in link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Dataset 𝐿1 𝐿2 lr dropout wd 𝛼1 𝛼2 Cora 2 3 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 5e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 Citeseer 1 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 1e-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 Pubmed 2 1 5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 Photo 1 1 5e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 Computer 2 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 CS 1 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 Physics 2 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 Table 7: Hyper-parameter setting details of SeeGera-v3 in attribute inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Dataset 𝐿1 𝐿2 lr dropout wd 𝛼1 𝛼2 Cora 1 1 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 5e-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 Citeseer 1 1 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='5 Physics 2 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='0 E VARIATIONAL LOWER BOUND In this section, we show the derivation on the variational lower bounds in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' L = Eℎ𝜙1 (ZV )Eℎ𝜙2 (ZF ) � log 𝑝 (ZV |A, X)𝑝 (ZF |X𝑇 )𝑝 (A, X) ℎ𝜙1 (ZV)ℎ𝜙2 (ZF) � = −DKL(ℎ𝜙1 (ZV) ||𝑝 (ZV |A, X)) − DKL(ℎ𝜙2 (ZF) ||𝑝 (ZF |X𝑇 )) + log𝑝 (A, X) ≥ −E𝜓1∼𝑞𝜙1 (𝜓1)DKL(𝑞1(ZV |𝜓1) ||𝑝 (ZV |A, X)) − E𝜓2∼𝑞𝜙2 (𝜓2)DKL(𝑞2(ZF |𝜓2) ||𝑝 (ZF |X𝑇 )) + log𝑝 (A, X) = E𝜓1∼𝑞𝜙1 (𝜓 )EZV∼𝑞1 (ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓 )EZF∼𝑞2 (ZF |𝜓2) � log 𝑝 (A, X, ZV, ZF) 𝑞1(ZV |𝜓1)𝑞2(ZF |𝜓2) � = L1, where DKL is the KL divergence and we employ DKL(E𝜓𝑞(Z|𝜓)||𝑝(Z)) ≤ E𝜓 DKL(𝑞(Z|𝜓)||𝑝(Z)) according to [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' To better understand L1, we decompose the joint distribution 𝑝(A, X, ZV, ZF) as 𝑝 (A, X, ZV, ZF) = 𝑝 (ZV)𝑝 (ZF) � 𝑖,𝑗∈V 𝑝 (A𝑖𝑗 |ZV 𝑖 , ZV 𝑗 ) � 𝑖∈V,𝑟∈F 𝑝 (X𝑖𝑟 |ZV 𝑖 , ZF 𝑟 ) and expand L1 to derive: L1 = E𝜓1∼𝑞𝜙1 (𝜓1)EZV∼𝑞1(ZV |𝜓1) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ∑︁ 𝑖,𝑗 ∈V log𝑝(A𝑖𝑗 |ZV 𝑖 , ZV 𝑗 ) \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + E𝜓1∼𝑞𝜙1 (𝜓1)EZV∼𝑞1(ZV |𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2)EZF∼𝑞2(ZF |𝜓2) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ∑︁ 𝑖 ∈V,𝑟 ∈F log𝑝(X𝑖𝑟 |ZV 𝑖 , ZF 𝑟 ) \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb − E𝜓1∼𝑞𝜙1 (𝜓1)DKL(𝑞1(ZV |𝜓1)||𝑝(ZV)) − E𝜓2∼𝑞𝜙2 (𝜓2)DKL(𝑞2(ZF|𝜓2)||𝑝(ZF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Here, 𝑞1(ZV |𝜓1) and 𝑞2(ZF|𝜓2) are encoders that generate em- beddings of nodes and features, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 𝑝(A𝑖𝑗 |ZV 𝑖 , ZV 𝑗 ) and 𝑝(X𝑖𝑟 |ZV 𝑖 , ZF𝑟 ) are decoders that reconstruct links and features from learned embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' The first two terms in the equation cor- respond to the negative reconstruction loss for links and features, while the last two terms are regularizers that promote the closeness between variational distributions and prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Similarly, we can expand L2 as: L2 = E𝜓1∼𝑞𝜙1 (𝜓1)E𝜓2∼𝑞𝜙2 (𝜓2 |𝜓1)E(ZV,ZF)∼𝑞(ZV,ZF |𝜓1,𝜓2) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ∑︁ 𝑖,𝑗 ∈V log𝑝(A𝑖𝑗 |ZV 𝑖 , ZV 𝑗 ) + ∑︁ 𝑖 ∈V,𝑟 ∈F log𝑝(X𝑖𝑟 |ZV 𝑖 , ZF 𝑟 ) \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb − E(𝜓1,𝜓2)∼𝑞𝜙 (𝜓1,𝜓2)DKL(𝑞(ZV, ZF|𝜓1,𝜓2)||𝑝(ZV, ZF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' 11 Conference’17, July 2017, Washington, DC, USA Xiang Li, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Table 8: Hyper-parameter setting details of SeeGera-v3 in node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content=' Dataset SeeGera Logistic Regression 𝐿1 𝐿2 lr dropout wd 𝛼1 𝛼2 lr dropout wd Cora 2 2 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='3 1e-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jtFMT4oBgHgl3EQf6DFB/content/2301.12458v1.pdf'} +page_content='1 1e-3 0.' metadata={'source': 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-0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b26bc825754b640f2cec1daa597d7634e6b0b310e1a7ed08b0c64490c50758aa +size 72291 diff --git a/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/2301.05447v1.pdf.txt b/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/2301.05447v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9421c9841962130cf8780be188aaf093ee5ea878 --- /dev/null +++ b/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/2301.05447v1.pdf.txt @@ -0,0 +1,1823 @@ +arXiv:2301.05447v1 [math.OC] 13 Jan 2023 +Noname manuscript No. +(will be inserted by the editor) +Modified Limited memory BFGS with displacement +aggregation +Manish Kumar Sahu, Suvendu Ranjan +Pattanaik +Received: date / Accepted: date +Abstract A displacement aggregation strategy is applied in modified limited +memory Broyden Fletcher Goldfarb Shanno (M-LBFGS) algorithm to solve the +large-scale unconstrained optimization problems. A displacement aggregation +helps in store less memory while updating a new iterate to approximate the +inverse Hessian matrix as it discards the generating linear dependence vectors. +It has been observed that M-LBFGS scheme achieves the same theoretical con- +vergence properties as the full memory scheme or the limited memory BFGS. +Also, Numerical results show that displacement aggregation in adaptive M- +LBFGS outperforms M-LBFGS. +Keywords Unconstrained Optimization · Quasi Newton Algorithm · +Modified Limited Memory Broyden Fletcher Goldfarb Shanno · Inexact Line +Search · Large scale Optimization +Mathematics Subject Classification (2010) 49M37 · 65K10 · 90C53 · +65K05 · 90C30 +1 Introduction +The Quasi-Newton method is the most effective optimization algorithm when +the objective function is non-linear, twice continuously differentiable and in- +volves a large number of variables. In this method, we approximate the inverse- +Hessian using gradient information iteratively. If one compares the Quasi- +Newton method with the Newton method (NM) or the steepest descent method +in terms of computational cost and storage space, then the quasi-Newton +method has a clear advantage over the other two methods [5]. The rate of +Manish Kumar Sahu +NIT ROURKELA CAMPUS +Tel.: +919090862721 +E-mail: manishkumarsahu132@gmail.com + +2 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +convergence of the steepest descent method is linear, while the rate of conver- +gence of the Newton method is quadratic [5]. It is known that the computation +of the Hessian matrix is quite challenging and time-consuming for the large- +scale problem. Therefore, Quasi-Newton is a helpful method where one has to +approximate the Hessian matrix. +Depending on the different ways of approximation to the Hessian, vari- +ous methods, i.e. Symmetric Rank 1 (SR1), Davidon–Fletcher–Powell (DFP) +and Broyden–Fletcher–Goldfarb–Shanno (BFGS) updates are introduced to +approximate Hessian matrix [5]. Among them, the BFGS update formula per- +forms better than the other two methods [5]. But, when the objective function +contains a large number of variables, then the limited memory scheme is quite +helpful. Further, Albert S. Berahas employs LBFGS with displacement aggre- +gation [1] and manifests that this strategy reduces the number of iterations +and function evaluations in the L-BFGS algorithm. +Dai [7] proposes a counterexample that the standard BFGS update fails +to converge for some non-convex functions. Some efficient attempts have been +made to modify the usual quasi-Newton equation and approximate the in- +verse Hessian. Then convergence analysis for non-convex objective function +is studied under certain assumptions. Li and Fukushima slightly modify the +standard BFGS update [2] and get the super-linear and global convergence of +the Modified BFGS without the convexity assumption for the unconstrained +minimization problems. Then, Yunhai Xiao [4] show that MLBFGS algorithm +performs better than the standard LBFGS for large scale optimization prob- +lems. Here, we propose a modified limited memory BFGS method with dis- +placement aggregation to solve unconstrained optimization problems with a +twice continuously differentiable objective function. The global convergence +of this proposed method is established under certain assumption. The nu- +merical experiment on some non-convex problems shows promising results of +M-LBFGS with displacement aggregation. +1.1 Contribution +Here, we analyse the displacement aggregation strategy [1] with the MLBFGS +algorithm for large-scale optimization problems. In particular, we show that +MLBFGS with displacement aggregation outperforms M-LBFGS when the +objective function is twice continuously differentiable and contains many vari- +ables. Therefore, M-LBFGS with displacement aggregation is more acceptable +than MLBFGS. +2 Background on M-LBFGS +Let xk denotes as the k-th iteration generated by the optimization algorithm. +We can get subsequent iterate xk+1 = xk + sk after computing sk. We have to +minimize the following quadratic model to get the subsequent iterate sk+1. The + +Modified Limited memory BFGS with displacement aggregation +3 +quadratic model is given by mk+1 = fk+1 + gT +k+1d + 1 +2dT Bk+1d, where Bk+1 +is the Hessian approximation, Wk+1 is the inverse Hessian approximation and +the descent direction d is calculated by minimizing the model mk+1(d) such +as dk+1 = −Wk+1gk+1 and sk = αk+1dk+1 for some αk+1 ≥ 0. We have to +choose the Hessian approximation matrix as a symmetric matrix such that it +satisfies the secant equation Bk+1sk = yk. In BFGS, we compute the inverse +Hessian matrix W in such a way that minW ∥W − Wk∥M having +(i)W = WT , i.e., symmetric matrix and +(ii)Wyk = sk with sT +k yk > 0, +where ∥ . ∥M is denoted as weighted Frobenius norm. But in MBFGS, we have +modified quasi-Newton equation of the form +Wk+1¯yk = sk, +(2.1) +where ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT +k sk +sT +k sk ] [2]. +Algorithm 1 Backtracking line search in M-BFGS +1. Take an initial guess x0 ∈ Rn and B0 ≻ 0 be the initial Hessian approximation and +constants m ∈ (0, 1) and n ∈ (0, 1). Let k = 0. +2. In order to get dk, we have to solve the equation Bkd = gk. +3. Take the smallest positive integer j that satisfy f(xk + mjdk) ≤ f(xk) + nmjgT +k dk and +λk = mjk. +4. Then the next iterate xk+1 = xk + λkdk. +5. Update Bkusing M-BFGS Hessian formula +Bk+1 = Bk − BksksT +k Bk +sT +k Bksk ++ ¯yk ¯yT +k +¯yT +k sk +6. where sk = xk+1 − xk, yk = gk+1 − gk, ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT +k sk +sT +k sk ]. +7. Then k = k + 1 and go to step-1. +2.1 Iterative and compact form of M-BFGS +There are numerous ways to construct an inverse Hessian approximation [5]. +But, we choose the iterative and compact forms because they are the most +practical methods for constructing such approximations. Let’s discuss a generic +notation for creating an inverse Hessian approximation ¯W ≻ 0 by dropping +the dependency on the optimization algorithm’s iteration count. +Let S = [s1, . . . , sm] ∈ Rn×m denotes as the difference of iterate displace- +ment, Y = [y1, . . . , ym] ∈ Rn×m denotes the difference of iterative gradient +and ̺ = [ +1 +sT +1 ¯y1 , . . . , +1 +sT +m ¯ym ]. Then the modified BFGS update is simplified by +Wj+1 = (I − ¯yjsT +j +sT +j ¯yj +)T Wj(I − ¯yjsT +j +sT +j ¯yj +) + sjsT +j +sT +j ¯yj += ET +j WjEj + ̺jsjsT +j , +(2.2) + +4 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +where ¯Y = [¯y1, . . . , ¯ym], ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT +k sk +sT +k sk ] ([2]). +From an initial matrix W ≻ 0, one compute inverse Hessian approximation +¯W and the displacement pairs in (S, ¯Y , ̺) by initializing ¯W ←− W. For all +j ∈ {1, 2, . . ., m}, +Ej ←− I − ̺j ¯yjsT +j , Fj ←− ̺jsjsT +j , ¯W ←− ET +j ¯WEj + Fj. +(2.3) +We compute matrix ¯W with previous iterate/gradient displacement vectors +using MBFGS optimization algorithm in each iteration. We denote output +function ¯W = MBFGS(W, S, ¯Y ). +We can construct an inverse Hessian approximation using the MBFGS al- +gorithm from the initial approximation by combining all the low-rank changes +directly instead of applying the updates iteratively. The compact form of in- +verse Hessian updates in Algorithm 2 generates the same output as (2.3). +Algorithm 2 Compact form of MBFGS algorithm in matrix notation +1. Choose an initial symmetric positive definite matrix W and (S, ¯Y , ̺) as in iterative form +of MBFGS . +2. Set ( ¯B, ¯C) ∈ Rm×m×Rm×m with ¯Bi,j ←− sT +i ¯yj ∀(i, j) such that 1 ≤ i ≤ j ≤ m and +¯Ci,i ←− sT +i ¯yi ∀i ∈ [1, ..., m] be the diagonal matrix, i.e., +¯B ←− + + +sT +1 ¯y1 . . . sT +1 ¯ym +0 +... +... +0 +0 sT +m¯ym + + , ¯C ←− + + +sT +1 ¯y1 +... +sT +m ¯ +ym + + . +(2.4) +3. Set +¯ +W ←− W + +�S W ¯Y � � ¯BT (C + ¯Y T W ¯Y ) ¯B−1 − ¯B−T +− ¯B−1 +0 +� � ST +¯Y T W +� +. +(2.5) +2.2 Global and superlinear convergence of MBFGS algorithm +In the following section, we show that Agg-MBFGS method generates the same +sequence of matrices as full-memory MBFGS. As a result, an optimization +technique using the above updating method maintains the same convergence +and convergence rate properties. Also, we show that Agg-MBFGS with limited +memory achieves global superlinear convergence under certain assumptions. +Theorem 1 Suppose that level set Ω = {x : f(x) ≤ f(x0)} is bounded and +f(x) is twice continuously differentiable near x∗ contained in Ω. Let xk → x∗ +where g(x∗) = 0, Hessian H is positive definite near x∗ and αk is satisfied +by backtracking Armijo line search method, then the sequence xk generated by +MBFGS algorithm converges to x∗ superlinearly as well as globally. +Proof : we can find the proof of this theorem in [2]. + +Modified Limited memory BFGS with displacement aggregation +5 +3 Displacement aggregation +3.1 Parallel iterative displacements in succession +In this section, we show that if we find in M-BFGS that a previously-stored +iterate is a multiple of a newly computed iterate displacement vector, then we +can omit the previously stored iterate displacement vector and still approxi- +mate inverse Hessian matrix, which is generated by the full memory scheme. +Theorem 2 Let S = [s1, . . . , sm] ∈ Rn×m ,Y = [y1, . . . , ym], ¯Y = [¯y1, . . . , ¯ym] ∈ +Rn×m with ¯yj = yj + rjsj , rj = max[0, +yT +j sk +sT +k sk ], ̺ = [ +1 +sT +1 ¯y1 , . . . , +1 +sT +m ¯ym ] ∈ Rm +>0 +and let sj = σsj+1 for some nonzero σ ∈ R, then with +¯S = [s1, . . . , sj−1, sj+1, . . . , sm] +ˆY = [¯y1, . . . , ¯yj−1, ¯yj+1, . . . , ¯ym] +Algorithm 2 gives +MBFGS(W, S, ¯Y ) = MBFGS(W, ¯S, ˆY ), +(3.1) +for any matrix W ≻ 0. +Proof Consider any matrix W ≻ 0. For any j ∈ [1, 2, ....m], assume that W1:j = +MBFGS(W, S1;j, ¯Y1:j) where +S1;j = [s1, . . . , sj], Y1:j = [y1, . . . , yj] and ¯Y1:j = [¯y1, . . . , ¯yj]. From (2.3), we +have +W1;j+1 = ET +j+1W1;jEj+1 + Fj+1 += ET +j+1ET +j W1;j−1EjEj+1 + ET +j+1FjEj+1 + Fj+1. +(3.2) +As sj = σsj+1, we have +EjEj+1 = (I − ̺j ¯yjsT +j )(I − ̺j+1¯yj+1sT +j+1) += (I − (σ ¯yjsT +j+1 +σsT +j+1 ¯yj +))(I − ̺j+1¯yj+1sT +j+1) += I − ̺j+1¯yj+1sT +j+1 − ¯yjsT +j+1 +sT +j+1 ¯yj ++ ̺j+1 +¯yj(sT +j+1¯yj+1)sT +j+1 +sT +j+1 ¯yj += (I − ̺j+1¯yj+1sT +j+1) = Ej+1. +FjEj+1 = ̺jsjsT +j (I − ̺j+1¯yj+1sT +j+1) += ( +1 +σsT +j+1¯yj +)σ2sj+1sT +j+1(I − ̺j+1¯yj+1sT +j+1) += 0. + +6 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +From equation (3.2), we get +W1;j+1 = ET +j+1W1;j−1Ej+1 + Fj+1. +(3.3) +Hence, the inverse Hessian matrix obtained by skipping the updates for (sj, ¯yj) +and simply using (sj+1, ¯yj+1) same as using (sj, ¯yj) and (sj+1, ¯yj+1). +According to Theorem 2, the later update replaces the earlier vector irrespec- +tive of the gradient displacement vectors if two iterate displacement vectors +are linearly dependent. We also notice that the same inverted Hessian matrix +results if the earlier update is skipped. Therefore, if we remove the linearly de- +pendent vector, we can free up some storage. In the next section, we consider +the case that a prior displacement iteration can be expressed as the linear +combination of the several subsequent displacements. +3.2 Aggregate MBFGS +Let iterate and gradient displacement information be represented by (S, Y ) = +([s1, . . . , sm], [y1, . . . , ym]) = (S1:m, Y1:m) and we also have a previous stored +curvature pair (s0, y0) ∈ Rn × Rn, ̺0 = +1 +sT +0 ¯y0 such that +s0 = S1:mσ, +(3.4) +for some σ ∈ Rm. where ¯Y1:m = [¯y1, . . . , ¯ym]. Here, we have the linear depen- +dence s0 on the span of the newly computed set S1:m. We aim to calculate +aggregated gradient displacement +ˆY = [ˆy1, . . . , ˆym], +(3.5) +in such a way that +MBFGS(W, S0:m, ¯Y0:m) = MBFGS(W, S1:m, ˆY1:m). +(3.6) +Here, the aim is to find ˆY1:min such a way that the inverse Hessian matrix +MBFGS(W, S0:m, ¯Y0:m) is equivalent with new one, which is generated by +skipping (s0, y0) and employing only (S1:m, ˆY1:m). +3.3 Existence of matrices A and real b in Agg-MBFGS +Now, we generalize the concept of linear dependence by proving the following +theorem. Remark:- We can show the existence of the matrix A and calculation +for b by the similar proof of theorem 3.2 in ([1]). +Theorem 3 Let one has an arbitrary positive definite matrix (W > 0) along +with (i) (S1:m, Y1:m, ¯Y1:m) = ([s1, . . . , sm], [y1, . . . , ym], [¯y1, . . . , ¯ym]) where we +accumulate all the linear independent columns of S1:m, ̺1:m = [̺1, . . . , ̺m], +(s0, ¯y0) and ̺0 = +1 +sT +0 ¯y0 such that s0 = S1:mσ for some σ ∈ Rn. Then there exist + +Modified Limited memory BFGS with displacement aggregation +7 +Algorithm 3 Basic View of Displacement Aggregation, +1 : Require: W ≻ 0 and (S1:m, ¯Y1:m, ̺1:m) = ([s1, . . . , sm], [¯y1, . . . , ¯ym], [̺1, . . . , ̺m]), +(s0, y0) and ̺0 = +1 +sT +0 ¯y0 +2 : Set +¯B0:m = +�sT +0 ¯y0 sT +0 ¯Y1:m +sT +0 ¯B1:m +� +, ¯C0:m = +� +sT +0 ¯y0 ¯C1:m +� +. +(3.7) +Find ˆY1.m = [ˆy1, . . . , ˆym] such that +ˆB1:m ←− + + +sT +1 ˆy1 . . . sT +1 ˆym +... +... +sT +mˆym + + , ˆC1:m ←− + + +sT +1 ˆ +y1 +... +sT +m ˆ +ym + + +(3.8) +and satisfies +MBF GS(W, S0:m, ¯Y0:m) =W + �S0:m W ¯Y0:m +� � ¯B−T +0:m( ¯C0:m + ¯Y T +0:mW ¯Y0:m) ¯B−1 +0:m − ¯B−T +0:m +− ¯B−1 +0:m +0 +� � ST +0:m +¯Y T +0:mW +� +=W + +� +S1:m W ˆY1:m +� � ˆB−T +1:m( ˆC1:m + ˆY T +1:mW ˆY1:m) ˆB−1 +1:m − ˆB−T +1:m +− ˆB−1 +1:m +0 +� � ST +1:m +ˆY T +1:mW +� +=MBF GS(W, S1:m, ˆY1:m). +3 return ˆY1:m +Algorithm 4 Detailed View of Displacement Aggregation +1 : Require: W ≻ 0 and (S1:m, ¯Y1:m, ̺1:m) = ([s1, . . . , sm], [¯y1, . . . , ¯ym], [̺1, . . . , ̺m]), +(s0, y0) and ̺0 = +1 +sT +0 ¯y0 . +2 : Set ˆY1:m = W−1S1:m +�A 0� + ¯y0 +�b +0 +�T ++ ¯Y1:m such that with χ0 = 1 + ̺0||¯y0||2 +W, one +finds +ˆB1:m = ¯B1:m. +(3.9) +�b +0 +� += −̺0(ST +1:m ¯Y1:m − ¯B1:m)T σ. +(3.10) +( ˆY1:m− ¯Y1:m)T W( ˆY1:m− ¯Y1:m) = χ0 +̺0 +�b +0 +� �b +0 +�T +− +�A 0�T (ST +1:m ¯Y1:m− ¯B1:m)−(ST +1:m ¯Y1:m− ¯B1:m)T �A 0� +. +(3.11) +3 return ˆY1:m +A ∈ Rm×m and b ∈ Rm−1such that ˆY1:m = W−1S1:m +�A 0� ++ ¯y0 +�b +0 +�T ++ ¯Y1:m +and the equations (3.9),(3.10),(3.11) hold. Consequently, for this ˆY1:m, one +can find sT +i ˆyi = sT +i ¯yi > 0 ∀ i ∈ {1, 2, . . ., m} and +MBFGS(W, S0:m, ¯Y0:m) = MBFGS(W, S1:m, ˆY1:m). + +8 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +Proof We have to replace y with ¯y. We have (m + 1)(m − 1) unknowns in +ˆY1:m = W−1S1:m +�A 0� ++ ¯y0 +� +b +0 +�T ++ ¯Y1:m, +(3.12) +for ˆY1:m. In particular, total number of unknowns in A and b are m(m − 1) +and m − 1 respectively. So total number of unknown is m(m − 1) + (m − 1) = +(m + 1)(m − 1). We can observed that (3.12) equation imposes ˆym = ¯ym. We +can define the submatrix ¯U by skipping the last column of ¯B1:m and ˆU with +size m × (m − 1) as a submatrix of ˆB1:m. +¯U = + + +sT +1 ¯y1 . . . +sT +1 ¯ym−1 +... +... +... +... +... sT +m−1¯ym−1 +0 +. . . +0 + + +. +(3.13) +We can further simplify (3.9-3.11) to the following form: +ˆU = ¯U, +(3.14) +b = −̺0(ST +1:m ¯Y1:m−1 − ¯U)T σ, +(3.15) +and +( ˆY1:m−1 − ¯Y1:m−1)T W( ˆY1:m−1 − ¯Y1:m−1) += χ0 +̺ bbT − AT (ST +1:m ¯Y1:m−1 − ¯U) − (ST +1:m ¯Y1:m−1 − ¯U)T A. +(3.16) +We can get the total number of equations by finding the number of nonzero +entries in (3.14) and recognizing the symmetry in (3.16). From (3.14), we +have m(m − 1)/2) equation. Similarly from (3.15) and (3.16) , we get m − 1 +and m(m − 1)/2 number of equations respectively. Hence the total number of +equations are +m(m − 1)/2 + (m − 1) + m(m − 1)/2 = (m + 1)(m − 1). +As a result, (3.14-3.16) is a square system of quadratic and linear equations +that must be solved for the unknowns in the matrices A and b. An equation +for b ∈ Bm is shown in equation (3.15). For the time being, let’s assume +that b is equal to the right-hand side of this equation, which leaves us with +the task of confirming the existence of a real solution for A. Let us write +A = [a1, ...am−1] where ai has length m for all ∀ i ∈ {1, 2, . . ., m − 1}. One +can get from (3.12) that the jth column of (3.14) needs ST +1:j ¯yj = ST +1:j ˆyj ⇐⇒ +ST +1:j ¯yj = ST +1:j(W−1S1:maj + bj ¯y0 + ¯yj) . Hence, equation (3.14) reduces to the +system of affine equations +ST +1:jW−1S1:maj = −bjST +1:j ¯y0, +(3.17) + +Modified Limited memory BFGS with displacement aggregation +9 +∀ j ∈ {1, 2, . . ., m − 1}. For each j ∈ {1, . . . , m − 1}, let us write +aj = M −1 +� +aj,1 +aj,2 +� +, +(3.18) +where M = ST +1:mW−1S1:m ≻ 0, with aj,1 having length j and aj,2 having length +m − j. We need the full column rank of S1:m for the positive definiteness of +M. Hence we have +aj,1 = −bjST +1,j ¯y0 ∈ Rj, +(3.19) +to satisfy (3.17). From (3.19), we can show that (3.14) and (3.17) are sat- +isfied for any aj,2. Then we have to show the existance of aj,2 ∈ Rm−j∀ +j ∈ {1, 2, . . ., m − 1} in such a way that (3.16) is satisfied which complete +the proof. From (3.16), we can write (3.12) as +AT MA + Ψ T A + AT Ψ − ̟̟T = 0, +(3.20) +where +Ψ = ST +1:m¯y0bT + ST +1:m ¯Y1:m−1 − ¯U ∈ Rm×(m−1), +(3.21) +and +̟ = +b +√̺0 +∈ Rm−1. +(3.22) +we can rewrite equation (3.20) as +(MA + Ψ)T M −1(MA + Ψ) = ̟̟T + Ψ T M −1Ψ. +(3.23) +we can rewrite the equations (3.23) to a particular form which will be useful +for our proof. Consider the matrix MA+Ψ in (3.23). By the defination of aj,1, +aj,2, M and Ψ in (3.18 − 3.22) as well as of ¯U from (3.13) , the jth column of +the matrix can be written as +[MA+Ψ]j = +�−bjST +1:j ¯y0 +aj,2 +� ++ +� bjST +1:j ¯y0 +bjST +j+1:m¯y0 +� ++ +� +0j +ST +j+1:m¯yj +� += +� +0j +aj,2 +� ++ +� +0j +ST +j+1:m(bj ¯y0 + ¯yj) +� +where 0j is a zero vector of length j. As M −1 is positive definite matrix, +there exist a matrix L ∈ Rm×m such that LT L = M −1. Let us define Z = +[z1, . . . , zm−1] ∈ R(m−1)×(m−1) such that ZT Z = ̟̟T +Ψ T M −1Ψ (whose ex- +istance follows as ̟̟T +Ψ T M −1Ψ ⪰ 0) and defining, for all j ∈ {1, 2, . . ., m− +1}, +γj(aj,2) = L +�� +0j +aj,2 +� ++ +� +0j +ST +j+1:m(bj ¯y0 + ¯yj) +�� +. +(3.24) +it follows that the (i, j) ∈ {1, 2, ...m − 1} × {1, 2, ...m − 1} element of equation +(3.23) is +γi(ai,2)T γj(aj,2) = zT +i zj. +(3.25) +we try to prove the existence of matrix A using an inductive argument in +reverse order {1, 2, . . ., m − 1}. Firstly we consider for the unknown am−1,2 +which is one dimensional. We find with a∗ +m−1,2 = −sT +m(bm−1¯y0 + ¯ym−1) ∈ R + +10 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +that γm−1(a∗ +m−1,2) = L +� +0m−1 +sT +m(bm−1¯y0 + ¯ym−1 − sT +m(bm−1¯y0 + ¯ym−1) +� += 0m. +Let am−1,2 = a∗ +m−1,2 + λm−1where λm−1 is one dimensional. From the left +hand side of the (i, j) = (m − 1, m − 1) equation in (3.25), one can find total +∥γm−1(am−1,2)∥2 +2 number of equation which is a strongly convex quadratic +with the unknown λm−1. As ∥γm−1(am−1,2)∥2 = 0 and zT +m−1zm−1 ≥ 0, there +exists λ∗ +m−1 ∈ R such that am−1,2 = a∗ +m−1,2 + λ∗ +m−1 ∈ R satisfies the (i, j) = +(m − 1, m − 1) equation in (3.25). +In order to show the existence of al,2 ∈ Rm−l satisfying (3.25) ∀(i, j) with +i ∈ {j, . . . , m − 1} and j = l ,let us assume that there exist real numbers +{al+1,2, . . . , am−1,2} in such a way that equation (3.25) hold ∀(i, j) with i ∈ +{l + 1, . . . , m − 1} and j ∈ {i, . . ., m − 1} i.e solving the following system of +equation for al,2: +γm−1(am−1,2)T γl(al,2) = zT +m−1zl, +(3.26) +. +. +. +γl+1(al+1,2)T γl(al,2) = zT +l+1zl, +(3.27) +γl(al,2)T γl(al,2) = zT +l zl. +(3.28) +Here equation (3.26 − 3.27) are affine equations in al,2, whereas (3.28) is a +quadratic equation in al,2. For all t ∈ {l + 1, . . . , m − 1}, let +Θl+1,t = +� +0t−(l+1) +at,2 + ST +t+1:m(bt¯y0 + ¯yt) +� +∈ Rm−(l+1). +(3.29) +By (3.24), we can write +γt(at,2) = L +� +0l+1 +Θl+1,t +� +∀t ∈ {l + 1, l + 2, . . . , m − 1}. +(3.30) +Our goal is to find a∗ +l,2 ∈ Rm−lthat satisfy (3.26 − 3.27)such that +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) ∈ span{ +� +0 +Θl+1,l+1 +� +, . . . , +� +0 +Θl+1,m−1 +� +}, +(3.31) +and from (3.28) +γl(a∗ +l,2)T γl(a∗ +l,2) ≤ zT +l zl. +(3.32) +After it is completed, we can show the existence of nonzero vector ¯ +al,2 ∈ Rm−l +such that al,2∗ + λl ¯ +al,2 satisfies (3.26-3.27) for arbitrary one dimensional λl. +As equation (3.32) hold and the left hand side of (3.28) is a strongly convex +quadratic in the unknown λl, we can argue that there existsλ∗ +l ∈ R such that +al,2 = a∗ +l,2 + λ∗ +l ¯al,2 ∈ Rm−l satisfies (3.26- 3.28). + +Modified Limited memory BFGS with displacement aggregation +11 +Let {Θl+1,l+1 . . . Θl+1,m−1} has column rank c so that their exists {tl, . . . , tc} ⊆ +{l + 1, . . . , m − 1} with +span{ +� +0 +Θl+1,t1 +� +, . . . , +� +0 +Θl+1,tc +� +} = span{ +� +0 +Θl+1,l+1 +� +, . . . , +� +0 +Θl+1,m−1 +� +}. +(3.33) +Now, Let us discuss the case when c = 0 +Θl+1,t = 0m−(l+1), γt(at,2) = 0m∀t ∈ {l + 1, l + 2, . . . , m − 1}. +(3.34) +from (3.25) and (3.34) using induction hypothesis, we have +zt = 0m−1∀t ∈ {l + 1, l + 2, . . . , m − 1}. +(3.35) +From (3.34) and (3.35), the affine equations (3.26 − 3.27) are satisfied by any +a∗ +l,2 ∈ Rm−l. We can choose a∗ +l,2 = −ST +l+1:m(bl¯y0 + ¯yl) and find by (3.24) that +γl( a∗ +l,2) = L +�� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +�� += 0m, +(3.36) +which shows that this choice satisfies (3.32). Let us discuss the case when +c > 0. For a∗ +l,2 to satisfy (3.31), it follows with (3.33) that we must have +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) = +� +0 +. . . +0 +Θl+1,tl . . . Θl+1,tc +� +βl, +(3.37) +where βl has length c. Choosing +βl = +�� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LT L +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +��−1 + + +zT +tl +. +. +. +zT +tc + + +zl ∈ Rc +(3.38) +From (3.37), we have for any t ∈ {t1, . . . , tc} +γt(at,2)T γl(a∗ +l,2) = +� 0l+1 +Θl+1,t +�T +LT L +� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +� += +� 0l+1 +Θl+1,t +�T +LT L +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +� +βl = zT +t zl. +(3.39) +Now we have to prove that γt(at,2)T γl(a∗ +l,2) = zT +t zl for any t ∈ ({l+1, . . ., m− +1}|{t1, . . . , tc}). From (3.33),we have Θl+1,t = [Θl+1,t1 . . . Θl+1,tc]γl,1 for any +t and towards this end, first notice that for any such t, we have from (3.33) +that Θl+1,t = [Θl+1,t1, . . . , Θl+1,tc]γl,t for some γl,1 ∈ Rc. Combining the rela- +tionship (3.33) along with inductive hypothesis that, for any pair (i, j) with +i ∈ {l + 1, . . . , m − 1} and j ∈ {i, . . . , m − 1}, we have +� 0l+1 +Θl+1,t +�T +LT L +� 0l+1 +Θl+1,t +� += γi(ai,2)T γj(aj,2) = zT +i zj. +(3.40) + +12 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +As M −1 = LT L is positive definite matrix, we have +rank([zl+1 . . . zm−1]) = rank([γl+1(al+1,2) . . . γm−1(am−1,2)]) += rank([γt1(at1,2) . . . γtc(atc,2)]) += rank([zt1 . . . ztc]) = c +. +(3.41) +There exists a vector ¯γl,1 ∈ Rc in a such way that zt = [zt1 . . . ztc]¯γl,1 for any +t ∈ ({l + 1, . . . , m − 1}|{t1, . . . , tc}) using (3.41). if we combine the definition +of γl,1 and ¯γl,1 with (3.40), we have from any such t that +γT +l,1 +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LT L +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +� += +� +0l+1 +Θl+1,t +�T +LT L +� +0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +� += zT +t [zt1 . . . ztc] += ¯γT +l,1[zt1 . . . ztc]T [zt1 . . . ztc] += ¯γT +l,1 +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LT L +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +� +From this, we have γl,t = ¯γl,t. From the definition of γl,1 = ¯γl,1, we have +γt(at,2)T γl(a∗ +l,2) = +� 0l+1 +Θl+1,t +�T +LT L +� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +� += γT +l,1 +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LTL +� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +� += γT +l,1[zt1, . . . , ztc]T zl += ¯γT +l,1[zt1, . . . , ztc]T zl = zT +t zl +for any t ∈ ({l + 1, . . . , m − 1}|{t1, . . . , tc}). Combining this with (3.39), we +can get a∗ +l,2 (from 3.37) along with βl (from 3.38) satisfies 3.32. From (3.37), + +Modified Limited memory BFGS with displacement aggregation +13 +(3.40) and (3.38), we have +γl(al,2)T γl(a∗ +l,2) += +� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +�T +LT L +� +0l +a∗ +l,2 + ST +l+1:m(bl¯y0 + ¯yl) +� += βT +l +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LT L +� 0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +� +βl += zT +l + + +zT +t1 +. +. +. +zT +tc + + +T +�� +0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +�T +LTL +� +0l+1 +. . . +0l+1 +Θl+1,tl . . . Θl+1,tc +��−1 + + +zT +t1 +. +. +. +zT +tc + + +zl += zT +l + + +zT +t1 +. +. +. +zT +tc + + +T  + + + + + + + +zT +t1 +. +. +. +zT +tc + + +�zT +t1 . . . zT +tc +� + + + + + + +−1  + +zT +t1 +. +. +. +zT +tc + + +zl ≤ zT +l zl, +As the eigenvalue of +�zT +t1 . . . zT +tc +� + + + + + + + + +zT +t1 +. +. +. +zT +tc + + +�zT +t1 . . . zT +tc +� + + + + + + +−1  + +zT +t1 +. +. +. +zT +tc + + +lie in {0, 1}, we can get the last inequality .this last inequality becomes strict +if zl /∈ span{zt1, . . . , ztc}). Hence, we have shown that a∗ +l,2 from (3.37) satisfies +(3.32). Our aim is to show the existence of a non zero a∗ +l,2 + λl¯al,2 in such +a way that a∗ +l,2 + λl¯al,2 satisfiy (3.26-3.43) for arbitrary λl. From (3.26-3.43), +such an ¯al,2 ∈ Rm−l must satisfy +� +0l+1 +. . . +0l+1 +Θl+1,l+1 . . . Θl+1,m−1 +� +LT L +� 0l +¯al,2 +� += 0m−(l+1). +(3.42) +Since +� +0l+1 +. . . +0l+1 +Θl+1,l+1 . . . Θl+1,m−1 +� +LT L ∈ R(m−(l+1))×m, +(3.43) +There are at least l + 1 linearly independent vectors in Rm that correspond +to the null space of this matrix, which means the nullity of this matrix is at +least l + 1 dimensions from the above equation. Let Nl+1 ∈ Rm×(l+1) has l + 1 +linearly independent vectors in Rm which lie in the null space of (3.43). As this +matrix has l + 1 linearly independent columns, there exists a nonzero vector +ζl+1 ∈ Rl+1the first l elements of Nl+1ζl+1 are zero. Consider +[¯al,2]t := [Nl+1ζl+1]l+1∀t ∈ {1, . . . , m − l}. + +14 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +one can find that ¯al,2 satisfies (3.42). from (3.32), we find that the left-hand side +of equation (3.28) is a strongly convex quadratic in the unknown λl . As (3.28) +holds, we can claim that there exists λ∗ +l ∈ R such that al,2 = a∗ +l,2 + λ∗ +l ¯al,2 ∈ +Rm−l which satisfies (3.26)-(3.28). +From all the previous discussion, we can be easily shown the existance +of A ∈ Rm×(m−1) and b ∈ Rm−1 such that , with ˆY1:m ∈ Rn×m defined +as in (3.12), the equation (3.9-3.11) hold. Hence one can easily show that +sT +i ˆyi = sT +i ¯yi > 0 ∀ i ∈ {1, 2, . . ., m} hold and MBFGS( W, S0:m, ¯Y0:m)= +MBFGS(W +, S1:m, ˆY1:m) as ˆY1:m ∈ Rn×m exist, we know that sT +i ˆyi = sT +i ¯yi > 0 ∀ i ∈ +{1, 2, . . ., m} are a subset of the equations in (3.26),and (3.9-3.11) was derived +explicitly to ensure that, with ˆY1:m satisfying (3.12), one would find that (3.6) +holds. +3.4 Implimentation of Agg-MBFGS +The implementation of the Agg-MBFGS technique to iteratively aggregate +displacement information using the MBFGS approximation is discussed here. +In the limited memory scheme, we have to use the most recent curvature pair +to approximate the inverse Hessian matrix. In aggregation scheme, we use +the curvature pairs that take from a subset of the prior iteration with the +changing gradient displacement vectors. Firstly, we have to store all iterate +displacement vectors in such a way that all of the displacement vectors are +linearly independent and accumulate in the set S = {sk0, . . . , skm−1} where +{ki}m−1 +i=0 +⊂ N with ki < ki+1∀i ∈ {0, . . . , m − 2} and the element of ¯Y = +{¯yk0, . . . , ¯ykm−1} are not same as the previously computed but they can be +computed by our aggregation scheme. Then, Let us take a newly computed +curvature pair (skm, ykm) for km ∈ N with km−1 < km . In this section, we want +to show how one can add a newly computed iterate displacement vector and if +needed, we may apply our aggregation scheme to form a new set ¯S ⊆ S ∪ skm +and ˆY in such a way that +1. Both ¯S and ˆY must have the same number of vectors i.e either m or m−1. +2. All the elements of the set ¯S should be linearly independent. +3. The curvature pairs (S ∪ skm, ¯Y ∪ ykm) generates the same inverse Hessian +approximation as generated by the curvature pairs ( ¯S, ˆY ) . +Till now, we denote the previous stored iterate/gradient displacement vector +as {so, . . . , sm−1} and {¯y0, . . . , ¯ym−1} respectively. We also denote newly com- +puted curvature pair as (sm, ¯ym). After computing (sm, ¯ym), three possible +cases arise which we discuss below. +1. When the iterate displacement vectors {s0, . . . , sm} and ¯Y is linearly in- +dependent, then simply we add the new curvature (sm, ¯ym) pairs by con- +tinuing optimization algorithm ¯S = {so, . . . , sm} and ˆY = {¯y0, . . . , ¯ym}. if +m = n , then this case is impossible. + +Modified Limited memory BFGS with displacement aggregation +15 +2. If sm−1 = σsm for some σ ∈ R, then we should discard the previously +stored pair (sm−1, ¯ym−1) and replace with newly computed pair (sm, ¯ym) +so that we can form newly updated set ¯S = {s0, . . . , sm−2, sm} and ˆY = +{¯y0, . . . , ¯ym−2, ¯ym}.The choice we can take so far has been justified by +theorem (??). +3. If sj ∈ span{sj+1, . . . , sm} for some j ∈ {0, . . . , m − 2}, then we can use +our aggregation scheme to compute {ˆyj+1, . . . , ˆym}, discard the pair (sj, yj) +and form new set ¯S = {s0, ., sj−1, sj+1, ., sm} and ˆY = {¯y0, ., ¯yj−1, ˆyj+1, ., ˆym}. +All the choice that we can take so far has been justified by theorem (3). +From a computational perspective, firstly, we have to identify which of the sce- +narios occurs. The best approach to demonstrate is to keep an inner product +matrix’s Cholesky factorization consistent with previously stored iterate dis- +placement vectors. Then add a freshly computed iterate displacement vector +and see if the procedure breaks down. Before computing the newly iterated dis- +placement vector, suppose that we have a lower triangular matrix L ∈ Rm×m +with the positive diagonal elements +[sm−1 . . . s0]T [sm−1 . . . s0] = LLT. +(3.44) +This decomposition is possible because the vectors [sm−1 . . . s0] are linearly +independent. Then we add the newly computed iterate vector and do Cholesky +factorization of the augmented inner product matrix. Then their exist a scalar +µ ∈ R>0, vector ζ ∈ Rm and a lower triangular matrix M ∈ Rm×m with +[sm . . . s0]T [sm . . . s0] = +�µ 0 +ζ M +� �µ ζT +0 M T +� +. +(3.45) +One can get µ = ∥sm∥, ζT = {sT +msm−1 . . . sT +ms0}/µ and MM T = LLT −ζζT by +equating the terms from (3.44) and (3.45) i.e. using a rank-one down date(see +[19]), one can easily compute M from L. Firstly, we discuss about the case +when this down date does not break down. If all the diagonal elements are +positive, then one can reach to case 1 . Then we compute the newly iterated +displacement vector. We get the newly updated Cholesky factorization after +a subsequent optimization algorithm. But if the computed diagonal element +being equal to zero, then there will be a strong possibility that down dates +does break down i.e one can get a lower triangular matrix Ξ ∈ Ri×i with +positive diagonal elements and a vector ξ ∈ Ri for smallest i ∈ {1, . . . , m} +such that +[smsm−1 . . . sm−i]T [smsm−1 . . . sm−i] = +�Ξ 0 +ξT 0 +� �ΞT ξ +0 0 +� +. +(3.46) +Letting σ ∈ Ri be the unique vector satisfying ΞT σ = ξ, one can find that the +vector [σT , −1]T lies in the null space of (3.46), from which it follows that +[smsm−1 . . . sm−i+1]σ = sm−i. +As the iterate displacement vectors {sm−1, . . . , sm−i} are linearly dependent, +then the first element of σ should be nonzero. When the breakdown happens + +16 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +for i = 1, our problem reduces to case-2. When the breakdown happens for +i > 1, then our problem reduces to case-3 with the vector σ in such a way that +one should apply our aggregation scheme to omit the pair (sm−i, ym−i). +We can apply standard Cholesky factorization during the updating proce- +dures to factorize +[sm . . . sm−i+1sm−i−1 . . . s0]T [sm . . . sm−i+1sm−i−1 . . . s0] +when the breakdown happens in the rank-one down date. +Now, we discuss how one may implement our aggregation scheme such +that, given S1:m with full column rank, ˆY1:m, ̺1:m > 0, σ ∈ Rm satisfying s0 = +S1:mσ, ¯y0 and ̺0 > 0, one may compute A ∈ Rm×m−1 and b ∈ Rm−1 in order +to obtain ˆY1:m. We show the existance of matrx A , matrix b from Theorem +3 and also calculate matrix b from (3.15). The main task is to compute the +matrix A. From the proof of Theorem 3, let A = [a1 . . . am−1] where al ∈ Rm +for all l ∈ {1, . . . , m − 1} and as in (3.18), Let +al = M −1 +� +al,1 +al,2 +� +where al,1 ∈ Rl, al,2 ∈ Rm−l, and M = ST +1:mW−1S1:m ≻ 0. The most expen- +sive operation is to the product M −1 with +�al,1 +al,2 +� +. It is better to calculate +the inverse of M using Cholesky factorization. Then we have to update the +iterate displacement set in each iteration by adding/deleting rows/columns to +continue the process of adding/deleting rows/columns. Further, it is pretty +easy to compute the product operation with M −1 as it seems in a triangular +fashion. One can compute A, which is described as Algorithm 5. +Algorithm 5 Computation ofA in Displacement Aggregation +1. For each l ∈ {1, . . . , m − 1}, calculate the l-element vector al,1 from (3.19). +2. Calculate am−1,2 by solving the quadratic equation (3.28). +3. for l = m − 2, . . . , 1 do +4. Calculate βl from (3.38 ). +5. Calculate a∗ +l,2 from (3.37 ). +6. Calculate ¯al,2 that satisfy (3.42). +7. Calculate λ∗ +l ∈ R in such a way that al,2 = a∗ +l,2 + λ∗ +l ¯al,2 solves the quadratic equation +(3.28). +8. end for +9. A = [a1 . . . am − 1]with {aj}m−1 +j=1 +defined by (3.18). +3.5 Space complexity of the proposed algorithm +Let X = [x0, . . . , xm−1], G = [∇fx0, . . . , ∇fxm−1], S = {s0, . . . , sm−1}, Y = +{y0, . . . , ym−1}, ¯Y = {¯y0, . . . , ¯ym−1} be defined in the above Algorithm 1. The + +Modified Limited memory BFGS with displacement aggregation +17 +total computational cost of Modified LBFGS is O(5mn) per iteration, where +n is the number of variables used in the optimization algorithm and m is the +desired memory allocation given by the user where (3 ≤ m ≤ 10), typically +taken in practice). The computational cost for computing the inner products +({sT +msm−1, . . . , sT +ms0}) is O(mn). The computation of Cholesky factorization +for calculating inverse of M and calculating σ in (case 2) along with (case 3) +is O(m2). +ˆYj+1:m = W−1 +0:j−1Sj+1:m +�A 0� ++ ¯y0 +�b +0 +�T ++ ¯Yj+1:m. +(3.47) +There is a computational cost to calculate W−1 +0:j−1Sj+1:m for every possible +value of j as required in (3.47). We can compute this matrix W−1 +0:j−1 without +using MBFGS inverse Hessian approximation. The total computational cost +for doing a matrix-vector product with a compact representation of this ap- +proximation is O(j(m − j)n) ≤ O(m2n). The cost of computing b in equation +(3.15) is O(m2). Then we have to calculate matrix A as in Algorithm 5. Let +the computational cost of ST +1:my0 is O(mn), the cost of computing {al,1}m−1 +l=1 +is O(m2). The reverse order computation is used to determine {al,2}m−1 +l=1 +to +solve the system of linear and quadratic equations that comprise (3.16). The +cost of computing am−1,2 is O(1) provided a factorization of M is known and +that the elements of the right-hand side (3.23) have already been computed at +the cost of O(m3). The QR factorization of the matrix in (3.43) is the most +expensive operation in each iteration of this scheme having a dimension of +each l is (m − (l + 1)) × m. Hence the total computational cost from l = m − 2 +to l = 1 is O(m4). Hence the total computational cost is O(m2n) + O(m4). +4 Numerical Experiment +The effectiveness of Algorithm-1 (M-LBFGS), Algorithm-2, and Algorithm-3 +are examined in this section (Aggregation Modified BFGS). A collection of 52 +nonlinear unconstrained problems is used in our experiment. +We have used the CUTEst environment to carry out our numerical ex- +periment, and all the test problems are taken from CUTEst. We have used +MATLAB 2020b interface to write the code. All the computational operations +are performed on a PC(Intel(R)Core(TM)i5-10210U CPU, 2.11GHz ) with the +UBUNTU Linux Operating system. The stopping criteria for our proposed al- +gorithms and M-LBFGS are that all the iteration continue until the gradient +vector reach ||gk||∞ ≤ 10−6max(1, ||g0||∞) where k ∈ N or surpass the limit +105 . +When it comes to the majority of cases, n ≫ m, which means that the +computational costs of doing our aggregation scheme are negligible in com- +parison to the computational costs of calculating search directions, which are +the same for all algorithms applying the standard two-loop recursion for Mod- +ified L-BFGS. Here we take the initial Hessian matrix as the identity matrix. +We tested the problems with dimensions ranging from 2 to 132,200. + +18 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +Table 1 number of iteration, function evaluation, and aggregation when MLBFGS and +AggMBFGS are applied to solve the problem from CUTEst set with n ∈ [2, 123200] +. +Name +Dim +AggMBFGS +MLBFGS +(n) +Iters +func +Agg +Iters +Func +ARGLINA +200 +2 +3 +0 +2 +3 +ARGLINB +200 +3 +49 +0 +3 +49 +ARGLINC +200 +3 +49 +0 +3 +49 +ARGTRIGLS +200 +669 +10362 +0 +669 +10362 +ARWHEAD +5000 +49 +212 +43 +59 +235 +BA-L1LS +57 +352 +8284 +0 +352 +8284 +BA-L1SPLS +57 +50 +1549 +0 +50 +1549 +BDQRTIC +5000 +62 +718 +0 +62 +718 +BOX +10000 +207 +1399 +106 +280 +1930 +BOXPOWER +20000 +10 +71 +1 +10 +71 +BROWNAL +200 +17 +114 +1 +17 +115 +BROYDN3DLS +5000 +89 +429 +0 +89 +429 +BROYDN7D +5000 +7830 +49931 +0 +7830 +49931 +BROYDNBDLS +5000 +104 +756 +0 +104 +756 +BRYBND +5000 +104 +756 +0 +104 +756 +CHAINWOO +4000 +353 +3630 +0 +353 +3630 +CHNROSNB +50 +335 +3065 +0 +335 +3065 +CHNRSNBM +50 +344 +3200 +0 +344 +3200 +CURLY10 +10000 +2183 +28121 +0 +2183 +28182 +CURLY20 +10000 +417 +6167 +0 +417 +6167 +CURLY30 +10000 +326 +5383 +0 +326 +5383 +DIXMAANA +3000 +15 +40 +9 +19 +52 +DIXMAANB +3000 +32 +63 +0 +32 +63 +DIXMAANC +3000 +25 +57 +0 +25 +57 +DIXMAAND +3000 +19 +50 +0 +19 +50 +DIXMAANE +3000 +268 +498 +0 +268 +498 +DIXMAANF +3000 +203 +250 +0 +203 +250 +DIXMAANG +3000 +132 +224 +0 +132 +224 +DIXMAANH +3000 +288 +1066 +0 +288 +1066 +DIXMAANI +3000 +201 +230 +0 +201 +230 + +Modified Limited memory BFGS with displacement aggregation +19 +Name +Dim +AggMBFGS +MLBFGS +(n) +Iters +func +Agg +Iters +Func +DIXMAANJ +3000 +128 +253 +0 +128 +253 +DIXMAANK +3000 +132 +301 +0 +132 +301 +DMN15333LS +99 +12 +253 +1 +13 +253 +DMN37142LS +66 +8531 +80501 +0 +8531 +80501 +ERRINROS +50 +60 +604 +0 +60 +604 +ERRINRSM +50 +74 +721 +0 +74 +721 +FLETBV3M +5000 +10 +86 +3 +12 +141 +HILBERTA +2 +13 +48 +10 +15 +50 +LIARWHD +5000 +59 +229 +51 +15 +47 +NONCVXU2 +5000 +34 +138 +6 +36 +142 +NONDQUAR +5000 +33 +231 +0 +33 +231 +POWELLSG +5000 +28 +269 +38 +39 +155 +POWER +10000 +55 +1841 +0 +55 +1841 +SPARSQVR +10000 +78 +1018 +0 +78 +1018 +TESTQUAD +5000 +4866 +95084 +0 +4866 +95084 +TQUARTIC +5000 +236 +2939 +633 +342 +1434 +YATP2LS +123200 +264 +3062 +0 +264 +3062 +YATP1LS +123200 +46 +400 +25 +56 +294 +5 Conclusion +We have shown that Modified Limited memory BFGS with displacement ag- +gregation performs well for the twice continuously differentiable function con- +taining many variables. So it is better to use a displacement aggregation strat- +egy while working with large-scale optimization. We also observed that Mod- +ified L-BFGS with displacement aggregation gives promising results for both +convex and non-convex functions with several variables under certain assump- +tions. + +20 +Manish Kumar Sahu, Suvendu Ranjan Pattanaik +References +1. Albert S. Berahas, Frank E. Curtis , Baoyu Zhou (2021) Limited-memory BFGS with +displacement aggregation, Mathematical Programming.,194:121-157 +2. Dong Hui Lia , Masao Fukushimab (2001) +A modified BFGS method and its global +convergence in nonconvex minimization, Journal of Computational and Applied Mathe- +matics,129: 15–35 +3. Y. Dai (2006) Convergence properties of the BFGS algorithm, SIAM Journal on Opti- +mization , 13: 693-701 +4. Yunhai Xiao, Zengxin Wei, Zhiguo Wang (2008) A limited memory BFGS-type method +for large-scale unconstrained optimization, Computers and Mathematics with Applica- +tions, 56: 1001–1009 +5. J.Nocedal, S.J Wright, “ Numerical Optimization ”, 2nd edition, Springer, New York +(2006) +6. G. Yuan, Z. Wang, and P. Li . “A modified Broyden family algorithm with global con- +vergence under a weak Wolfe Powell line search for unconstrained nonconvex problems.” +volume 57 ,pages 1-21 .(2020) +7. Y. Dai. “Convergence properties of the BFGS algorithm,” SIAM Journal on Optimiza- +tion, volume 13, pages 693-701(2006) +8. Y. Yuan. 1991 .“ A modified BFGS algorithm for unconstrained optimization.” IMA +Journal of Numerical Analysis, Volume 11, Pages 325-332(1991) +9. P. Li, J . Lu and H. Feng, The Global Convergence of a Modified BFGS Method under +Inexact Line Search for Nonconvex Functions, Mathematical Problems in Engineering. +Hindawi, volume 2021, pages 1-9 (2021) +10. M. JD Powell .“Some global convergence properties of a variable metric algorithm for +minimization without exact line searches,” Nonlinear Programming, SIAM-AMS proceed- +ings, Volume 9 (1976) +11. R. H. Byoyden, J. Nocedal, Y. Yuan . “ Global convergence of a class of quasi- +Newton methods on convex problems.” SIAM Journal on Numerical Analysis. Volume +24, 1171–1189 (1987). +12. W. F. Mascarenhas. “BFGS method with exact line searches fails for nonconvex objec- +tive functions.” Mathematical Programming, volume 99, pages 49-61 (2004) +13. G. Yuan and Z. Wei . “Convergence analysis of a modified BFGS method on convex +minimizations .” Computational Optimization and Applications, volume 47, pages 237- +255. (2010) +14. D .H. Lia, M. Fukushima. “A modified BFGS method and its global convergence in +nonconvex minimization.” Journal of Computational and Applied Mathematics, Volume +129, Pages 15-35 (2001) +15. M. Kim, S. Kwon, S. Oh. “The Performance of A Modified Armijo Line Search Rule in +BFGS Optimization Method .”.Journal of the Chungcheong Mathematical Society, volume +21, pages 117-117 (2008) +16. J. F Bonnans, J. C Gilbert, C. Lemar´echal, C. Sagastiz´abal. 1995. “ A family of variable +metric proximal methods.” Mathematical Programming. volume 68, pages 15-47 +17. C. G. Broyden. “The convergence of a class of double-rank minimization algo- +rithms.”.IMA Journal of Applied Mathematics, Volume 6, Pages 76-90 (1970) +18. Q. Guo , J. Liu , D. Wang. “A modified BFGS method and its superlinear convergence in +nonconvex minimization with general line search rule,” Korean Society for Computational +and Applied Mathematics, volume 28, pages 435-446(2008) +19. P.E Gill, G.H Golub, W. Murray, M. Saunders. “Methods for modifying matrix factor- +izations “, Mathematics of computation, volume 28, pages 505-535(1974) + diff --git a/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/load_file.txt b/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d6386409b74142a0a8d017d6a4a75673f520e44 --- /dev/null +++ b/n9E5T4oBgHgl3EQfIQ5K/content/tmp_files/load_file.txt @@ -0,0 +1,1448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf,len=1447 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='05447v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='OC] 13 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (will be inserted by the editor) Modified Limited memory BFGS with displacement aggregation Manish Kumar Sahu, Suvendu Ranjan Pattanaik Received: date / Accepted: date Abstract A displacement aggregation strategy is applied in modified limited memory Broyden Fletcher Goldfarb Shanno (M-LBFGS) algorithm to solve the large-scale unconstrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' A displacement aggregation helps in store less memory while updating a new iterate to approximate the inverse Hessian matrix as it discards the generating linear dependence vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' It has been observed that M-LBFGS scheme achieves the same theoretical con- vergence properties as the full memory scheme or the limited memory BFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Also, Numerical results show that displacement aggregation in adaptive M- LBFGS outperforms M-LBFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Keywords Unconstrained Optimization · Quasi Newton Algorithm · Modified Limited Memory Broyden Fletcher Goldfarb Shanno · Inexact Line Search · Large scale Optimization Mathematics Subject Classification (2010) 49M37 · 65K10 · 90C53 · 65K05 · 90C30 1 Introduction The Quasi-Newton method is the most effective optimization algorithm when the objective function is non-linear, twice continuously differentiable and in- volves a large number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In this method, we approximate the inverse- Hessian using gradient information iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' If one compares the Quasi- Newton method with the Newton method (NM) or the steepest descent method in terms of computational cost and storage space, then the quasi-Newton method has a clear advantage over the other two methods [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The rate of Manish Kumar Sahu NIT ROURKELA CAMPUS Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' : +919090862721 E-mail: manishkumarsahu132@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='com 2 Manish Kumar Sahu, Suvendu Ranjan Pattanaik convergence of the steepest descent method is linear, while the rate of conver- gence of the Newton method is quadratic [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' It is known that the computation of the Hessian matrix is quite challenging and time-consuming for the large- scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Therefore, Quasi-Newton is a helpful method where one has to approximate the Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Depending on the different ways of approximation to the Hessian, vari- ous methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Symmetric Rank 1 (SR1), Davidon–Fletcher–Powell (DFP) and Broyden–Fletcher–Goldfarb–Shanno (BFGS) updates are introduced to approximate Hessian matrix [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Among them, the BFGS update formula per- forms better than the other two methods [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' But, when the objective function contains a large number of variables, then the limited memory scheme is quite helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Further, Albert S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Berahas employs LBFGS with displacement aggre- gation [1] and manifests that this strategy reduces the number of iterations and function evaluations in the L-BFGS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Dai [7] proposes a counterexample that the standard BFGS update fails to converge for some non-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Some efficient attempts have been made to modify the usual quasi-Newton equation and approximate the in- verse Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then convergence analysis for non-convex objective function is studied under certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Li and Fukushima slightly modify the standard BFGS update [2] and get the super-linear and global convergence of the Modified BFGS without the convexity assumption for the unconstrained minimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then, Yunhai Xiao [4] show that MLBFGS algorithm performs better than the standard LBFGS for large scale optimization prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Here, we propose a modified limited memory BFGS method with dis- placement aggregation to solve unconstrained optimization problems with a twice continuously differentiable objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The global convergence of this proposed method is established under certain assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The nu- merical experiment on some non-convex problems shows promising results of M-LBFGS with displacement aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='1 Contribution Here, we analyse the displacement aggregation strategy [1] with the MLBFGS algorithm for large-scale optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In particular, we show that MLBFGS with displacement aggregation outperforms M-LBFGS when the objective function is twice continuously differentiable and contains many vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Therefore, M-LBFGS with displacement aggregation is more acceptable than MLBFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2 Background on M-LBFGS Let xk denotes as the k-th iteration generated by the optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can get subsequent iterate xk+1 = xk + sk after computing sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We have to minimize the following quadratic model to get the subsequent iterate sk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The Modified Limited memory BFGS with displacement aggregation 3 quadratic model is given by mk+1 = fk+1 + gT k+1d + 1 2dT Bk+1d, where Bk+1 is the Hessian approximation, Wk+1 is the inverse Hessian approximation and the descent direction d is calculated by minimizing the model mk+1(d) such as dk+1 = −Wk+1gk+1 and sk = αk+1dk+1 for some αk+1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We have to choose the Hessian approximation matrix as a symmetric matrix such that it satisfies the secant equation Bk+1sk = yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In BFGS, we compute the inverse Hessian matrix W in such a way that minW ∥W − Wk∥M having (i)W = WT , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', symmetric matrix and (ii)Wyk = sk with sT k yk > 0, where ∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ∥M is denoted as weighted Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' But in MBFGS, we have modified quasi-Newton equation of the form Wk+1¯yk = sk, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='1) where ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT k sk sT k sk ] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Algorithm 1 Backtracking line search in M-BFGS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Take an initial guess x0 ∈ Rn and B0 ≻ 0 be the initial Hessian approximation and constants m ∈ (0, 1) and n ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In order to get dk, we have to solve the equation Bkd = gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Take the smallest positive integer j that satisfy f(xk + mjdk) ≤ f(xk) + nmjgT k dk and λk = mjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then the next iterate xk+1 = xk + λkdk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Update Bkusing M-BFGS Hessian formula Bk+1 = Bk − BksksT k Bk sT k Bksk + ¯yk ¯yT k ¯yT k sk 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' where sk = xk+1 − xk, yk = gk+1 − gk, ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT k sk sT k sk ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then k = k + 1 and go to step-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='1 Iterative and compact form of M-BFGS There are numerous ways to construct an inverse Hessian approximation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' But, we choose the iterative and compact forms because they are the most practical methods for constructing such approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let’s discuss a generic notation for creating an inverse Hessian approximation ¯W ≻ 0 by dropping the dependency on the optimization algorithm’s iteration count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let S = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm] ∈ Rn×m denotes as the difference of iterate displace- ment, Y = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ym] ∈ Rn×m denotes the difference of iterative gradient and ̺ = [ 1 sT 1 ¯y1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , 1 sT m ¯ym ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then the modified BFGS update is simplified by Wj+1 = (I − ¯yjsT j sT j ¯yj )T Wj(I − ¯yjsT j sT j ¯yj ) + sjsT j sT j ¯yj = ET j WjEj + ̺jsjsT j , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2) 4 Manish Kumar Sahu, Suvendu Ranjan Pattanaik where ¯Y = [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym], ¯yk = yk + rk||gk||sk and rk = 1 + max[0, yT k sk sT k sk ] ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From an initial matrix W ≻ 0, one compute inverse Hessian approximation ¯W and the displacement pairs in (S, ¯Y , ̺) by initializing ¯W ←− W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For all j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m}, Ej ←− I − ̺j ¯yjsT j , Fj ←− ̺jsjsT j , ¯W ←− ET j ¯WEj + Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='3) We compute matrix ¯W with previous iterate/gradient displacement vectors using MBFGS optimization algorithm in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We denote output function ¯W = MBFGS(W, S, ¯Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can construct an inverse Hessian approximation using the MBFGS al- gorithm from the initial approximation by combining all the low-rank changes directly instead of applying the updates iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The compact form of in- verse Hessian updates in Algorithm 2 generates the same output as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Algorithm 2 Compact form of MBFGS algorithm in matrix notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Choose an initial symmetric positive definite matrix W and (S, ¯Y , ̺) as in iterative form of MBFGS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Set ( ¯B, ¯C) ∈ Rm×m×Rm×m with ¯Bi,j ←− sT i ¯yj ∀(i, j) such that 1 ≤ i ≤ j ≤ m and ¯Ci,i ←− sT i ¯yi ∀i ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m] be the diagonal matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', ¯B ←− \uf8ee \uf8ef\uf8f0 sT 1 ¯y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT 1 ¯ym 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0 0 sT m¯ym \uf8f9 \uf8fa\uf8fb , ¯C ←− \uf8ee \uf8ef\uf8f0 sT 1 ¯y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT m ¯ ym \uf8f9 \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Set ¯ W ←− W + �S W ¯Y � � ¯BT (C + ¯Y T W ¯Y ) ¯B−1 − ¯B−T − ¯B−1 0 � � ST ¯Y T W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2 Global and superlinear convergence of MBFGS algorithm In the following section, we show that Agg-MBFGS method generates the same sequence of matrices as full-memory MBFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As a result, an optimization technique using the above updating method maintains the same convergence and convergence rate properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Also, we show that Agg-MBFGS with limited memory achieves global superlinear convergence under certain assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Theorem 1 Suppose that level set Ω = {x : f(x) ≤ f(x0)} is bounded and f(x) is twice continuously differentiable near x∗ contained in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let xk → x∗ where g(x∗) = 0, Hessian H is positive definite near x∗ and αk is satisfied by backtracking Armijo line search method, then the sequence xk generated by MBFGS algorithm converges to x∗ superlinearly as well as globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Proof : we can find the proof of this theorem in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Modified Limited memory BFGS with displacement aggregation 5 3 Displacement aggregation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='1 Parallel iterative displacements in succession In this section, we show that if we find in M-BFGS that a previously-stored iterate is a multiple of a newly computed iterate displacement vector, then we can omit the previously stored iterate displacement vector and still approxi- mate inverse Hessian matrix, which is generated by the full memory scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Theorem 2 Let S = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm] ∈ Rn×m ,Y = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ym], ¯Y = [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym] ∈ Rn×m with ¯yj = yj + rjsj , rj = max[0, yT j sk sT k sk ], ̺ = [ 1 sT 1 ¯y1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , 1 sT m ¯ym ] ∈ Rm >0 and let sj = σsj+1 for some nonzero σ ∈ R, then with ¯S = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sj−1, sj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm] ˆY = [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯yj−1, ¯yj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym] Algorithm 2 gives MBFGS(W, S, ¯Y ) = MBFGS(W, ¯S, ˆY ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='1) for any matrix W ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Proof Consider any matrix W ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For any j ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='.m], assume that W1:j = MBFGS(W, S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j, ¯Y1:j) where S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sj], Y1:j = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , yj] and ¯Y1:j = [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯yj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='3), we have W1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j+1 = ET j+1W1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='jEj+1 + Fj+1 = ET j+1ET j W1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j−1EjEj+1 + ET j+1FjEj+1 + Fj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2) As sj = σsj+1, we have EjEj+1 = (I − ̺j ¯yjsT j )(I − ̺j+1¯yj+1sT j+1) = (I − (σ ¯yjsT j+1 σsT j+1 ¯yj ))(I − ̺j+1¯yj+1sT j+1) = I − ̺j+1¯yj+1sT j+1 − ¯yjsT j+1 sT j+1 ¯yj + ̺j+1 ¯yj(sT j+1¯yj+1)sT j+1 sT j+1 ¯yj = (I − ̺j+1¯yj+1sT j+1) = Ej+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' FjEj+1 = ̺jsjsT j (I − ̺j+1¯yj+1sT j+1) = ( 1 σsT j+1¯yj )σ2sj+1sT j+1(I − ̺j+1¯yj+1sT j+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 6 Manish Kumar Sahu, Suvendu Ranjan Pattanaik From equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2), we get W1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j+1 = ET j+1W1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='j−1Ej+1 + Fj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='3) Hence, the inverse Hessian matrix obtained by skipping the updates for (sj, ¯yj) and simply using (sj+1, ¯yj+1) same as using (sj, ¯yj) and (sj+1, ¯yj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' According to Theorem 2, the later update replaces the earlier vector irrespec- tive of the gradient displacement vectors if two iterate displacement vectors are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We also notice that the same inverted Hessian matrix results if the earlier update is skipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Therefore, if we remove the linearly de- pendent vector, we can free up some storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In the next section, we consider the case that a prior displacement iteration can be expressed as the linear combination of the several subsequent displacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2 Aggregate MBFGS Let iterate and gradient displacement information be represented by (S, Y ) = ([s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm], [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ym]) = (S1:m, Y1:m) and we also have a previous stored curvature pair (s0, y0) ∈ Rn × Rn, ̺0 = 1 sT 0 ¯y0 such that s0 = S1:mσ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='4) for some σ ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' where ¯Y1:m = [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Here, we have the linear depen- dence s0 on the span of the newly computed set S1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We aim to calculate aggregated gradient displacement ˆY = [ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ˆym], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='5) in such a way that MBFGS(W, S0:m, ¯Y0:m) = MBFGS(W, S1:m, ˆY1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='6) Here, the aim is to find ˆY1:min such a way that the inverse Hessian matrix MBFGS(W, S0:m, ¯Y0:m) is equivalent with new one, which is generated by skipping (s0, y0) and employing only (S1:m, ˆY1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='3 Existence of matrices A and real b in Agg-MBFGS Now, we generalize the concept of linear dependence by proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Remark:- We can show the existence of the matrix A and calculation for b by the similar proof of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='2 in ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Theorem 3 Let one has an arbitrary positive definite matrix (W > 0) along with (i) (S1:m, Y1:m, ¯Y1:m) = ([s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm], [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ym], [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym]) where we accumulate all the linear independent columns of S1:m, ̺1:m = [̺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ̺m], (s0, ¯y0) and ̺0 = 1 sT 0 ¯y0 such that s0 = S1:mσ for some σ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then there exist Modified Limited memory BFGS with displacement aggregation 7 Algorithm 3 Basic View of Displacement Aggregation, 1 : Require: W ≻ 0 and (S1:m, ¯Y1:m, ̺1:m) = ([s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm], [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym], [̺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ̺m]), (s0, y0) and ̺0 = 1 sT 0 ¯y0 2 : Set ¯B0:m = �sT 0 ¯y0 sT 0 ¯Y1:m sT 0 ¯B1:m � , ¯C0:m = � sT 0 ¯y0 ¯C1:m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='7) Find ˆY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='m = [ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ˆym] such that ˆB1:m ←− \uf8ee \uf8ef\uf8f0 sT 1 ˆy1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT 1 ˆym .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT mˆym \uf8f9 \uf8fa\uf8fb , ˆC1:m ←− \uf8ee \uf8ef\uf8f0 sT 1 ˆ y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT m ˆ ym \uf8f9 \uf8fa\uf8fb (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='8) and satisfies MBF GS(W, S0:m, ¯Y0:m) =W + �S0:m W ¯Y0:m � � ¯B−T 0:m( ¯C0:m + ¯Y T 0:mW ¯Y0:m) ¯B−1 0:m − ¯B−T 0:m − ¯B−1 0:m 0 � � ST 0:m ¯Y T 0:mW � =W + � S1:m W ˆY1:m � � ˆB−T 1:m( ˆC1:m + ˆY T 1:mW ˆY1:m) ˆB−1 1:m − ˆB−T 1:m − ˆB−1 1:m 0 � � ST 1:m ˆY T 1:mW � =MBF GS(W, S1:m, ˆY1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3 return ˆY1:m Algorithm 4 Detailed View of Displacement Aggregation 1 : Require: W ≻ 0 and (S1:m, ¯Y1:m, ̺1:m) = ([s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm], [¯y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym], [̺1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ̺m]), (s0, y0) and ̺0 = 1 sT 0 ¯y0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2 : Set ˆY1:m = W−1S1:m �A 0� + ¯y0 �b 0 �T + ¯Y1:m such that with χ0 = 1 + ̺0||¯y0||2 W, one finds ˆB1:m = ¯B1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='9) �b 0 � = −̺0(ST 1:m ¯Y1:m − ¯B1:m)T σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='10) ( ˆY1:m− ¯Y1:m)T W( ˆY1:m− ¯Y1:m) = χ0 ̺0 �b 0 � �b 0 �T − �A 0�T (ST 1:m ¯Y1:m− ¯B1:m)−(ST 1:m ¯Y1:m− ¯B1:m)T �A 0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11) 3 return ˆY1:m A ∈ Rm×m and b ∈ Rm−1such that ˆY1:m = W−1S1:m �A 0� + ¯y0 �b 0 �T + ¯Y1:m and the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='9),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='10),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Consequently, for this ˆY1:m, one can find sT i ˆyi = sT i ¯yi > 0 ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m} and MBFGS(W, S0:m, ¯Y0:m) = MBFGS(W, S1:m, ˆY1:m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 8 Manish Kumar Sahu, Suvendu Ranjan Pattanaik Proof We have to replace y with ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We have (m + 1)(m − 1) unknowns in ˆY1:m = W−1S1:m �A 0� + ¯y0 � b 0 �T + ¯Y1:m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12) for ˆY1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In particular, total number of unknowns in A and b are m(m − 1) and m − 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' So total number of unknown is m(m − 1) + (m − 1) = (m + 1)(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can observed that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12) equation imposes ˆym = ¯ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can define the submatrix ¯U by skipping the last column of ¯B1:m and ˆU with size m × (m − 1) as a submatrix of ˆB1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ¯U = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sT 1 ¯y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT 1 ¯ym−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT m−1¯ym−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='13) We can further simplify (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11) to the following form: ˆU = ¯U, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14) b = −̺0(ST 1:m ¯Y1:m−1 − ¯U)T σ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='15) and ( ˆY1:m−1 − ¯Y1:m−1)T W( ˆY1:m−1 − ¯Y1:m−1) = χ0 ̺ bbT − AT (ST 1:m ¯Y1:m−1 − ¯U) − (ST 1:m ¯Y1:m−1 − ¯U)T A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16) We can get the total number of equations by finding the number of nonzero entries in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14) and recognizing the symmetry in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14), we have m(m − 1)/2) equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Similarly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16) , we get m − 1 and m(m − 1)/2 number of equations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence the total number of equations are m(m − 1)/2 + (m − 1) + m(m − 1)/2 = (m + 1)(m − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As a result, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16) is a square system of quadratic and linear equations that must be solved for the unknowns in the matrices A and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' An equation for b ∈ Bm is shown in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For the time being, let’s assume that b is equal to the right-hand side of this equation, which leaves us with the task of confirming the existence of a real solution for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let us write A = [a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='am−1] where ai has length m for all ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' One can get from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12) that the jth column of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14) needs ST 1:j ¯yj = ST 1:j ˆyj ⇐⇒ ST 1:j ¯yj = ST 1:j(W−1S1:maj + bj ¯y0 + ¯yj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence, equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14) reduces to the system of affine equations ST 1:jW−1S1:maj = −bjST 1:j ¯y0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='17) Modified Limited memory BFGS with displacement aggregation 9 ∀ j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}, let us write aj = M −1 � aj,1 aj,2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='18) where M = ST 1:mW−1S1:m ≻ 0, with aj,1 having length j and aj,2 having length m − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We need the full column rank of S1:m for the positive definiteness of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence we have aj,1 = −bjST 1,j ¯y0 ∈ Rj, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='19) to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='19), we can show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='17) are sat- isfied for any aj,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then we have to show the existance of aj,2 ∈ Rm−j∀ j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m − 1} in such a way that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16) is satisfied which complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16), we can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12) as AT MA + Ψ T A + AT Ψ − ̟̟T = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='20) where Ψ = ST 1:m¯y0bT + ST 1:m ¯Y1:m−1 − ¯U ∈ Rm×(m−1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='21) and ̟ = b √̺0 ∈ Rm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='22) we can rewrite equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='20) as (MA + Ψ)T M −1(MA + Ψ) = ̟̟T + Ψ T M −1Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='23) we can rewrite the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='23) to a particular form which will be useful for our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Consider the matrix MA+Ψ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' By the defination of aj,1, aj,2, M and Ψ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='18 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='22) as well as of ¯U from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='13) , the jth column of the matrix can be written as [MA+Ψ]j = �−bjST 1:j ¯y0 aj,2 � + � bjST 1:j ¯y0 bjST j+1:m¯y0 � + � 0j ST j+1:m¯yj � = � 0j aj,2 � + � 0j ST j+1:m(bj ¯y0 + ¯yj) � where 0j is a zero vector of length j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As M −1 is positive definite matrix, there exist a matrix L ∈ Rm×m such that LT L = M −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let us define Z = [z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , zm−1] ∈ R(m−1)×(m−1) such that ZT Z = ̟̟T +Ψ T M −1Ψ (whose ex- istance follows as ̟̟T +Ψ T M −1Ψ ⪰ 0) and defining, for all j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m− 1}, γj(aj,2) = L �� 0j aj,2 � + � 0j ST j+1:m(bj ¯y0 + ¯yj) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='24) it follows that the (i, j) ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='m − 1} × {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='m − 1} element of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='23) is γi(ai,2)T γj(aj,2) = zT i zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25) we try to prove the existence of matrix A using an inductive argument in reverse order {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Firstly we consider for the unknown am−1,2 which is one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We find with a∗ m−1,2 = −sT m(bm−1¯y0 + ¯ym−1) ∈ R 10 Manish Kumar Sahu, Suvendu Ranjan Pattanaik that γm−1(a∗ m−1,2) = L � 0m−1 sT m(bm−1¯y0 + ¯ym−1 − sT m(bm−1¯y0 + ¯ym−1) � = 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let am−1,2 = a∗ m−1,2 + λm−1where λm−1 is one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From the left hand side of the (i, j) = (m − 1, m − 1) equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25), one can find total ∥γm−1(am−1,2)∥2 2 number of equation which is a strongly convex quadratic with the unknown λm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As ∥γm−1(am−1,2)∥2 = 0 and zT m−1zm−1 ≥ 0, there exists λ∗ m−1 ∈ R such that am−1,2 = a∗ m−1,2 + λ∗ m−1 ∈ R satisfies the (i, j) = (m − 1, m − 1) equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In order to show the existence of al,2 ∈ Rm−l satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25) ∀(i, j) with i ∈ {j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1} and j = l ,let us assume that there exist real numbers {al+1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , am−1,2} in such a way that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25) hold ∀(i, j) with i ∈ {l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1} and j ∈ {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m − 1} i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e solving the following system of equation for al,2: γm−1(am−1,2)T γl(al,2) = zT m−1zl, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' γl+1(al+1,2)T γl(al,2) = zT l+1zl, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='27) γl(al,2)T γl(al,2) = zT l zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) Here equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='27) are affine equations in al,2, whereas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) is a quadratic equation in al,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For all t ∈ {l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}, let Θl+1,t = � 0t−(l+1) at,2 + ST t+1:m(bt¯y0 + ¯yt) � ∈ Rm−(l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='29) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='24), we can write γt(at,2) = L � 0l+1 Θl+1,t � ∀t ∈ {l + 1, l + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='30) Our goal is to find a∗ l,2 ∈ Rm−lthat satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='27)such that a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) ∈ span{ � 0 Θl+1,l+1 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , � 0 Θl+1,m−1 � }, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='31) and from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) γl(a∗ l,2)T γl(a∗ l,2) ≤ zT l zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32) After it is completed, we can show the existence of nonzero vector ¯ al,2 ∈ Rm−l such that al,2∗ + λl ¯ al,2 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='27) for arbitrary one dimensional λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32) hold and the left hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) is a strongly convex quadratic in the unknown λl, we can argue that there existsλ∗ l ∈ R such that al,2 = a∗ l,2 + λ∗ l ¯al,2 ∈ Rm−l satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Modified Limited memory BFGS with displacement aggregation 11 Let {Θl+1,l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,m−1} has column rank c so that their exists {tl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , tc} ⊆ {l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1} with span{ � 0 Θl+1,t1 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , � 0 Θl+1,tc � } = span{ � 0 Θl+1,l+1 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , � 0 Θl+1,m−1 � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='33) Now, Let us discuss the case when c = 0 Θl+1,t = 0m−(l+1), γt(at,2) = 0m∀t ∈ {l + 1, l + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='34) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='34) using induction hypothesis, we have zt = 0m−1∀t ∈ {l + 1, l + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='35) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='35), the affine equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='27) are satisfied by any a∗ l,2 ∈ Rm−l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can choose a∗ l,2 = −ST l+1:m(bl¯y0 + ¯yl) and find by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='24) that γl( a∗ l,2) = L �� 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) �� = 0m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='36) which shows that this choice satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let us discuss the case when c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For a∗ l,2 to satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='31), it follows with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='33) that we must have a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) = � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � βl, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37) where βl has length c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Choosing βl = �� 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc ��−1 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb zl ∈ Rc (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='38) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37), we have for any t ∈ {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , tc} γt(at,2)T γl(a∗ l,2) = � 0l+1 Θl+1,t �T LT L � 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) � = � 0l+1 Θl+1,t �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � βl = zT t zl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='39) Now we have to prove that γt(at,2)T γl(a∗ l,2) = zT t zl for any t ∈ ({l+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m− 1}|{t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , tc}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='33),we have Θl+1,t = [Θl+1,t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc]γl,1 for any t and towards this end, first notice that for any such t, we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='33) that Θl+1,t = [Θl+1,t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , Θl+1,tc]γl,t for some γl,1 ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Combining the rela- tionship (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='33) along with inductive hypothesis that, for any pair (i, j) with i ∈ {l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1} and j ∈ {i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}, we have � 0l+1 Θl+1,t �T LT L � 0l+1 Θl+1,t � = γi(ai,2)T γj(aj,2) = zT i zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='40) 12 Manish Kumar Sahu, Suvendu Ranjan Pattanaik As M −1 = LT L is positive definite matrix, we have rank([zl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zm−1]) = rank([γl+1(al+1,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' γm−1(am−1,2)]) = rank([γt1(at1,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' γtc(atc,2)]) = rank([zt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ztc]) = c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='41) There exists a vector ¯γl,1 ∈ Rc in a such way that zt = [zt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ztc]¯γl,1 for any t ∈ ({l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}|{t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , tc}) using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' if we combine the definition of γl,1 and ¯γl,1 with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='40), we have from any such t that γT l,1 � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � = � 0l+1 Θl+1,t �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � = zT t [zt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ztc] = ¯γT l,1[zt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ztc]T [zt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ztc] = ¯γT l,1 � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � From this, we have γl,t = ¯γl,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From the definition of γl,1 = ¯γl,1, we have γt(at,2)T γl(a∗ l,2) = � 0l+1 Θl+1,t �T LT L � 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) � = γT l,1 � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LTL � 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) � = γT l,1[zt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ztc]T zl = ¯γT l,1[zt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ztc]T zl = zT t zl for any t ∈ ({l + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}|{t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , tc}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Combining this with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='39), we can get a∗ l,2 (from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37) along with βl (from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='38) satisfies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37), Modified Limited memory BFGS with displacement aggregation 13 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='40) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='38), we have γl(al,2)T γl(a∗ l,2) = � 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) �T LT L � 0l a∗ l,2 + ST l+1:m(bl¯y0 + ¯yl) � = βT l � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LT L � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc � βl = zT l \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb T �� 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc �T LTL � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,tc ��−1 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb zl = zT l \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb T \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 −1 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb zl ≤ zT l zl, As the eigenvalue of �zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc � \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 −1 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 zT t1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' zT tc \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb lie in {0, 1}, we can get the last inequality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='this last inequality becomes strict if zl /∈ span{zt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ztc}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence, we have shown that a∗ l,2 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Our aim is to show the existence of a non zero a∗ l,2 + λl¯al,2 in such a way that a∗ l,2 + λl¯al,2 satisfiy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='43) for arbitrary λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='43), such an ¯al,2 ∈ Rm−l must satisfy � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,m−1 � LT L � 0l ¯al,2 � = 0m−(l+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='42) Since � 0l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 0l+1 Θl+1,l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Θl+1,m−1 � LT L ∈ R(m−(l+1))×m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='43) There are at least l + 1 linearly independent vectors in Rm that correspond to the null space of this matrix, which means the nullity of this matrix is at least l + 1 dimensions from the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let Nl+1 ∈ Rm×(l+1) has l + 1 linearly independent vectors in Rm which lie in the null space of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As this matrix has l + 1 linearly independent columns, there exists a nonzero vector ζl+1 ∈ Rl+1the first l elements of Nl+1ζl+1 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Consider [¯al,2]t := [Nl+1ζl+1]l+1∀t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 14 Manish Kumar Sahu, Suvendu Ranjan Pattanaik one can find that ¯al,2 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='32), we find that the left-hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) is a strongly convex quadratic in the unknown λl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28) holds, we can claim that there exists λ∗ l ∈ R such that al,2 = a∗ l,2 + λ∗ l ¯al,2 ∈ Rm−l which satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From all the previous discussion, we can be easily shown the existance of A ∈ Rm×(m−1) and b ∈ Rm−1 such that , with ˆY1:m ∈ Rn×m defined as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12), the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence one can easily show that sT i ˆyi = sT i ¯yi > 0 ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m} hold and MBFGS( W, S0:m, ¯Y0:m)= MBFGS(W , S1:m, ˆY1:m) as ˆY1:m ∈ Rn×m exist, we know that sT i ˆyi = sT i ¯yi > 0 ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', m} are a subset of the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='26),and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11) was derived explicitly to ensure that, with ˆY1:m satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='12), one would find that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='4 Implimentation of Agg-MBFGS The implementation of the Agg-MBFGS technique to iteratively aggregate displacement information using the MBFGS approximation is discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In the limited memory scheme, we have to use the most recent curvature pair to approximate the inverse Hessian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In aggregation scheme, we use the curvature pairs that take from a subset of the prior iteration with the changing gradient displacement vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Firstly, we have to store all iterate displacement vectors in such a way that all of the displacement vectors are linearly independent and accumulate in the set S = {sk0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , skm−1} where {ki}m−1 i=0 ⊂ N with ki < ki+1∀i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 2} and the element of ¯Y = {¯yk0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ykm−1} are not same as the previously computed but they can be computed by our aggregation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then, Let us take a newly computed curvature pair (skm, ykm) for km ∈ N with km−1 < km .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' In this section, we want to show how one can add a newly computed iterate displacement vector and if needed, we may apply our aggregation scheme to form a new set ¯S ⊆ S ∪ skm and ˆY in such a way that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Both ¯S and ˆY must have the same number of vectors i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e either m or m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' All the elements of the set ¯S should be linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The curvature pairs (S ∪ skm, ¯Y ∪ ykm) generates the same inverse Hessian approximation as generated by the curvature pairs ( ¯S, ˆY ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Till now, we denote the previous stored iterate/gradient displacement vector as {so, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm−1} and {¯y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym−1} respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We also denote newly com- puted curvature pair as (sm, ¯ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' After computing (sm, ¯ym), three possible cases arise which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' When the iterate displacement vectors {s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm} and ¯Y is linearly in- dependent, then simply we add the new curvature (sm, ¯ym) pairs by con- tinuing optimization algorithm ¯S = {so, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm} and ˆY = {¯y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' if m = n , then this case is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Modified Limited memory BFGS with displacement aggregation 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' If sm−1 = σsm for some σ ∈ R, then we should discard the previously stored pair (sm−1, ¯ym−1) and replace with newly computed pair (sm, ¯ym) so that we can form newly updated set ¯S = {s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm−2, sm} and ˆY = {¯y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym−2, ¯ym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='The choice we can take so far has been justified by theorem (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' If sj ∈ span{sj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm} for some j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 2}, then we can use our aggregation scheme to compute {ˆyj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ˆym}, discard the pair (sj, yj) and form new set ¯S = {s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', sj−1, sj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', sm} and ˆY = {¯y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', ¯yj−1, ˆyj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=', ˆym}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' All the choice that we can take so far has been justified by theorem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From a computational perspective, firstly, we have to identify which of the sce- narios occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The best approach to demonstrate is to keep an inner product matrix’s Cholesky factorization consistent with previously stored iterate dis- placement vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then add a freshly computed iterate displacement vector and see if the procedure breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Before computing the newly iterated dis- placement vector, suppose that we have a lower triangular matrix L ∈ Rm×m with the positive diagonal elements [sm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0]T [sm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0] = LLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='44) This decomposition is possible because the vectors [sm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0] are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then we add the newly computed iterate vector and do Cholesky factorization of the augmented inner product matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then their exist a scalar µ ∈ R>0, vector ζ ∈ Rm and a lower triangular matrix M ∈ Rm×m with [sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0]T [sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0] = �µ 0 ζ M � �µ ζT 0 M T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='45) One can get µ = ∥sm∥, ζT = {sT msm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sT ms0}/µ and MM T = LLT −ζζT by equating the terms from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='44) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='45) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' using a rank-one down date(see [19]), one can easily compute M from L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Firstly, we discuss about the case when this down date does not break down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' If all the diagonal elements are positive, then one can reach to case 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then we compute the newly iterated displacement vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We get the newly updated Cholesky factorization after a subsequent optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' But if the computed diagonal element being equal to zero, then there will be a strong possibility that down dates does break down i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='e one can get a lower triangular matrix Ξ ∈ Ri×i with positive diagonal elements and a vector ξ ∈ Ri for smallest i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m} such that [smsm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sm−i]T [smsm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sm−i] = �Ξ 0 ξT 0 � �ΞT ξ 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='46) Letting σ ∈ Ri be the unique vector satisfying ΞT σ = ξ, one can find that the vector [σT , −1]T lies in the null space of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='46), from which it follows that [smsm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sm−i+1]σ = sm−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' As the iterate displacement vectors {sm−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm−i} are linearly dependent, then the first element of σ should be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' When the breakdown happens 16 Manish Kumar Sahu, Suvendu Ranjan Pattanaik for i = 1, our problem reduces to case-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' When the breakdown happens for i > 1, then our problem reduces to case-3 with the vector σ in such a way that one should apply our aggregation scheme to omit the pair (sm−i, ym−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can apply standard Cholesky factorization during the updating proce- dures to factorize [sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sm−i+1sm−i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0]T [sm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' sm−i+1sm−i−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' s0] when the breakdown happens in the rank-one down date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Now, we discuss how one may implement our aggregation scheme such that, given S1:m with full column rank, ˆY1:m, ̺1:m > 0, σ ∈ Rm satisfying s0 = S1:mσ, ¯y0 and ̺0 > 0, one may compute A ∈ Rm×m−1 and b ∈ Rm−1 in order to obtain ˆY1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We show the existance of matrx A , matrix b from Theorem 3 and also calculate matrix b from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The main task is to compute the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' From the proof of Theorem 3, let A = [a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' am−1] where al ∈ Rm for all l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1} and as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='18), Let al = M −1 � al,1 al,2 � where al,1 ∈ Rl, al,2 ∈ Rm−l, and M = ST 1:mW−1S1:m ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The most expen- sive operation is to the product M −1 with �al,1 al,2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' It is better to calculate the inverse of M using Cholesky factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then we have to update the iterate displacement set in each iteration by adding/deleting rows/columns to continue the process of adding/deleting rows/columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Further, it is pretty easy to compute the product operation with M −1 as it seems in a triangular fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' One can compute A, which is described as Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Algorithm 5 Computation ofA in Displacement Aggregation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' For each l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , m − 1}, calculate the l-element vector al,1 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Calculate am−1,2 by solving the quadratic equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' for l = m − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , 1 do 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Calculate βl from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='38 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Calculate a∗ l,2 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='37 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Calculate ¯al,2 that satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Calculate λ∗ l ∈ R in such a way that al,2 = a∗ l,2 + λ∗ l ¯al,2 solves the quadratic equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' end for 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' A = [a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' am − 1]with {aj}m−1 j=1 defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='5 Space complexity of the proposed algorithm Let X = [x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , xm−1], G = [∇fx0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ∇fxm−1], S = {s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sm−1}, Y = {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ym−1}, ¯Y = {¯y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , ¯ym−1} be defined in the above Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The Modified Limited memory BFGS with displacement aggregation 17 total computational cost of Modified LBFGS is O(5mn) per iteration, where n is the number of variables used in the optimization algorithm and m is the desired memory allocation given by the user where (3 ≤ m ≤ 10), typically taken in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The computational cost for computing the inner products ({sT msm−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' , sT ms0}) is O(mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The computation of Cholesky factorization for calculating inverse of M and calculating σ in (case 2) along with (case 3) is O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ˆYj+1:m = W−1 0:j−1Sj+1:m �A 0� + ¯y0 �b 0 �T + ¯Yj+1:m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='47) There is a computational cost to calculate W−1 0:j−1Sj+1:m for every possible value of j as required in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We can compute this matrix W−1 0:j−1 without using MBFGS inverse Hessian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The total computational cost for doing a matrix-vector product with a compact representation of this ap- proximation is O(j(m − j)n) ≤ O(m2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The cost of computing b in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='15) is O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Then we have to calculate matrix A as in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Let the computational cost of ST 1:my0 is O(mn), the cost of computing {al,1}m−1 l=1 is O(m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The reverse order computation is used to determine {al,2}m−1 l=1 to solve the system of linear and quadratic equations that comprise (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The cost of computing am−1,2 is O(1) provided a factorization of M is known and that the elements of the right-hand side (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='23) have already been computed at the cost of O(m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The QR factorization of the matrix in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='43) is the most expensive operation in each iteration of this scheme having a dimension of each l is (m − (l + 1)) × m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence the total computational cost from l = m − 2 to l = 1 is O(m4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Hence the total computational cost is O(m2n) + O(m4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 4 Numerical Experiment The effectiveness of Algorithm-1 (M-LBFGS), Algorithm-2, and Algorithm-3 are examined in this section (Aggregation Modified BFGS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' A collection of 52 nonlinear unconstrained problems is used in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We have used the CUTEst environment to carry out our numerical ex- periment, and all the test problems are taken from CUTEst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We have used MATLAB 2020b interface to write the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' All the computational operations are performed on a PC(Intel(R)Core(TM)i5-10210U CPU, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='11GHz ) with the UBUNTU Linux Operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' The stopping criteria for our proposed al- gorithms and M-LBFGS are that all the iteration continue until the gradient vector reach ||gk||∞ ≤ 10−6max(1, ||g0||∞) where k ∈ N or surpass the limit 105 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' When it comes to the majority of cases, n ≫ m, which means that the computational costs of doing our aggregation scheme are negligible in com- parison to the computational costs of calculating search directions, which are the same for all algorithms applying the standard two-loop recursion for Mod- ified L-BFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' Here we take the initial Hessian matrix as the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We tested the problems with dimensions ranging from 2 to 132,200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' 18 Manish Kumar Sahu, Suvendu Ranjan Pattanaik Table 1 number of iteration, function evaluation, and aggregation when MLBFGS and AggMBFGS are applied to solve the problem from CUTEst set with n ∈ [2, 123200] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='Dim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='AggMBFGS ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='294 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='5 Conclusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='We have shown that Modified Limited memory BFGS with displacement ag- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='gregation performs well for the twice continuously differentiable function con- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content='taining many variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' So it is better to use a displacement aggregation strat- egy while working with large-scale optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E5T4oBgHgl3EQfIQ5K/content/2301.05447v1.pdf'} +page_content=' We also observed that Mod- ified L-BFGS with displacement aggregation gives promising results for both convex and non-convex functions with several variables under certain assump- tions.' metadata={'source': 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a/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/2301.02095v1.pdf.txt b/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/2301.02095v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ff902e57a682cba26a262e0f0b9b50aea7f873d --- /dev/null +++ b/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/2301.02095v1.pdf.txt @@ -0,0 +1,4442 @@ +Generic transversality of travelling fronts, +standing fronts, and standing pulses for +parabolic gradient systems +Romain Joly and Emmanuel Risler +January 6, 2023 +For nonlinear parabolic systems of the form +∂tw(x, t) = ∂2 +xw(x, t) − ∇V +�w(x, t) +� , +the following conclusions are proved to hold generically with respect to the +potential V : every travelling front invading a minimum point of V is bistable, +there is no standing front, every standing pulse is stable at infinity, the profiles +of these fronts and pulses approach their limits at ±∞ tangentially to the +eigenspaces corresponding to the smallest eigenvalues of D2V at these points, +these fronts and pulses are robust with respect to small perturbations of +the potential, and the set of their profiles is discrete. These conclusions are +obtained as consequences of generic transversality results for heteroclinic +and homoclinic solutions of the differential systems governing the profiles of +such fronts and pulses. Among these results, it is proved that, for a generic +Hamiltonian system of the form +¨u = ∇V (u) , +every asymmetric homoclinic orbit is transverse and every symmetric homo- +clinic orbit is elementary. +2020 Mathematics Subject Classification: 35K57, 37C20, 37C29, 37J46. +Key words and phrases: parabolic gradient systems, travelling fronts, standing fronts and pulses, homoclinic +and heteroclinic orbits of Hamiltonian systems, generic transversality, Morse–Smale theorem. +arXiv:2301.02095v1 [math.AP] 5 Jan 2023 + +Contents +1 +Introduction +1 +1.1 +Travelling fronts, standing fronts and standing pulses . . . . . . . . . . . . +1 +1.2 +Differential system governing the profiles of fronts and pulses . . . . . . . +3 +1.3 +Transversality of fronts and pulses +. . . . . . . . . . . . . . . . . . . . . . +6 +1.4 +The space of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +1.5 +Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +1.6 +Short historical review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2 +Stable and unstable manifolds of equilibria +11 +2.1 +Linearization around an equilibrium point . . . . . . . . . . . . . . . . . . +12 +2.2 +Local stable and unstable manifolds when the speed c is positive +. . . . . +13 +2.3 +Local stable and unstable manifolds when the speed c equals 0 +. . . . . . +16 +3 +Preliminary properties of travelling fronts and standing fronts and pulses +18 +3.1 +Proof of Proposition 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +3.2 +Equivalent definitions of a symmetric standing pulse . . . . . . . . . . . . +19 +3.3 +Values reached only once by profiles of travelling fronts / standing pulses +20 +4 +Tools for genericity +22 +4.1 +An instance of the Sard–Smale transversality theorem +. . . . . . . . . . . +22 +4.2 +Extending local genericity to global genericity . . . . . . . . . . . . . . . . +24 +4.3 +Potentials that are quadratic past a given radius +. . . . . . . . . . . . . . +24 +4.4 +Topological properties of restriction maps . . . . . . . . . . . . . . . . . . +25 +5 +Generic transversality of travelling fronts +27 +5.1 +Notation and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +5.2 +Reduction to a local statement +. . . . . . . . . . . . . . . . . . . . . . . . +27 +5.3 +Proof of the local statement (Proposition 5.2) . . . . . . . . . . . . . . . . +29 +5.3.1 +Setting +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +5.3.2 +Additional conditions on ν and r . . . . . . . . . . . . . . . . . . . +30 +5.3.3 +Equivalent characterizations of transversality . . . . . . . . . . . . +31 +5.3.4 +Checking hypothesis 1 of Theorem 4.2 . . . . . . . . . . . . . . . . +34 +5.3.5 +Checking hypothesis 2 of Theorem 4.2 . . . . . . . . . . . . . . . . +34 +5.3.6 +Conclusion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +6 +Generic elementarity of symmetric standing pulses +40 +6.1 +Notation and statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +6.2 +Proof of Proposition 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +6.2.1 +Application of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . +41 +6.2.2 +Checking hypothesis 1 of Theorem 4.2 . . . . . . . . . . . . . . . . +41 +6.2.3 +Checking hypothesis 2 of Theorem 4.2 . . . . . . . . . . . . . . . . +42 + +7 +Generic transversality of asymmetric standing pulses +44 +7.1 +Notation and statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +7.2 +Asymmetric standing pulses of bounded length and away from Ssym +. . . +45 +7.3 +Openness of ν⋔ asym stand pulses(¯ξ, ε) . . . . . . . . . . . . . . . . . . . . . . . +46 +7.4 +Density of ν⋔ asym stand pulses(¯ξ, ε) . . . . . . . . . . . . . . . . . . . . . . . . +47 +7.4.1 +Application of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . +47 +7.4.2 +Checking hypothesis 1 of Theorem 4.2 . . . . . . . . . . . . . . . . +48 +7.4.3 +Checking hypothesis 2 of Theorem 4.2 . . . . . . . . . . . . . . . . +48 +7.4.4 +Conclusion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +7.5 +Transversality of symmetric standing pulses? +. . . . . . . . . . . . . . . . +50 +8 +Generic non-existence of standing fronts +51 +9 +Proof of the main results +51 +9.1 +Proof of conclusion 1 of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . +51 +9.2 +Proof of conclusions 2 and 3 of Theorem 1.7 . . . . . . . . . . . . . . . . . +56 +9.3 +Proof of conclusion 4 of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . +57 +9.4 +Proof of conclusions 1 to 4 of Corollary 1.1 +. . . . . . . . . . . . . . . . . +57 +10 Generic asymptotic behaviour for the profiles of bistable travelling fronts +and of standing pulses stable at infinity +58 +10.1 Asymptotic behaviour of profiles +. . . . . . . . . . . . . . . . . . . . . . . +58 +10.2 Local strongly stable and unstable manifolds when the speed c is positive +59 +10.3 Idea of the proof of conclusion 2 of Theorem 1.8 +. . . . . . . . . . . . . . +61 +10.4 Proof of conclusion 1 of Theorem 1.8 . . . . . . . . . . . . . . . . . . . . . +63 +10.5 Proof of conclusion 2 of Theorem 1.8 for bistable travelling fronts . . . . . +64 +10.5.1 Reduction to a local statement . . . . . . . . . . . . . . . . . . . . +64 +10.5.2 Proof of the local statement . . . . . . . . . . . . . . . . . . . . . . +65 +10.5.3 Extension to all potentials . . . . . . . . . . . . . . . . . . . . . . . +65 + +1 Introduction +The purpose of this paper is to address the generic properties of travelling fronts and +standing fronts/pulses of nonlinear parabolic systems of the form +(1.1) +∂tw(x, t) = ∂2 +xw(x, t) − ∇V +�w(x, t) +� , +where time variable t and space variable x are real, the spatial domain is the whole real +line, the function (x, t) �→ w(x, t) takes its values in Rd with d a positive integer, and +the nonlinearity is the gradient of a potential function V : Rd → R, which is assumed +to be regular (of class at least C2). Travelling fronts and standing fronts/pulses are the +solutions of system (1.1) of the form w(x, t) = u(x−ct) that are stationary in a travelling +(c > 0) or standing (c = 0) frame and that approach critical points of V at the two ends +of space. An insight into the main results of this paper (Theorem 1.7, completed with +Theorem 1.8) is provided by the following corollary, illustrated by Figure 1.1. Its terms +are precisely defined in the following subsections. +Corollary 1.1. For a generic potential V the following conclusions hold: +1. every travelling front invading a minimum point of V is bistable; +2. there is no standing front, and every standing pulse is stable at infinity; +3. the set of all bistable travelling fronts and all standing pulses is discrete; +4. every travelling front and every standing pulse (considered individually) is robust +with respect to small perturbations of V ; +5. the profile of every bistable travelling front or standing pulse stable at infinity +approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to +the smallest eigenvalue of D2V at this point. +1.1 Travelling fronts, standing fronts and standing pulses +Let c be a real quantity. A function +u : R → Rd, +ξ �→ u(ξ) +is the profile of a wave travelling at speed c (if c is nonzero), respectively a stationary +solution (if c equals 0), for system (1.1) if the function w : (x, t) �→ u(x − ct) is a solution +of this system, that is if u is a solution of the second order differential system +(1.2) +¨u = −c ˙u + ∇V (u) , +where ˙u and ¨u denote the first and second derivatives of u. Up to applying the transform +(ξ, c) �→ (−ξ, −c), which leaves system (1.2) unchanged, we may assume that that the +speed c is nonnegative (and will always do so). Let us recall that a critical point of the +1 + +c1 +c2 +e3,− +e2,− +e1,− +c = 0 +c3 +x +Rd +e+ +e+ +Figure 1.1: Illustration of Corollary 1.1. The coloured lines represent the profiles of +travelling fronts or standing fronts/pulses wi(x, t) = ui(x − cit) approaching a minimum +point e+ of a given potential at the right end of space. If this potential is generic, the +critical point ei,− approached at the left end of space by every such profile must be a +minimum point, and for the standing profile (speed c = 0) this minimum point must be +e+. In addition, these profiles (up to translation of the argument) are isolated from each +other, so that the set of such profiles (up to translation of the argument) is countable +with respect to both speed and profile, and robust with respect to small perturbations +of the potential. Additionally (conclusion 5), these profiles approach their limits e+ +(ei,−) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V (e+) +(D2V (ei,−)), but this last feature is not displayed on the figure. +potential V is a point e of Rd such that ∇V (e) = 0, and that a non-degenerate local +minimum point of V is a critical point m of V such that D2V (m) is positive definite. +If e− and e+ are two critical points of V , and if u is a non-constant global solution of +system (1.2) such that the following limits hold +(1.3) +u(ξ) −−−−→ +ξ→−∞ e− +and +u(ξ) −−−−→ +ξ→−∞ e+ , +then the solution (x, t) �→ u(x − ct) of (1.1) is said to connect e− to e+ and is called: +• a travelling front if c ̸= 0 and e− ̸= e+, +• a standing front if c = 0 and e− ̸= e+, +• a standing pulse if c = 0 and e− = e+. +In addition, a travelling or standing front connecting a critical point e− to a critical +point e+ is said to be bistable if both these critical points are non-degenerate (local or +global) minimum points of V . Accordingly, a standing pulse connecting a critical point +to itself is said to be stable at infinity if this critical point is a non-degenerate (local +or global) minimum point of V . Among standing pulses, it is relevant to distinguish +symmetric pulses, which are even with respect to some time (the solution goes back and +forth following the same path), from asymmetric pulses which are not. +Travelling fronts and standing fronts and pulses can be interpreted in terms of energy as +follows. Let us denote by ˜V the opposite potential −V . Then, in system (1.2) (where the +argument ξ plays the role of a time), the speed plays the role of a damping coefficient, and +2 + +the nonlinear conservative force derives from the potential ˜V . In other words, the system +governs the motion of a ball rolling on the graph of ˜V , submitted to the gravitational +force and to a friction force −c ˙u. Its Hamiltonian energy is the function HV defined as: +(1.4) +HV : R2d → R , +(u, v) �→ 1 +2|v|2 − V (u) = 1 +2|v|2 + ˜V (u) , +and, for every solution ξ �→ u(ξ) of this system and every time ξ where this solution is +defined, the time derivative of HV along this solution reads +(1.5) +d +dξ HV +�u(ξ), ˙u(ξ) +� = −c| ˙u(ξ)|2 . +As a consequence, if such a solution satisfies the limits (1.3), +• if c is positive then e− and e+ must differ and V (e−) must be smaller than +V (e+); then, +– from the point of view of the parabolic system (1.1), the travelling front will +be said to invade the “higher” (with respect to V ) critical point e+ (which is +“replaced” with the “lower” one e−); +– from the point of view of the Hamiltonian system (1.2) the damping “absorbs” +the positive lag ˜V (e−) − ˜V (e+) (the “higher” critical point with respect to ˜V +is e− and the “lower” one is e+); +• and if c is zero then e− and e+ must belong to the same level set of V . +In addition, as explained on Figures 1.2 to 1.4, the mechanical interpretation provides an +intuitive explanation of Corollary 1.1. +1.2 Differential system governing the profiles of fronts and pulses +Keeping the previous notation, let us consider the vector field +(1.6) +Fc,V : R2d → R2d, +� +u +v +� +�→ +� +v +∇V (u) − cv +� +. +The second order differential system (1.2) is equivalent to the first order differential +system +(1.7) +� +˙u = v +˙v = ∇V (u) − cv +or equivalently +˙U = Fc,V (U) with U = (u, v) ∈ R2d . +A point E of R2d is an equilibrium point of system (1.7) if and only if there exists a +critical point e of V such that E equals (e, 0). Assume that e is non-degenerate, or in +other words that 0 is not in the spectrum of the symmetric matrix D2V (e). Let W s +c,V (E) +and W u +c,V (E) denote the stable and unstable manifolds of E for the differential system +(1.7). Recall that each of these manifolds is defined as the union of the images of the +3 + +e1 +e2 +˜V = −V +e3 u(ξ) +Figure 1.2: Heteroclinic connections between critical points belonging to different level +sets of V for system (1.2) (dimension d equals 2). This system governs the motion of +a ball rolling on the surface u �→ ˜V (u) = −V (u), submitted to the gravitational force +and to a friction force −c ˙u. The minimum points e1 and e3 of V are maximum points +for −V , whereas e2 denotes a saddle point. A travelling front connecting e1 or e2 to e3 +corresponds to the ball leaving e1 or e2 with speed zero at time −∞, and rolling towards +e3 with the suitable damping c such that is reaches e3 at rest when time goes to +∞. +Roughly speaking, this asymptotic behaviour in the future requires two conditions: the +right direction (towards e3) and the right damping (to reach e3 and stop there). As +can be intuitively seen on the figure, starting from e1 provides two degrees of freedom +(direction and damping), whereas starting from e2 provides only one (damping). For +that reason, connections between e1 and e3 (bistable travelling fronts invading e3) are +expected to occur generically and to be a robust feature, by contrast with connections +between e2 and e3 (non bistable travelling fronts invading e3), which should occur only +for rare potentials. Conclusions 1 and 3 of Corollary 1.1 above and Theorem 1.7 below +formally confirm these expectations. +4 + +e +Figure 1.3: A symmetric standing pulse. A ball is dropped with speed zero at the same +level of V as the critical point e, and there is no damping. Because the Hamiltonian +(energy) is conserved, reaching e as time goes to +∞ only requires to adjust the “direction” +towards e. If e is a minimum point of V (a maximum point of −V ) as on the figure, this +condition can be fulfilled by choosing the adequate dropping point on the one-dimensional +level set V −1�{e} +�. If by contrast e was a saddle point, the dropping point should also +lie on the one-dimensional stable manifold of e, adding an additional condition. For +that reason, symmetric standing pulses stable at infinity are expected to be a generic +and robust feature, whereas those not stable at infinity should not occur but for rare +potentials. Conclusions 2 and 3 of Corollary 1.1 above and Theorem 1.7 below confirm +these expectations. +u(ξ) +e +Figure 1.4: An asymmetric standing pulse. A ball “leaves” the critical point e with +speed zero at time −∞, and there is no damping. Because the Hamiltonian (energy) is +conserved, going back to e as time goes to +∞ only requires to adjust the “direction” +towards e. If e is a minimum point of V (a maximum point of −V ) as on the figure, +this condition can be fulfilled by leaving e in the adequate direction. If by contrast +e was a saddle point, there would be no choice for the direction of leaving (and in +addition, going back to e would require to do so through a particular direction). For +that reason, asymmetric standing pulses stable at infinity are expected to be a generic +and robust feature, whereas those not stable at infinity should not occur but for rare +potentials. Conclusions 2 and 3 of Corollary 1.1 above and Theorem 1.7 below confirm +these expectations. +5 + +solutions ξ �→ U(ξ) that converge to E at an exponential rate as ξ goes to +∞/−∞, +tangentially to the stable/unstable linear space of this equilibrium (see section 2). The +following statement, proved in subsection 3.1, formalizes the correspondence between the +profiles of travelling fronts and standing fronts/pulses and the intersections between such +manifolds. +Proposition 1.2. Let e− and e+ be two (possibly equal) non-degenerate critical points of +V , let E− and E+ denote the corresponding equilibria for system (1.7), and let c denote a +real (zero or nonzero) quantity. For every profile ξ �→ u(ξ) of a front/pulse connecting e− +to e+ and travelling at speed c (or standing if c equals zero), the image of the corresponding +solution ξ �→ +�u(ξ), ˙u(ξ) +� of system (1.7) belongs to W u +c,V (E−) ∩ W s +c,V (E+). +The meaning of this proposition is twofold. First, it states that the convergence of +u(ξ) towards e± at ±∞ yields the convergence of +�u(ξ), ˙u(ξ) +� towards (e±, 0). In other +words, every profile of a travelling or standing front of the partial differential system (1.1) +corresponds to a heteroclinic orbit of system (1.7), and every profile of a standing pulse +corresponds to a homoclinic orbit of this system. Second, those convergences occur at an +exponential rate, thus not along a centre manifold (which exists for a non-degenerate +critical point which is not a minimum point when c vanishes, see subsection 2.1). +1.3 Transversality of fronts and pulses +Usually, the transversality of a heteroclinic orbit connecting two equilibria E− and E+ +is defined as the transversality of the intersection between the unstable manifold of E− +and the stable manifold of E+. For travelling fronts, however, the freedom of moving the +speed c must be taken into account, and leads to the following definition. +Definition 1.3 (transversality of a travelling front). Let e− and e+ be two non-degenerate +critical points of V and let E− and E+ denote the corresponding equilibria for system +(1.7). A front with profile ξ �→ u(ξ) travelling at a positive speed c and connecting e− to +e+ is said to be transverse if the intersection +� +� � +c′>0 +{c′} × W u +c′,V (E−) +� +� ∩ +� +� � +c′>0 +{c′} × W s +c′,V (E+) +� +� +is transverse, in R2d+1, along the set {c} × U(R). +For a standing pulse (connecting a critical point e of V to itself) the speed c equals 0, +so that system (1.7) preserves the Hamiltonian HV defined in (1.4). As a consequence, +the stable and unstable manifolds of the equilibrium E corresponding to e belong to the +same level set of HV , so that the transversality between those manifolds cannot hold in +R2d, but only in this level set (which is a 2d − 1-manifold of class Ck+1 outside of the set +of equilibria). This leads to the following definition. +6 + +Definition 1.4 (transversality of a standing pulse). Let e denote a non-degenerate +critical point of V and let E = (e, 0). A standing pulse with profile ξ �→ u(ξ) and +connecting e to itself is said to be transverse if the intersection +W u +0,V (E) ∩ W s +0,V (E) +is transverse, inside the level set H−1 +V +�−V (e) +� deprived of E, along the trajectory U(R). +As mentioned above, standing pulses divide into two classes (symmetric and asymmetric, +see Figures 1.3 and 1.4), which will require separate treatments in the proofs. Here is a +more precise definition. +Definition 1.5 (symmetric standing pulse, turning time). Let e denote a non-degenerate +critical point of V . A standing pulse with profile ξ �→ u(ξ) connecting e to itself is said +to be symmetric if there exists a time ξturn, called the turning time of the pulse, such +that ˙u(ξturn) vanishes; or equivalently, such that U(ξturn) belongs to Rd × {0Rd}. This +subspace Rd × {0Rd}, often called the reversibility or symmetry subspace, will be denoted +by Ssym. +If such a turning time exists then it is unique and the profile of the pulse is indeed +symmetric with respect to this turning time, see Lemma 3.2. Note that in the scalar case +d = 1 every standing pulse is symmetric (the derivative ˙u must vanish if the solution +approaches the same limits at both ends of R). For symmetric standing pulses (for any +value of the dimension d), instead of the transversality defined in Definition 1.4, the +following weaker transversality property ([17, 23, 51]) will be required. +Definition 1.6 (elementary symmetric standing pulse). Assume that the standing pulse +ξ �→ u(ξ) is symmetric with turning time ξturn. This pulse is said to be elementary if the +intersection +W u +0,V (E) ∩ Ssym +is transverse, in R2d, at the point U(ξturn). The feature of being elementary, for a standing +pulse, will be called elementarity. +Note that every transverse symmetric standing pulse is elementary: due to the time +reversibility when c is zero, a non-transverse intersection between W u +0,V (E) and Ssym +induces a non-transverse intersection between W u +0,V (E) and W s +0,V (E). But the converse +is false: for a symmetric standing pulse, the intersection W u +0,V (E) ∩ W s +0,V (E) may be non- +transverse in the sense of Definition 1.4 while this intersection still crosses transversally +the reversibility subspace Ssym. This may occur, for instance, if a symmetric standing +pulse is the limit of a one-parameter family of asymmetric ones. +1.4 The space of potentials +For the remaining of the paper, let us take and fix an integer k ≥ 1. Let us denote by +Ck+1 +b +(Rd, R) the space of functions Rd → R of class Ck+1 which are bounded, as well as +their derivatives of order not larger than k + 1, equipped with the norm +∥W∥Ck+1 +b += +max +α multi-index, |α|≤k+1 ∥∂|α| +uαW∥L∞(Rd,R) . +7 + +Let us embed the larger space Ck+1(Rd, R) with the following topology: for V in this space, +a basis of neighbourhoods of V is given by the sets V + O, where O is an open subset of +Ck+1 +b +(Rd, R) embedded with the topology defined by ∥·∥Ck+1 +b +. This topology (which can +be viewed as the one of an extended metric) is convenient, since local properties may be +studied in Banach spaces of the form +V + +� +Ck+1 +b +(Rd, R), ∥·∥Ck+1 +b +� +, +with ∥·∥Ck+1 +b +viewed as a classical norm. In this paper, the space Ck+1(Rd, R) will always +be embedded with this topology (if a topology is needed) and +� +Ck+1(Rd, R), ∥·∥Ck+1 +b +� +will be denoted simply by Ck+1(Rd, R). +Let us recall that a subset A of a topological set B is said to be a generic subset of B +if it contains a countable intersection of dense open subsets of B; accordingly, a property +is said to hold for a generic potential if it holds for every potential in a generic subset +of Ck+1(Rd, R). It is important to notice that Ck+1(Rd, R) is a Baire space because it +is locally equal to the Baire space Ck+1 +b +(Rd, R). Thus, the notion of genericity provides +relevant definitions of “large” subsets and “almost everywhere satisfied” properties. Other +definitions exist and the results stated in this paper presumably still hold for those (the +interested reader may consider [2, 3, 25, 34]). +Actually, the results stated in this paper also hold with other natural topologies, such +as Whitney’s topology. However the space Ck+1(Rd, R) is not locally a metric space for +Whitney’s topology (which is not characterized by sequences) and this leads to technical +difficulties. The framework chosen above is thus convenient to state the main arguments +while avoiding unessential technicalities, but the choice of the topology is not a key issue. +To finish, let us recall that a function having only non-degenerate critical points is +commonly called a Morse function. According to a classical result (see for instance [24]), +the set of Morse functions is a generic subset of Ck+1(Rd, R). Since the intersection of two +generic subsets is still a generic subset, and since our purpose is to state results which +hold generically, assuming that the potential V under consideration is a Morse function +does not present any inconvenience. As a consequence, only nondegenerate critical points +will be considered in the following, and the potential V will often be assumed to be a +Morse function. +1.5 Main results +The following generic transversality statement is the main result of this paper. +Theorem 1.7 (generic transversality of fronts and pulses). +There exists a generic subset of Ck+1(Rd, R) such that, for every potential function V in +this subset, V is a Morse function and the following conclusions hold for the fronts and +pulses defined by V : +1. every travelling front invading a minimum point of V is transverse; +2. every symmetric standing pulse is elementary; +8 + +3. every asymmetric standing pulse is transverse; +4. there is no standing front. +The core of this paper (sections 5 to 8) is devoted to the proof of this result among +potentials which are quadratic past some radius (see their definition in (4.2)). For such +potentials, conclusion 1 is proved by Proposition 5.1, conclusion 2, by Proposition 6.1, +conclusion 3 by Proposition 7.1, and conclusion 4 by Proposition 8.1. Sections 5 and 6 +are devoted, respectively, to the proofs of these propositions. In section 9, the proof +of Theorem 1.7 is completed by extending these conclusions to general potentials of +Ck+1(Rd, R) (not necessarily quadratic past some radius), and the qualitative conclusions +1 to 4 of Corollary 1.1 are derived from Theorem 1.7. +Using the same techniques, the following extension of Theorem 1.7 (and of conclusions +1 to 4 of Corollary 1.1) is proved in section 10. The second conclusion of this extension +is nothing but the last conclusion 5 of Corollary 1.1. +Theorem 1.8. There exists a generic subset of Ck+1(Rd, R) such that, for every potential +function V in this subset, in addition to the conclusions of Theorem 1.7 (and to the +conclusions 1 to 4 of Corollary 1.1), the following two additional conclusions hold: +1. for every minimum point of V , the smallest eigenvalue of the Hessian D2V at this +minimum point is simple; +2. the profile of every bistable travelling front or standing pulse stable at infinity +approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to +the smallest eigenvalue of D2V at this point. +As explained in subsection 1.1, conclusions 2 to 4 of Theorem 1.7 can be interpreted in +terms of homoclinic and heteroclinic orbits of the Hamiltonian system +(1.8) +� +˙u = ∂vH(u, v) +˙v = −∂uH(u, v) +where +H(u, v) = 1 +2|v|2 + ˜V (u) +and +˜V = −V . +The following statement explicitly provides this interpretation (for conclusions 2 and 3 +only, since conclusion 4 is actually elementary and well known, see section 8). No proof +is given since it is exactly a translation of these conclusions, with obvious meanings for +(a)symmetry and elementarity of homoclinic orbits. +Theorem 1.9 (the Hamiltonian point of view). There exists a generic subset of Ck+1(Rd, R) +such that, for every potential function ˜V in this subset, +1. every asymmetric homoclinic orbit of the Hamiltonian system (1.8) is transverse; +2. every symmetric homoclinic orbit of the Hamiltonian system (1.8) is elementary. +9 + +1.6 Short historical review +Theorem 1.7 and its proof rely on transversality theorems, also known as Sard–Smale or +Thom’s theorems, and are closely related to classical transversality results for differential +systems, see for instance [1, 31, 36, 45, 48]. Significant differences with respect to previous +works still deserve to be mentioned. +First, genericity in Theorem 1.7 holds with respect to the sole potential function V , +not general vector fields in R2d. Thus, perturbations of a given potential only provide a +partial control on the dynamics (in other words, differential systems of the form (1.7) do +not generate all possible flows in R2d). This constraint is balanced by the peculiarities of +the systems considered, which will have to be taken into account. To our best knowledge, +the first genericity result about the dynamics of a special class of differential systems +goes back to [45], and deals with polynomial flows. +Concerning Hamiltonian flows, homoclinic orbits play an important role, both from +theoretical and physical points of view, see for instance the reviews [12, 16] and articles +[4, 15, 23, 32, 50, 51]. The transversality/elementarity of such orbits has important +dynamical consequences, as the presence of Smale horseshoes associated to complex +dynamics. +In [30, 50], the genericity of these properties is considered in a general +abstract framework, and obtained only under sufficient conditions corresponding to the +assumptions of the transversality Theorem 4.2. In [32], this genericity is proved, but in the +case of non-autonomous systems. Other references dealing with the generic transversality +of connecting orbits include [33, 37, 38, 46] and references therein. In [46], genericity +holds with respect to all Hamiltonian flows, and not only second order conservative +systems as (1.8). In [33, 37, 38], genericity holds with respect to the potential ˜V only, in +a more general setting where the “kinetic energy” |v|2 /2 of the Hamiltonian in (1.8) is +replaced by a more general expression. But the transversality of homoclinic orbits is not +considered in these papers. In [33], the transversality of heteroclinic orbits is derived from +a perturbation result of [13]. The others results of [33, 37, 38] are concerned with closed +orbits. Thus, to the best of our knowledge, even Theorem 1.9 (the results concerning +standing pulses, in the language of Hamiltonian systems) is new. +Concerning the nonzero dissipation case (conclusion 1 of Theorem 1.7), the statement +differs from usual genericity properties. If c is fixed (and nonzero), heteroclinic connections +corresponding to travelling fronts invading a minimum point of V do generically not exist +for the flow of system (1.7). But the freedom provided by the parameter c ensures the +generic existence, transversality, and robustness of heteroclinic connections corresponding +to bistable travelling fronts. This parameter c will thus have to be taken into account in +the setting where transversality theorems will be applied, a significant difference with +classical genericity results about the flows of differential systems. +The initial motivation for this paper actually relates to parabolic systems of the form +(1.1). For such systems, the global dynamics of bistable solutions, that is solutions close +at both ends of space to local minimum points of the potential V , has been described +under rather general (assumed to be generic) hypotheses on V by the second author +in [39, 40, 43]. Every such solutions must approach, as time goes to +∞, far to the +left in space a stacked family of bistable fronts travelling to the left, far to the right +10 + +in space a stacked family of bistable fronts travelling to the right, and in between a +pattern of standing pulses/fronts going slowly away from one another (this extends to +gradient systems the program initiated in the late seventies by Fife and McLeod for scalar +equations [19–21]). The present paper provides a rigorous proof of the genericity of the +hypotheses made on the potential V in [39, 40, 43]. The same hypotheses yield similar +conclusions for hyperbolic gradient systems [41] and for radially symmetric solutions +of parabolic gradient systems in higher space dimension [42]. The results obtained in +this last reference rely on an additional hypothesis, which is the higher space dimension +analogue of conclusion 2 of Theorem 1.7 (elementarity of symmetric standing pulses). +The genericity of this hypothesis is proved in the companion paper [44], using the same +approach as in the present paper. +The extension Theorem 1.8 of Theorem 1.7 (comprising the last conclusion 5 of +Corollary 1.1) is motivated by the study of the long-range interaction between fronts and +pulses of the parabolic system (1.1). The long-range interaction between such “localized +structures” is the object of a large body of literature, both in Mathematics and Physics, see +for instance [6, 11, 14, 18, 28, 52] among many other possible references. The conclusions +of Theorem 1.8 are especially relevant in conjunction with this topic, for the following +reason. Consider a solution of the parabolic system (1.1) close to, say, two standing fronts +or pulses or two fronts travelling at the same speed, far away from one another. Let us +denote by uleft(·) and uright(·) their profiles, so that the solution is close to a translate +of uleft on R− and to a translate of uright on R+. Then, the (large) distance between +these two translates is expected to vary slowly, according to a (long-range) interaction +law that can be computed at first order, and which is related to the asymptotics of uleft +at +∞ and of uright at −∞. Basically, when (as in the present context) the tails of uleft +and uright are not oscillating, this first order long-range interaction can be attractive or +repulsive or neutral, depending on the sign of a scalar product between the (oriented) +directions through which uleft and uright approach their (common) limit (at +∞ and +at −∞ respectively), see for instance the conjecture at the bottom of p. 59 of [5], or +expressions (2.12) and (2.13) in Theorem 2.3 of [18]. In the present context, according to +the conclusions of Theorem 1.8 and for a generic potential, these two oriented directions +are aligned with the one-dimensional eigenspace associated with the smallest eigenvalue +of the Hessian D2V of the potential at the minimum point which is the common limit +mentioned above. Among the consequences, the first order long-range interaction is thus +either attractive or repulsive, but not neutral. +2 Stable and unstable manifolds of equilibria +Throughout all this section V denotes a potential function in Ck+1(Rd, R) and c denotes +a non-negative quantity (speed). As stated in Proposition 1.2, the travelling fronts and +standing fronts/pulses of the parabolic equation (1.1) correspond to heteroclinic and +homoclinic connections for the flow in R2d generated by the first order differential system +(1.7). Let Ω be the maximal (open) subset of R × R2d where this flow is defined and let +11 + +us consider its flow Sc,V defined as +(2.1) +Sc,V : Ω → R2d , +(ξ, U0) �→ U(ξ) , +where U(ξ) is the solution of (1.7) with U(0) = U0. By definition, for every (ξ, U0) in Ω, +∂ +∂ξ Sc,V (ξ, U0) = Fc,V +�Sc,V (ξ, U0) +� +where +Fc,V : +� +u +v +� +�→ +� +v +∇V (u) − cv +� +. +Although the variable ξ denotes primarily the space variable in a frame travelling at +speed c for the initial partial differential system (1.1), this variable also plays the role of +a time in the differential systems (1.2) and (1.7) prescribing the profiles of travelling and +standing waves. In the following, this variable will thus often be referred to as a “time”. +2.1 Linearization around an equilibrium point +Let e denote a non-degenerate critical point of V . Let (u1, . . . , ud) denote an orthonormal +basis of Rd made of eigenvectors of the Hessian D2V (e) and let µ1, . . . , µd denote the +corresponding (real) eigenvalues. +Definition 2.1. Let us call Morse index of e, denoted by m(e), the number of negative +eigenvalues of D2V (e), counted with their algebraic multiplicity. +Since the critical point e is assumed to be non-degenerate, it is: a minimum point +if m(e) equals 0, a saddle point if m(e) is between 1 and d − 1, and a maximum point +if m(e) equals d. In addition, none of the quantities µ1, . . . , µd vanishes, and we may +assume that +µ1 ≤ · · · ≤ µm(e) < 0 < µm(e)+1 ≤ · · · ≤ µd +if +m(e) > 0 , +and +0 < µ1 ≤ · · · ≤ µd +if +m(e) = 0 . +Now, let us consider the equilibrium point E = (e, 0Rd) of Sc,V corresponding to e. The +linearized differential system (1.7) at E reads: +(2.2) +˙U = DFc,V (E)U , +or equivalently +� ˙u = v +˙v = D2V (e)u − cv +. +Observe that a complex quantity λ is an eigenvalue for the linear system (2.2) if and only +if the quantity λ(λ + c) is an eigenvalue for the Hessian D2V (e), that is if λ(λ + c) is +equal to one of the quantities µ1, . . . , µd. For j in {1, . . . , d}, let +(2.3) +λj,+ = −c +2 + +� +c2 +4 + µj +and +λj,− = −c +2 − +� +c2 +4 + µj +denote the two (real or complex) eigenvalues of the linear system (2.2) corresponding to +µj, and let +(2.4) +Uj,+ = +� +uj +λj,+uj +� +and +Uj,− = +� +uj +λj,−uj +� +12 + +denote the corresponding eigenvectors. Let +(2.5) +Es +c,V (E) +and +Ec +c,V (E) +and +Eu +c,V (E) +denote the stable, centre, and unstable subspaces of R2d for the linear operator DFc,V +defined in (2.2), that is the eigenspaces corresponding to eigenvalues with negative, +zero and positive real parts respectively. The dimensions of those spaces and of the +corresponding invariant manifolds (defined below) derive from expressions (2.3), and +are as summarized in Table 2.1. The case of a negative speed c can be derived by the +c = 0 +c > 0 +Dimension of Eu +c,V (E) and W u +c,V (E) +d − m(e) +d − m(e) +Dimension of Es +c,V (E) and W s +c,V (E) +d − m(e) +d + m(e) +Dimension of Ec +c,V (E) and W c +loc,c,V(E) +2m(e) +0 +Table 2.1: Dimensions of stable, unstable, and centre manifolds for an equilibrium point +E = (e, 0) of the differential system (1.7), corresponding to a critical point e of the +potential with Morse index m(e). +transformation (c, ξ) �→ (−c, −ξ) which leaves the systems (1.2) and (1.7) unchanged +(and exchanges the stable and unstable dimensions). +The dimension of Eu +c,V (E) is also commonly called the Morse index of E. To avoid +any confusion, the denomination Morse index will only be used for critical points of the +potential, not for the corresponding equilibria in R2d. +2.2 Local stable and unstable manifolds when the speed c is positive +The construction of the local stable (unstable) manifold of an equilibrium of a differential +system is classical. A historical reference is Kelley’s article [29], comprising the construc- +tion and the dependence on the parameters, however with a slightly non-optimal regularity. +A complete construction can be found in many textbooks, for example Theorem 3.2.1 of +[22] or Theorem 9.4 of [49]. The goal of this subsection and of subsection 2.3 below is to +provide precise statements (Proposition 2.2 below and Proposition 2.4 in subsection 2.3 +when the speed c equals 0) concerning these manifolds (for the differential system (1.7)), +and the associated notation (without proofs); those statements and notation will be +called upon in the sequel. +Take V0 in Ck+1(Rd, R), let e0 denote a non-degenerate critical point of V0, and let c0 +denote a positive quantity. According to Table 2.1, the point (e0, 0), which will be denoted +by E0, is a hyperbolic equilibrium point and the subspaces Eu +c0,V0(E0) and Es +c0,V0(E0) +introduced in (2.5) generate the whole space R2d (or in other words the central part +Ec +c0,V0(E0) reduces to {0R2d}). Let +(2.6) +βu = min {Re(λ) : λ eigenvalue of DFc0,V0(E0) with Re(λ) > 0} +and +βs = max {Re(λ) : λ eigenvalue of DFc0,V0(E0) with Re(λ) < 0} . +13 + +There exist norms ∥·∥u on the unstable subspace Eu +c0,V0(E0) and ∥·∥s on the stable subspace +Es +c0,V0(E0) such that, for every non negative quantity ξ, +(2.7) +���exp +� +−ξDFc0,V0(E0)|Eu +c0,V0(E−) +���� +u ≤ exp +� +−βu +2 ξ +� +, +and +���exp +� +ξDFc0,V0(E0)|Es +c0,V0(E+) +���� +s ≤ exp +�βs +2 ξ +� +. +For every positive quantity r, let +(2.8) +Bu +E0(r) = +�Uu ∈ Eu +c0,V0(E0) : ∥Uu∥u ≤ r +� , +and +Bs +E0(r) = +�Us ∈ Es +c0,V0(E0) : ∥Us∥s ≤ r +� , +and +BE0(r) = +�Uu + Us : Uu ∈ Bu +E0(r) and Us ∈ Bs +E0(r) +� . +Proposition 2.2 (local stable and unstable manifolds). There exist a neighbourhood ν +of V0 in Ck+1(Rd, R), a neighbourhood C of c0 in (0, +∞) and a positive quantity r such +that, for every (c, V ) in C × ν, the following statements hold. +1. There exists a unique critical point e(V ) of V such that E(V ) = (e(V ), 0) belongs +to E0 + BE0(r). In addition, e(V ) has the same Morse index as e0 and the map +ν → Rd, V �→ e(V ) is of class Ck. +2. There exist Ck-functions +wu +loc, c, V : Bu +E0(r) → Bs +E0(r) +and +ws +loc, c, V : Bs +E0(r) → Bu +E0(r) +such that, if we consider the sets +W u +loc, c, V +�E(V ) +� = +� +E(V ) + Uu + wu +loc, c, V (Uu) : Uu ∈ Bu +E0(r) +� +and +W s +loc, c, V +�E(V ) +� = +� +E(V ) + Us + ws +loc, c, V (Us) : Us ∈ Bs +E0(r) +� +, +then, for every U in BE0(r) the following two assertions are equivalent: +a) U is in W u +loc, c, V +�E(V ) +�; +b) Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in (−∞, 0] and Sc,V (ξ, U) → +E(V ) as ξ → −∞; +and for every U in BE0(r) the following two assertions are equivalent: +c) U ∈ W s +loc, c, V +�E(V ) +�; +d) Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in [0, +∞) and Sc,V (ξ, U) → +E(V ) as ξ → +∞. +3. Both differentials Dwu +loc, c0, V0(0) and Dws +loc, c0, V0(0) vanish, and both maps +C × ν × Bu +E0(r) → Bs +E0(r), +(c, V, Uu) �→ wu +loc, c, V (Uu) +and +C × ν × Bs +E0(r) → Bu +E0(r), +(c, V, Us) �→ ws +loc, c, V (Us) +are of class Ck. +14 + +With the notation provided by Proposition 2.2, for every (c, V ) in C×ν, let us introduce +the maps +ˆwu +loc, c, V : Bu +E0(r) → R2d, +Uu �→ E(V ) + Uu + wu +loc, c, V (Uu) , +and +ˆws +loc, c, V : Bs +E0(r) → R2d, +Us �→ E(V ) + Us + ws +loc, c, V (Us) . +Local unstable and stable manifolds of E(V ) can be defined as +(2.9) +W u +loc, c, V +�E(V ) +� = ˆwu +loc, c, V +�Bu +E0(r) +� , +and +W s +loc, c, V +�E(V ) +� = ˆws +loc, c, V +�Bs +E0(r) +� . +Those manifolds depend smoothly of c and V . The global unstable and stable manifolds +W u +c,V +�E(V ) +� = {U ∈ R2d : Sc,V (ξ, U) → E(V ) when ξ → −∞} +and +W s +c,V +�E(V ) +� = {U ∈ R2d : Sc,V (ξ, U) → E(V ) when ξ → +∞} +can then be derived from those local manifolds through the flow Sc,V as follows: +W u +c,V +�E(V ) +� = Sc,V +� +R × W u +loc, c, V +�E(V ) +�� +, +and +W s +c,V +�E(V ) +� = Sc,V +� +R × W s +loc, c, V +�E(V ) +�� +. +Remark. Here are two observations that will turn out to play some role in the forthcoming +proofs. +• According to the characterization provided by this proposition (equivalence between +2a and 2b and between 2c and 2d), for every solution ξ �→ U(ξ) of system (1.7), +if this solution belongs to the stable (unstable) manifold of E(V ) then it crosses +exactly once the border ∂W s +loc, c, V +�E(V ) +� of the local stable manifold (the border +∂W u +loc, c, V +�E(V ) +� of the local unstable manifold) of E(V ). In addition, according +to the the conditions (2.7) satisfied by the norms ∥·∥u and ∥·∥s, up to replacing the +radius r by a smaller quantity, this intersection between the trajectory of ξ �→ U(ξ) +and the border of the local stable (unstable) manifold of E(V ) is transverse inside +the full stable (unstable) manifold. Although the transversality of this intersection +is not formally required in the following, assuming that it holds helps figuring out +the broad scheme, see for instance Figure 5.1. +• The functions wu +loc, c, V and ws +loc, c, V are uniquely defined by the characterization +provided by Proposition 2.2 once the radius r and the departure sets of these two +functions are chosen. As a consequence, those two functions and the local stable +and unstable manifolds W u +loc, c, V +�E(V ) +� and W s +loc, c, V +�E(V ) +� remain unchanged if +the potential function V is modified outside a neighbourhood of the set +πpos +� +W u +loc, c, V +�E(V ) +� ∪ W s +loc, c, V +�E(V ) +�� +, +where πpos is the projection onto the position coordinates: +(2.10) +πpos : R2d → Rd, +(u, v) �→ u . +15 + +2.3 Local stable and unstable manifolds when the speed c equals 0 +As Table 2.1 shows, an equilibrium E is hyperbolic except if c vanishes and m(e) is +positive. In this case, there exists, in addition to the stable and unstable manifolds of +E, a centre manifold with dimension 2m(e) (corresponding to the central part of the +spectrum of the linear system (2.2) at E). However, as shown by the following lemma, +a solution ξ �→ U(ξ) of system (1.7) cannot asymptotically approach E through such a +centre manifold. The statement of Proposition 2.4 below and the proof of Proposition 1.2, +provided in subsection 3.1, rely on this lemma. +Lemma 2.3 (approach of critical points through stable/unstable manifolds). Assume +that c equals 0. For every critical point e of V such that the Morse index m(e) is positive, +and for every (maximal) solution ξ �→ U(ξ) of the differential system (1.7), if E denotes +the point (e, 0), the following conclusions hold: +1. if U(ξ) goes to E as ξ goes to +∞, then the trajectory of ξ �→ U(ξ) converges to E +tangentially to the stable space Es +V (E). +2. if U(ξ) goes to E as ξ goes to −∞, then the trajectory of ξ �→ U(ξ) converges to E +tangentially to the unstable space Eu +V (E), +Proof. Let ξ �→ U(ξ) = (u, v)(ξ) denote a solution of the differential system (1.7) for a +speed c equal to 0, and let us assume that U(ξ) goes to E as ξ goes to +∞. It follows +from the invariance of the Hamiltonian function HV (defined in (1.4)) along U(·) that +HV (U(ξ)) = HV (E), or in other words that +(2.11) +1 +2 |v(ξ)|2 − V +�u(ξ) +� = −V (e) . +Let us proceed by contradiction and assume that this solution does not belong to the +stable manifold of E. With the notation of subsection 2.1, it follows that, as ξ goes to ++∞, the component of U(ξ) − E along the centre subspace Ec +V (E) is dominant compared +to its component along the hyperbolic subspace Es +V (E) + Eu +V (E); with symbols, if πcent +denotes the projection along Es +V (E) + Eu +V (E) onto Ec +V (E) in R2d, +(2.12) +U(ξ) − E = πcent +�U(ξ) − E +� + oξ→+∞ +� +πcent +�U(ξ) − E +�� +. +It follows from the expressions (2.3) and (2.4) of the eigenvalues and eigenvectors of +DF0,V (E) that +Ec +V (E) = span +�U1,+, U1,−, . . . , Um(e),+, Um(e),− +� . +As a consequence, applying the projection πpos (projection onto the position coordinates, +defined in (2.10)) to equality (2.12), it follows that, if we denote by πm(e) the orthogonal +projection onto span{u1, . . . , um(e)} in Rd, +u(ξ) − e = πm(e) +�u(ξ) − e +� + oξ→+∞ +� +πm(e) +�u(ξ) − e +�� +. +Since the restriction of D2V (e) to the image of πm(e) is negative definite, it follows that, +for ξ sufficiently large, V +�u(ξ) +� is smaller than V (e), thus −V +�u(ξ) +� is larger than −V (e), +contradicting equality (2.11). Lemma 2.3 is proved. +16 + +As for Proposition 2.2 in the case c > 0, the aim of the next Proposition 2.4 is to +provide (in the case c = 0) a precise statement and the associated notation concerning +the local stable and unstable manifolds of an equilibrium for the differential system (1.2). +In this case c = 0, the conclusions of Lemma 2.3 show that centre manifolds are not +relevant for homoclinic and heteroclinic solutions; for that reason, those centre manifolds +are ignored in Proposition 2.4. Concerning the construction and properties of the local +stable and unstable manifolds, there is no difference with respect to the positive speed +case considered in Proposition 2.2, see again [22, 29, 49]. Observe that, by contrast with +the statements that can be found in textbooks, the characterization of these local stable +and unstable manifolds does not require an exponential rate of convergence towards E, +again due to the conclusions of Lemma 2.3 (see the equivalence between assertions 2a +and 2b and between assertions 2c and 2d in Proposition 2.4 below). +Notation. For the remaining of this paper, when the speed c vanishes, it will be omitted +in the notation. Thus, concerning the previously introduced notation, +FV +SV +Es +V +Ec +V +Eu +V +W s +V +W u +V +stand for: +F0,V +S0,V +Es +0,V +Ec +0,V +Eu +0,V +W s +0,V +W u +0,V . +Take V0 in Ck+1(Rd, R) and let e0 denote a non-degenerate critical point of V0 and let +E0 = (e0, 0) (which is not necessarily hyperbolic). Let βu and βs be as in (2.6). As in +the case c > 0, there exist norms ∥·∥u on the unstable subspace Eu +V0(E0) and ∥·∥s on the +stable subspace Es +V0(E0) such that inequalities (2.7) hold for every non negative quantity +ξ. Let ∥·∥c denote any norm on the centre subspace Ec +V0(E0) (for instance the euclidean +norm). For every positive quantity r, let +Bu +E0(r) = {Uu ∈ Eu +V0(E0) : ∥Uu∥u ≤ r} , +Bs +E0(r) = {Us ∈ Es +V0(E0) : ∥Us∥s ≤ r} , +Bc +E0(r) = {Uc ∈ Ec +V0(E0) : ∥Uc∥c ≤ r} , +and +BE0(r) = {Uu + Us + Uc : Uu ∈ Bu +E0(r) and Us ∈ Bs +E0(r) and Uc ∈ Bc +E0(r)} . +Proposition 2.4 (local stable and unstable manifolds). There exist a neighbourhood ν +of V0 in Ck+1(Rd, R) and a positive quantity r such that, for every V in ν, the following +statements hold. +1. There exists a unique critical point e(V ) of V such that E(V ) = (e(V ), 0) belongs +to E0 + BE0(r). In addition, e(V ) has the same Morse index as e0 and the map +ν → Rd, V �→ e(V ) is of class Ck. +2. There exist Ck-functions +wu +loc, V : Bu +E0(r) → Bs +E0(r) + Bc +E0(r) +and +ws +loc, V : Bs +E0(r) → Bu +E0(r) + Bc +E0(r) +such that, if we consider the sets +W u +loc, V +�E(V ) +� = +� +E(V ) + Uu + wu +loc, V (Uu) : Uu ∈ Bu +E0(r) +� +and +W s +loc, V +�E(V ) +� = +� +E(V ) + Us + ws +loc, V (Us) : Us ∈ Bs +E0(r) +� +, +17 + +then, for every U in BE0(r), the following two assertions are equivalent: +a) U is in W u +loc, V +�E(V ) +�; +b) SV (ξ, U)−E(V ) remains in BE0(r) for all ξ in (−∞, 0] and SV (ξ, U) → E(V ) +as ξ → −∞, +and for every U in BE0(r), the following two assertions are equivalent: +c) U ∈ W s +loc, V +�E(V ) +�; +d) SV (ξ, U)−E(V ) remains in BE0(r) for all ξ in [0, +∞) and SV (ξ, U) → E(V ) +as ξ → +∞. +3. Both differentials Dwu +loc, V0(0) and Dws +loc, V0(0) vanish, and both maps +ν × Bu +E0(r) → Bs +E0(r), +(V, Uu) �→ W u +loc, V (Uu) +and +ν × Bs +E0(r) → Bu +E0(r), +(V, Us) �→ W s +loc, V (Us) +are of class Ck. +With the notation provided by Proposition 2.4, for every V in ν, let us introduce the +maps +ˆwu +loc, V : Bu +E0(r) → R2d, +Uu �→ E(V ) + Uu + wu +loc, V (Uu) , +and +ˆws +loc, V : Bs +E0(r) → R2d, +Us �→ E(V ) + Us + ws +loc, V (Us) . +Local unstable and stable manifolds of E(V ) can be defined as +(2.13) +W u +loc, V +�E(V ) +� = ˆwu +loc, V +�Bu +E0(r) +� , +and +W s +loc, V +�E(V ) +� = ˆws +loc, V +�Bs +E0(r) +� . +Those manifolds depend smoothly of V . As in subsection 2.2, the global unstable/stable +manifolds of E(V ), denoted by W u +V (E(V )) and W s +V (E(V )) can be expressed in terms of +those local manifolds and of the flow SV . Both observations made in the remark ending +the previous subsection are still valid in the present case of zero speed and potential +existence of a centre manifold. +3 Preliminary properties of travelling fronts and standing fronts +and pulses +Let us take and fix, for this whole section, a potential function V in Ck+1(Rd, R). +3.1 Proof of Proposition 1.2 +Let e− and e+ be two (possibly equal) non-degenerate critical points of V , let c denote a +non negative quantity (speed), and let ξ �→ u(ξ) denote the profile of a front or pulse +connecting e− to e+ and travelling at speed c (or standing if c equals zero) for the +potential V . +18 + +Lemma 3.1. The derivative ˙u(ξ) goes to 0 as ξ goes to ±∞. +Proof. If the speed c is positive, then ξ �→ u(ξ) is the profile of a travelling front. It +follows from integrating (1.5) that +(3.1) +lim +ξ→+∞ HV (u(ξ)) − lim +ξ→−∞ HV (u(ξ)) = −c +� +R +| ˙u(ξ)|2 dξ +and thus that ˙u(·) is in L2(R). Thus 0 is an adherent value of the kinetic part of the +Hamiltonian function ξ �→ HV +�U(ξ) +� as ξ goes to ±∞, meaning that V (e±) are adherent +values of HV (U(ξ)). Since according to (1.5) this last function decreases with ξ, it follows +that HV +�U(ξ) +� goes to V (e±) as ξ goes to ±∞, and the intended conclusion follows. +If the speed c equals 0, it follows from the differential system (1.2) and the convergence +of u(·) to critical points that ¨u(ξ) goes to 0 as ξ goes to ±∞. Thus ˙u(·) is uniformly +continuous and the convergence of u yields the intended conclusion. +Proof of Proposition 1.2. Let us use the notation of Proposition 1.2. If c is non zero or +if c equals 0 and both Morse indices m(e−) and m(e+) of e− and e+ vanish, then E− +and E+ are hyperbolic equilibria of the differential system (1.7) and the conclusions of +Proposition 1.2 follow from Lemma 3.1. +If c equals 0 and the Morse indices m(e−) and m(e+) are any, then the equilibria E− +and E+ are not necessarily hyperbolic, but again in this case it follows from Lemma 3.1 +that U(ξ) goes to E± as ξ → ±∞; and it follows from Lemma 2.3 that the values of +ξ �→ U(ξ) belong to the unstable manifold of E− and to the stable manifold of E+. +3.2 Equivalent definitions of a symmetric standing pulse +Let e denote a non-degenerate critical point of V , and let ξ �→ u(ξ) denote the profile of a +standing pulse connecting e to itself. In Definition 1.5, the symmetry of such a pulse was +defined by the existence of a “turning time” where ˙u vanishes. The following standard +result (see for instance [17]) completes this definition. +Lemma 3.2 (equivalent definitions of a symmetric standing pulse). For every real +quantity ξturn, the following properties are equivalent: +1. ξturn is a turning time in the sense of Definition 1.5, that is ˙u(ξturn) = 0; +2. for every ξ in R, +(3.2) +u(ξturn − ξ) = u(ξturn + ξ) ; +3. there exists ξ in R such that +(3.3) +u(ξturn − ξ) = u(ξturn + ξ) +and +˙u(ξturn − ξ) = − ˙u(ξturn + ξ) . +In addition, these statements hold for at most one real quantity ξturn. +19 + +Proof. Differentiating equality (3.2) with respect to ξ yields equalities (3.3) for all ξ, +so that property 2 implies property 3, and property 3 for ξ equal to 0 is equivalent to +property 1, so that property 2 implies property 1 and property 1 implies property 3. +It remains to prove that property 3 implies property 2. Assume that property 3 holds, +and, for every real quantity ξ, let us write +u1(ξ) = u(ξturn + ξ) +and +u2(ξ) = u(ξturn − ξ) . +Since ξ �→ u(ξ) is a solution of the second order differential system (1.2) with c equal to +zero, both ξ �→ U1(ξ) and ξ �→ U2(ξ) are solutions of the first order differential system +(1.7) (again with c equal to zero). According to property 3, there exists ξ such that U1(ξ) +is equal to U2(ξ). Thus U1(ξ) must be equal to U2(ξ) for every real time ξ, and property +2 follows. Thus the three properties of Lemma 3.2 are equivalent. +In addition, if property 2 holds for two different turning times ξturn and ξ′ +turn, then +ξ �→ u(ξ) is periodic with a period equal to 2(ξ′ +turn − ξturn), a contradiction with the +assumption that u is a standing pulse connecting e to itself. Lemma 3.2 is proved. +3.3 Values reached only once by profiles of travelling fronts / standing +pulses +The proofs carried on in the sections 5 to 7 below rely on the construction of suitable +perturbations of the potential V . Whereas the uniqueness of the solutions of differential +system (1.7) ensures that the function ξ �→ +�u(ξ), ˙u(ξ) +� defined by such a solution is +one-to-one, this is not necessarily true for the function ξ �→ u(ξ) (as shown by Figure 3.1). +As a consequence, a perturbation of the potential V may affect this solution at different +times. The goal of the following proposition is to avoid this inconvenience, by providing +in each case under consideration a time interval where u(ξ) is reached only once. +Proposition 3.3. +1. For every profile ξ �→ u(ξ) of a front travelling at a positive speed c and connecting +two non-degenerate critical points, there exists a time ξonce such that, for all times +ξ∗ in (−∞, ξonce] and ξ in R, +(3.4) +u(ξ) = u(ξ∗) =⇒ ξ = ξ∗ . +2. For every profile ξ �→ u(ξ) of an asymmetric standing pulse and for every nonempty +open interval I of R, there exists a nonempty open interval Ionce, included in I, +such that, for all times ξ∗ in Ionce and ξ in R, implication (3.4) holds. +3. For every profile ξ �→ u(ξ) of a symmetric standing pulse, if ξturn denotes the +turning time of this pulse (see Lemma 3.2), then, for every nonempty open interval +I included in (−∞, ξturn], there exists a nonempty open interval Ionce, included in +I, such that, for all times ξ∗ in Ionce and ξ in (−∞, ξturn], implication (3.4) holds. +20 + +u +0 +−V +e− +e+ +u(ξ) +Figure 3.1: The one-dimensional example of this figure shows that property 1 of Proposi- +tion 3.3 may not hold outside a small neighbourhood of the critical point e−. +Proof of statement 1 of Proposition 3.3. Let ξ �→ u(ξ) denote the profile of a front trav- +elling at a positive speed c for the potential V , and let e− denote the critical point, +assumed to be non-degenerate, approached by u(ξ) as ξ goes to −∞. Since all eigenval- +ues of DFc,V (E−)|Eu +c,V (E−) are real and positive (see subsection 2.1), the corresponding +solution U(ξ) of system (1.7) must approach E− tangentially to some (real, unstable) +eigenvector Ueig of DFc,V (E−) as ξ goes to −∞. If λ denotes the corresponding (positive) +eigenvalue, then Ueig is of the form (ueig, λueig), where ueig is an eigenvector of D2V (e−), +see expression (2.4). Thus there must exist a nonzero scalar function ξ �→ α(ξ) so that, +as ξ goes to −∞, +U(ξ) = E− + α(ξ)Ueig + o +�α(ξ) +� , +that is +� +u(ξ) = e− + α(ξ)ueig + o +�α(ξ) +� , +˙u(ξ) = α(ξ)λueig + o +�α(ξ) +� . +It follows that there exists a large negative time ξ0 such that, for every ξ in (−∞, ξ0], +d +dξ |u(ξ) − e−|2 = 2(u(ξ) − e−) · ˙u(ξ) > 0 . +In particular, the function +(3.5) +(−∞, ξ0] → Rd, +ξ �→ u(ξ) +is a C1-diffeomorphism onto its image. According to the decrease (1.5) of the Hamiltonian, +the quantity HV +�U(ξ0) +� is smaller than −V (e−). As a consequence, there exists a time +ξonce in (−∞, ξ0) such that, for every ξ∗ in (−∞, ξonce], +(3.6) +HV +�U(ξ0) +� < −V +�u(ξ∗) +� . +Take a time ξ∗ in (−∞, ξonce] and a time ξ in R and let us assume that u(ξ) equals u(ξ∗). +If ξ was larger than ξ0 then it would follow from the expression (1.4) of the Hamiltonian, +its decrease (1.5) and inequality (3.6) that +−V +�u(ξ) +� ≤ HV +�U(ξ) +� ≤ HV +�U(ξ0) +� < −V +�u(ξ∗) +� , +a contradiction with the equality of u(ξ) and u(ξ∗). Thus ξ is not larger than ξ0, and it +follows from the one-to-one property of the function (3.5) that ξ must be equal to ξ∗. +Statement 1 of Proposition 3.3 is proved. +21 + +Proof of statement 2 of Proposition 3.3. Let ξ �→ u(ξ) be the profile of an asymmetric +standing pulse for the potential V , let e denote the critical point approached by u(ξ) +as ξ goes to ±∞, and let I be a nonempty open interval of R. In view of the intended +conclusion (statement 2), we may assume that I is bounded. According to the invariance +(1.5) of the Hamiltonian HV , for every ξ in R, the difference V +�u(ξ) +� − V (e) is equal to +| ˙u(ξ)|2/2 and is therefore nonzero, so that the critical point e is never reached by the +function ξ �→ u(ξ) on R. As a consequence there exists a (small) positive quantity r such +that |u(ξ) − e| is larger than r for all ξ in I; and since u(ξ) approaches e as ξ goes to +±∞, there exists a (large) positive quantity M such that |u(ξ) − e| is smaller than r +outside of [−M, M]. +Assume that there exist two different times ξ and ξ′ in R such that u(ξ) equals u(ξ′). +Then, again according to the invariance (1.5) of the Hamiltonian HV , the time derivatives +˙u(ξ) and ˙u(ξ′) must have the same norm. Besides, these two vectors cannot be equal (or +else the profile u would be periodic) nor opposite (or else according to Lemma 3.2 the +pulse would be symmetric), thus they are not proportional. Thus the couples (ξ, ξ′) such +that u(ξ) is equal to u(ξ′) are isolated in R2. In addition, if (ξ, ξ′) is such a couple and ξ +is in I then ξ′ must belong to [−M, M]. This shows by compactness that there exists +only a finite number of couples (ξ, ξ′) in I × R such that u(ξ) equals u(ξ′). Statement 2 +of Proposition 3.3 follows. +Proof of statement 3 of Proposition 3.3. The arguments are the same as in the proof of +statement 2 above. Let ξ �→ u(ξ) be the profile of a symmetric pulse with turning time +ξturn for the potential V , let I be a nonempty open interval of (−∞, ξturn], assumed to be +bounded. If there exist two different times ξ and ξ′ in (−∞, ξturn] such that u(ξ) equals +u(ξ′), again the time derivatives ˙u(ξ) and ˙u(ξ′) have the same norm. These two vectors +cannot be equal (or else the profile u would be periodic) nor opposite (or else, according +to statement 3 of Lemma 3.2, (ξ + ξ′)/2 would be a second turning time — smaller than +ξturn — for u, a contradiction with the conclusion of Lemma 3.2). Thus again, ˙u(ξ) and +˙u(ξ′) cannot be proportional, and the same arguments as in the proof of statement 2 +above show that there exists only a finite number of couples (ξ, ξ′) in I × (−∞, ξturn] +such that u(ξ) = u(ξ′). +4 Tools for genericity +4.1 An instance of the Sard–Smale transversality theorem +To prove that a given property generically holds, a standard method is to express this +property as a transversality problem and to use one instance among the family of theorems +known as Sard–Smale theorem (or Thom’s theorems, or transversality theorems), see [1, +7, 24, 36]. In this paper the following instance will be used (Theorem 4.2 below). Let us +consider a function +Φ : M × Λ → N , +where M and N are two finite-dimensional manifolds and Λ (“parameter space”) is a +Banach manifold, together with a submanifold W of N (see Figure 4.1). Let us assume +22 + +that these four manifolds and the function Φ are of class Ck (as everywhere in the paper +k denotes an integer which is not smaller than 1). Finally, let codim(W) denote the +codimension of W in N. +Definition 4.1. With the notation above, the image of Φ is said to be transverse to W, +if, for every (m, λ) in M × Λ such that Φ(m, λ) is in W, the following equality holds: +DΦ(TmM × TλΛ) + TΦ(m,λ)W = TΦ(m,λ)N +(here DΦ denotes the differential of Φ at (m, λ)). Accordingly, for every λ in Λ, if Φλ +denotes the function: +M → N, +m �→ Φ(m, λ) , +then the image of Φλ is said to be transverse to W if, for every m in M such that Φ(m, λ) +is in W, denoting DΦλ the differential of Φλ at m, +DΦλ(TmM) + TΦ(m,λ)W = TΦ(m,λ)N . +Theorem 4.2 (Sard–Smale transversality theorem). With the notation above, if +1. k > dim(M) − codim(W), +2. and the image of Φ is transverse to W, +then there exists a generic subset Λgen of Λ such that, for every λ in Λgen, the image of +Φλ is transverse to W. +The proof of this result can be found in [1] or in [24]. The key hypothesis, which is +often difficult to check, is the transversality hypothesis 2. Notice that the conclusion +is stronger than this hypothesis since it states that the transversality holds for a fixed +generic parameter λ, whereas hypothesis 2 uses the freedom of moving λ. +λ moves +Φ +M +Φ(M, λ) +W +Figure 4.1: Geometric interpretation of Theorem 4.2. Assume that for a given parameter +λ0, DΦλ0(TM) + TW is not the whole tangent space TN, but that the dependence of +Φ on λ provides the missing directions. Then for almost every λ close to λ0, the image +Φ(M, λ) intersects W transversally. +23 + +4.2 Extending local genericity to global genericity +Theorem 4.2 (under the form above or another) is the standard tool to prove that a +property generically holds. However, it turns out that is is often difficult, in practice, +to express a property using a single function Φ as above; thus one is often led to patch +together several conclusions provided by this theorem. The following lemma provides a +way to carry out this patching process. This lemma is identical to Lemma 3.3 of Chapter +3 of [35], where a proof can be found. +Lemma 4.3 (local genericity implies global genericity in a separable Baire space). Let +V be a separable Baire space and Vdense be a dense subset of V. For every subset Vgen of +V, the following two assertions are equivalent: +1. the subset Vgen is generic in V; +2. for every V0 in Vdense, there exists a neighbourhood ν of V0 in V such that Vgen ∩ ν +is generic in ν. +4.3 Potentials that are quadratic past a given radius +The whole space Ck+1(Rd, R) of potentials is somewhat difficult to handle, for various +reasons: it is not separable, even locally, and the flow of system (1.7) is not globally +well-defined for some of the potentials V in this space. To get around these difficulties, +the proofs of the next sections 5 and 6 will be carried out on a more restricted class +Vquad-R of potentials, after what the results will be extended to the full set Ck+1(Rd, R) +in the final section 9. Let +(4.1) +Vfull = Ck+1(Rd, R) , +and, for a positive quantity R, let +(4.2) +Vquad-R = +� +V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = |u|2 +2 +� +. +By contrast with Vfull, the affine subspace Vquad-R of Vfull is separable, and therefore +provides a framework where Lemma 4.3 may be applied. The next lemma states some +(nice) properties of the flow of system (1.7) for a potential V in Vquad-R. It is followed +by another one (Corollary 4.6 below) providing the adequate tools to proceed with the +extension mentioned above and carried out in section 9. +Notation. For every non negative quantity r, let BRd(0, r) and BRd(0, r) denote the open +ball and the closed ball centred at the origin and of radius r in Rd. +Lemma 4.4. For every positive quantity R and for every potential V in Vquad-R, the +following conclusions hold. +1. For every speed c, the flow defined by the differential system (1.7) is global. +24 + +2. Every profile ξ �→ u(ξ) of a travelling front or a standing front or a standing pulse, +for this potential, satisfies the following bound: +(4.3) +sup +ξ∈R +|u(ξ)| < R . +Proof. Let V be in Vquad-R and let c be a real quantity. According to the definition (4.2) +of Vquad-R, there exists a positive quantity K such that, for every u in Rd, +|∇V (u)| ≤ K + |u| . +As a consequence, it follows from the expression (1.6) of Fc,V that, for every solution +U = (u, v) of (1.7) in R2d, +��� ˙U(ξ) +��� = |Fc,V (u, v)| = |(v, ∇V (u) − cv)| = O|U|→∞ +� |U(ξ)| +� . +This bound prevents solutions from blowing up in finite time, proving conclusion 1. +Now let ξ �→ u(ξ) denote a solution of (1.2) approaching critical points of V at both +ends of R. Let us write q = |u|2/2, so that +(4.4) +˙q = u · ˙u +and +¨q = −c ˙q + ˙u2 + u · ∇V (u) , +and so that, since V is in Vquad-R, for every real quantity ξ, +|u(ξ)| ≥ R =⇒ +d +dξ |ξ +�ecξ ˙q(ξ) +� = ecξ� ˙u2(ξ) + u2(ξ) +� > 0 . +Since V is quadratic outside the ball BRd(0, R), its critical points must belong to the +interior of BRd(0, R), and the same must be true for u(ξ) when |ξ| is large. Now, if |u(·)| +were to reach the value R at some (finite) time ξ0, then (if ξ0 is the first time when this +happens) ˙q(ξ0) would be nonnegative; the implications above show that, from this time +on, the quantity ecξ ˙q(ξ) (and thus also the quantity ˙q(ξ)) would remain positive; so that +q(ξ) and |u(ξ)| would keep increasing with ξ, a contradiction with the fact that u(ξ) must +be back inside BRd(0, R) for ξ large. Conclusion 2 is proved. +4.4 Topological properties of restriction maps +Let R denote a positive quantity and let us consider the set +(4.5) +Vres-R = Ck+1�BRd(0, R), R +� . +The next Lemma 4.5 will be used to carry out, in section 9, the extension mentioned +at the beginning of this subsection. To ease its formulation, let us adopt Vquad-∞ as an +alternative notation for the space Vfull. Let R′ denote either a quantity larger than R or +∞, and let us consider the restriction operator: +(4.6) +resR,R′ : Vquad-R′ → Vres-R , +V �→ V|BRd(0,R) . +25 + +Lemma 4.5. The restriction map resR,R′ is continuous, surjective and open. +Proof. If two potentials of Vquad-R′ are Ck-close, then their restrictions to the closed ball +BRd(0, R) are still Ck-close on this ball, so that the map resR,R′ is continuous. +To prove that the map resR,R′ is surjective and open, it is sufficient to construct a +continuous right inverse for this map. For this purpose we may consider Seeley’s extension +ext∞,R : Vres-R → Vfull , +which is a right inverse for resR,∞ (that is resR,∞ ◦ ext∞,R is the identity map of Vres-R). +The map defined in Seeley’s original paper [47] extends to the whole space Rd a function +initially defined on a half space, but using spherical coordinates the same definition leads +to this extension ext∞,R. This map ext∞,R is linear and continuous for the usual topology +for the departure set Vres-R and the topology of uniform convergence of derivatives up +to order k on compact subsets of Rd for the arrival set Vfull. Now, if χ : [0, +∞) → R +denotes a smooth truncation function satisfying +χ ≡ 1 on [0, R] +and +χ ≡ 0 on +�min(R + 1, R′), +∞ +� , +then the map extR′,R : Vres-R → Vquad-R defined, for every V in Vres-R, by +extR′,R(V )(u) = χ(|u|) ext∞,R(V )(u) + +�1 − χ(|u|) +�|u|2 +2 +, +is a right inverse of resR,R′ and is continuous (for the topologies of uniform convergence +of derivatives up to order k for the departure and arrival sets). Lemma 4.5 is proved. +Corollary 4.6. For every couple (A, B) of subsets of Vres-R, let A′ = res−1 +R,R′(A) and +B′ = res−1 +R,R′(B) denote the sets of the potentials of Vquad-R′ whose restrictions to BRd(0, R) +belong to A and B respectively. Then the following equivalences hold: +A is open in Vres-R ⇐⇒ A′ is open in Vquad-R′ , +(4.7) +A is dense in B ⇐⇒ A′ is dense in B′ , +(4.8) +A is dense in Vres-R ⇐⇒ A′ is dense in Vquad-R′ . +(4.9) +Proof. Equivalence (4.7) follows from the continuity and the openness of resR,R′. +According to the surjectivity of resR,R′, the set resR,R′(A′) is equal to A and the set +resR,R′(B′) is equal to B. Since the image of a dense set by a continuous map is dense in +its image, if A′ is dense in B′ then A is dense in B. Implication “ ⇐= ” of (4.8) is proved. +On the other hand, if A is dense in B, then, for every open subset Ω′ of B′, its image +Ω := resR,R′(Ω′) is, according to Lemma 4.5, open in B so that the intersection A ∩ Ω is +nonempty. According to the surjectivity of resR,R′, the set res−1 +R,R′(A∩Ω) is also nonempty +and it is by construction included in A′ ∩ Ω′, which is a fortiori nonempty. This proves +that A′ is dense in B′ and completes the proof of equivalence (4.8). +Finally, equivalence (4.9) follows from (4.8) by setting B′ equal to Vquad-R′ and B equal +to Vres-R. Corollary 4.6 is proved. +26 + +5 Generic transversality of travelling fronts +5.1 Notation and statement +Notation. Let us recall the notation Vfull and Vquad-R introduced in (4.1) and (4.2). For +every potential function V in Vfull, let Σcrit(V ) and Σmin(V ) denote the set of non- +degenerate critical points and of non-degenerate minimum points of V , respectively, and +let us consider the set +(5.1) +FV = +�(c, u) ∈ (0, +∞) × Ck+1(R, Rd) : ξ �→ u(ξ) is a global solution of the +system ¨u = −c ˙u + ∇V (u) and there exists (e−, e+) in +Σcrit(V ) × Σmin(V ) such that +lim +ξ→−∞ u(ξ) = e− and +lim +ξ→+∞ u(ξ) = e+ +� . +In other words, (c, u) is in FV if and only if c is a positive quantity and ξ �→ u(ξ) is the +profile of a front travelling at speed c and connecting a non-degenerate critical point (at +the left end) to a non-degenerate minimum point (at the right end), for the potential V . +Let us take and fix a positive quantity R. +The goal of this section is to prove +Proposition 5.1 below, which is a weaker version of statement 1 of Theorem 1.7 since the +potentials under consideration belong to the subspace Vquad-R and not to the full space +Vfull. The reasons for first proving the intended genericity result in this restricted setting +are explained at the beginning of subsection 4.3, and the extension from Vquad-R to Vfull +will be carried out in the last section 9. As a reminder, the transversality of a travelling +front was defined in Definition 1.3. +Proposition 5.1. For every positive quantity R, there exists a generic subset of Vquad-R +such that, for every potential V in this subset, every travelling front (c, u) in FV is +transverse. +5.2 Reduction to a local statement +Let V0 denote a potential function in Vquad-R, and let e−,0 and e+,0 denote a non-degenerate +critical point and a non-degenerate minimum point of V0, respectively. +According +to Proposition 2.2 (or simply to the implicit function theorem), there exists a small +neighbourhood νrobust(V0, e−,0, e+,0) of V0 in Vquad-R and two Ck+1-functions e−(·) and +e+(·), defined on νrobust(V0, e−,0, e+,0) and with values in Rd, such that e−(V0) equals +e−,0 and e+(V0) equals e+,0 and, for every V in νrobust(V0, e−,0, e+,0), both e−(V ) and +e+(V ) are critical point of V close to e+,0. The following local generic transversality +statement, which calls upon this notation, yields Proposition 5.1 (as shown below). +Proposition 5.2. For every positive speed c0, there exist a neighbourhood νV0, e−,0, e+,0, c0 +of V0 in Vquad-R, included in νrobust(V0, e−,0, e+,0), a neighbourhood CV0, e−,0, e+,0, c0 of c0 in +(0, +∞), and a generic subset νV0, e−,0, e+,0, c0, gen of νV0, e−,0, e+,0, c0 such that, for every V +in νV0, e−,0, e+,0, c0, gen, every front travelling at a speed c in CV0, e−,0, e+,0, c0 and connecting +e−(V ) to e+(V ), for the potential V , is transverse. +27 + +Proof that Proposition 5.2 yields Proposition 5.1. Let us denote by Vquad-R-Morse the +dense open subset of Vquad-R defined by the Morse property (see [24]): +(5.2) +Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are non-degenerate} . +Let V0 denote a potential function in Vquad-R-Morse. Its critical points are non-degenerate +and thus isolated and, since V0 is in Vquad-R, they belong to the open ball BRd(0, R), +so that those critical points are in finite number. Assume that Proposition 5.2 holds. +With the notation of this proposition, let us consider the following three intersections, at +each time over all couples (e−,0, e+,0) with e−,0 a non-degenerate critical point and e+,0 +a non-degenerate minimum point of V0: +(5.3) +νV0, c0 = νrobust(V0) ∩ +�� +νV0, e−,0, e+,0, c0 +� +, +CV0, c0 = +� +CV0, e−,0, e+,0, c0 +and +νV0, c0, gen = νrobust(V0) ∩ +�� +νV0, e−,0, e+,0, c0, gen +� +. +Those are finite intersections, so that νV0, c0 is still a neighbourhood of V0 in Vquad-R, +CV0, c0 is still a neighbourhood of c0 in (0, +∞) and the set νV0, c0, gen is still a generic +subset of νV0, c0. Let I denote a compact sub-interval of (0, +∞); the three sets defined +above in (5.3) can be constructed likewise for every c0 in I. Since I is compact, it can be +covered by a finite union of sets CV0,c0,i, corresponding to a finite set {c0,1, . . . , c0,p} of +speeds. Again the intersections +νV0, I = +� +1≤i≤p +νV0, c0,i +and +νV0, I, gen = +� +1≤i≤p +νV0, c0,i, gen . +are finite and thus νV0, I, gen is still a generic subset of νV0, I, which is a neighbourhood of +V0 in Vquad-R-Morse. By construction, for every potential function V in νV0, I, gen, all fronts +travelling at a speed belonging to I and connecting a critical point of V to a minimum +point of V are transverse. In other words, the set +Vquad-R-Morse-⋔-F-I = +�V ∈ Vquad-R-Morse : for every travelling front (c, u) in FV , +if c is in I then (c, u) is transverse +� , +is locally generic in the sense that Vquad-R-Morse-⋔-F-I ∩ νV0, I is generic in νV0, I. Since +Vquad-R is separable, applying Lemma 4.3 with V = Vquad-R, Vdense = Vquad-R-Morse, +Vgen = Vquad-R-Morse-⋔-F-I and ν = νV0, I shows that the set Vquad-R-Morse-⋔-F-I is generic +in the whole set Vquad-R. As a consequence, the set +� +q∈N∗ +Vquad-R-Morse-⋔-F-[−1/q,q] +is still generic in Vquad-R. For every potential V in this set, all travelling fronts belonging +to FV are transverse, so that this set fulfils the conclusions of Proposition 5.1. +The remaining part of section 5 will thus be devoted to the proof of Proposition 5.2. +28 + +5.3 Proof of the local statement (Proposition 5.2) +5.3.1 Setting +For the remaining part of this section, let us fix a potential function V0 in Vquad-R, a +non-degenerate critical point e−,0 of V0 and a non-degenerate minimum point e+,0 of V0, +differing from e−,0. According to Proposition 2.2, there exist a neighbourhood ν of V0 +in Vquad-R, included in νrobust(V0, e−,0, e+,0), a neighbourhood C of c0 in (0, +∞), and a +positive quantity r such that, for every (c, V ) in C × ν, there exist Ck+1-functions +ˆwu +loc, c, V : Bu +E−,0(r) → R2d +and +ˆws +loc, c, V : Bs +E+,0(r) → R2d +such that the sets +W u +loc, c, V +�E−(V ) +� = ˆwu +loc, c, V +�Bu +E−,0(r) +� +and +W s +loc, c, V +�E+(V ) +� = ˆws +loc, c, V +�Bs +E+,0(r) +� +define a local unstable manifold of E−(V ) and a local stable manifold of E+(V ), respec- +tively (see the conclusions of Proposition 2.2 and equalities (2.9)). +Here is the setting where Sard–Smale theorem (Theorem 4.2) will be applied (see +Figure 5.1). Let +W u +loc, c, V (E−(V )) +freedom when bu moves +freedom when ξ moves +ˆwu +loc, c, V (bu) +Bu +bu +Φu(bu, ξ, c, V ) +freedom when c moves +ˆws +loc, c, V (bs) +E+(V ) +Φs(bs, c, V ) +bs +E−(V ) +flow Sc,V (ξ, ·) +Figure 5.1: The function ˆwu +loc, c, V (·) maps Bu onto the boundary of the local unstable +manifold W u +loc, c, V +�E−(V ) +�. A point ˆwu +loc, c, V (bu) of this boundary is pushed forward +during a time ξ by the flow Sc,V (ξ, ·) to give the image Φu(bu), which still belongs to the +global unstable manifold of E−(V ). On the other hand, Φs maps Bs onto the boundary +of the local stable manifold W s +loc, c, V (E+). The dependence of Φu on the time ξ and +the point bu provides a number of degrees of freedom equal to the dimension of the +unstable manifold, while an additional degree of freedom is provided by the speed c. This +additional dependence makes the difference between the transversality of a travelling +front as defined in Definition 1.3 and the classical transversality of stable and unstable +manifolds. +Bu = ∂Bu +E−,0(r) , +Bs = ∂Bs +E+,0(r) , +M = Bu × Bs × R × C , +Λ = ν , +N = (R2d)2 , +and +W = {(A, B) ∈ N : A = B} . +Notice that W is the diagonal of N. Let us consider the functions +Φu : Bu × R × C × Λ +→ R2d , +(bu, ξ, c, V ) +�−→ Sc,V +�ξ, ˆwu +loc, c, V (bu) +� +and +Φs : Bs × C × Λ +→ R2d , +(bs, c, V ) +�−→ ˆws +loc, c, V (bs) . +29 + +For every V in Λ and c in C, the image of Φu(·, ·, c, V ) is the global unstable manifold +W u +c,V +�E−(V ) +� (except the point E−(V ) itself), whereas the image of Φs(·, c, V ) is the +boundary of the local stable manifold W s +loc, c, V +�E+(V ) +�. Finally, let +(5.4) +Φ : M × Λ → N , +(m, V ) = +�(bu, bs, ξ, c), V +� �→ +�Φu(bu, ξ, c, V ), Φs(bs, c, V ) +� . +5.3.2 Additional conditions on ν and r +The main step in the proof of Proposition 5.2 is the construction of a suitable perturbation +W of V (carried out in subsection 5.3.5 below). This construction requires more accurate +conditions on the setting above. +First, since e−,0 and e+,0 differ, we may assume that ν and C and r are small enough +so that, for every V in ν, +(5.5) +πpos +� +W u +loc, c, V +�E−(V ) +�� +∩ πpos +� +W s +loc, c, V +�E+(V ) +�� += ∅ , +where πpos is the projection on the first component defined in (2.10). +Next, the following lemma is a more uniform version of assertion 1 of Proposition 3.3, +the key difference being that r can be chosen small enough such that Ionce contains +positive times. +Lemma 5.3. Up to replacing ν by a smaller neighbourhood of V0 in Vquad-R, and C by +a smaller neighbourhood of c0 in (0, +∞), and r by a smaller radius, we may assume +that the following conclusions hold. For every V in ν, every c in C, and every solution +ξ �→ U(ξ) = +�u(ξ), ˙u(ξ) +� of system (1.7) such that U(0) belongs to the boundary of +W u +loc, c, V +�E−(V ) +� (in other words there exists bu in Bu such that U(0) equals ˆwu +loc, c, V (bu)), +there exists a a compact interval with nonempty interior Ionce, included in (0, +∞), such +that: +1. the function ξ �→ |u(ξ) − e−(V )| is increasing on Ionce (so that u|Ionce is a diffeo- +morphism onto its image), +2. and for all ξ∗ in Ionce and ξ in R, u(ξ) = u(ξ∗) implies ξ = ξ∗, +3. and condition (5.5) holds, and in addition, +u(Ionce) ∩ πpos +� +W u +loc, c, V +�E−(V ) +� ∪ W s +loc, c, V +�E+(V ) +�� += ∅ . +Proof. Consider for now that ν and C and r are as in the previous subsection, and, for +some bu in Bu, let us consider the solution ξ �→ U(ξ) = +�u(ξ), ˙u(ξ) +� of system (1.7) defined +as: +U(ξ) = Sc0,V0 +�ξ, ˆwu +loc, c0, V0(bu) +� +(so that +U(0) = ˆwu +loc, c0, V0(bu)). +The same arguments as in the proof of statement 1 of Proposition 3.3 yield the following +conclusions. First, there exists a (large, negative) time ξ0(bu) such that the function ξ �→ +30 + +|u(ξ) − e−,0| is increasing on +�−∞, ξ0(bu) +�. Then, there exists ξonce(bu) in +�−∞, ξ0(bu) +� +such that, for every ξ∗ in +�−∞, ξonce(bu) +�, +HV +� +U +�ξ0(bu) +�� +< −V +�u(ξ∗) +� +(which is nothing but inequality (3.6)). Then, it follows from statement 1 of Proposition 3.3 +that, for the interval Ionce equal to +�ξonce(bu) − 2, ξonce(bu) − 1 +�, conclusions 1 and 2 of +Lemma 5.3 hold for the solution U (and they still hold if ξonce(bu) is replaced by a smaller +quantity). +Now, observe that, due to the smooth dependence of the map (−∞, 0] → R2d, ξ �→ +Sc,V +�ξ, ˆwu +loc, c, V (bu) +� on V and c and bu, this construction can be made uniform with +respect to bu in a (small) open subset Ω of Bu and V in a (small) neighbourhood νΩ +(included in ν) of V0, and to c in a (small) neighbourhood CΩ (included in C) of c0; in +other words, there exists a (sufficiently large negative) quantity ξonce(Ω) such that the +conclusions above hold for all such V and c and bu. Since Bu is compact, it can be +covered by a finite number Ω1 . . . Ωn of such open subsets. Thus, replacing +ν by +n +� +i=1 +νΩi +and +C by +n +� +i=1 +CΩi , +and choosing +ξonce = +min +i∈{1,...,n} ξonce(Ωi) +and +Ionce = [ξonce − 2, ξonce − 1] , +conclusions 1 and 2 of Lemma 5.3 hold. Up to replacing r by a smaller positive quantity, +we may assume in addition that Ionce belongs to (0, +∞). Finally, again up to replacing +r by a smaller positive quantity, we may assume that conclusion 3 also holds. +5.3.3 Equivalent characterizations of transversality +Let us consider the set +FΛ, C = +�(V, c, u) : V ∈ Λ and c ∈ C and u is the profile of a front travelling +at speed c and connecting e−(V ) to e+(V ), for the potential V +� , +and let us denote by �FΛ, C the set of equivalence classes of FΛ, C for the equivalence +relation: (V, c, u) ∼ (V †, c†, u†) if and only if V = V † and c = c† and u = u† up to a +translation of the time. The aim of this subsection is to prove Proposition 5.5 below, +relating the transversality of the intersection Φ(M × Λ) ∩ W to the transversality of +travelling fronts belonging to FΛ, C. To begin with, the next Proposition 5.4 formalizes +the correspondence between the intersection of the image of Φ with the diagonal W and +the profiles of such travelling fronts. +Proposition 5.4. The map +(5.6) +Φ−1(W) → FΛ, C , +(bu, bs, ξ, c, V ) �→ +� +V, c, ξ′ �→ πpos +� +Sc,V +�ξ′, ˆwu +loc, c, V (bu) +��� +defines a a one-to-one correspondence between Φ−1(W) and the quotient set �FΛ, C. +31 + +Proof. The image by Φ of a point (bu, bs, ξ, c, V ) of M × Λ belongs to the diagonal W +of N if and only if Φu(bu, ξ, c, V ) = Φs(bs, c, V ). If this last equality holds, the function +u : ξ′ �→ Φu(bu, ξ′, c, V ) is a solution belonging to the unstable manifold W u +c,V +�E−(V ) +� +such that u(ξ) = Φs(bs, c, V ) belongs to the local stable manifold of E+(V ). Thus u +defines the profile of a front travelling at speed c and connecting e−(V ) to e+(V ). The +map (5.6) is thus well defined. +Now, if ξ �→ u(ξ) is the profile of a front travelling at a speed c in C for the potential +V and connecting e−(V ) to e+(V ), then, according to Proposition 1.2, the image of +ξ �→ +�u(ξ), ˙u(ξ) +� belongs to the intersection W u +c,V +�E−(V ) +� ∩ W s +c,V +�E+(V ) +�. As a con- +sequence, this image must cross the boundary of W u +loc, c, V +�E−(V ) +� at a time ξ− and +the boundary of W s +loc, c, V +�E+(V ) +� at a time ξ+: there exists bu in Bu and bs in Bs such +that +�u(ξ−), ˙u(ξ−) +� = ˆwu +loc, c, V (bu) and +�u(ξ+), ˙u(ξ+) +� = ˆws +loc, c, V (bs). By construction, +Φu(bu, ξ+ − ξ−, c, V ) = Φs(bs, c, V ) and thus Φ(bu, bs, ξ+ − ξ−, c, V )) is in W. In addition, +according to the remark at the end of subsection 2.2, the times ξ− and ξ+ at which these +intersections occur are unique (for a given profile ξ �→ u(ξ)), thus so are the points bu in +Bu and bs in Bs and the time lag ξ+ − ξ−. This completes the proof of this one-to-one +correspondence. +Both corresponding notions of transversality are related as follows. +Proposition 5.5. For every potential function V in Λ, the following two statements are +equivalent. +1. The image of the function M → N, m �→ Φ(m, V ) is transverse to W. +2. Every profile ξ �→ u(ξ) of a front travelling at a speed c in C and connecting e−(V ) +to e+(V ), for the potential V , is transverse. +Proof. Let us take (m1, V1) in M × Λ such that Φ(m1, V1) is in W, let (bu +1, bs +1, ξ1, c1) +denote the point m1 and let ξ �→ u1(ξ) denote the profile of the corresponding travelling +front. In other words, +for all ξ in R , +U1(ξ) = Φu(bu +1, ξ, c1, V1) , +where +U1(ξ) = +�u1(ξ), ˙u1(ξ) +� . +Let us consider the maps +ΓΦ : (Bu × R × C) × (Bs × C) → R × R2d +�(bu, ξ, cu), (bs, cs) +� �→ +�cu, Φu(bu, ξ, cu, V1) +� + +�cs, Φs(bs, cs, V1) +� , +and +∆Φ : M → R2d , +(bu, bs, ξ, c) �→ Φu(bu, ξ, c, V1) − Φs(bs, c, V1) . +and let us write, only for this proof, DΦ for DTm1MΦ, and similarly DΦu and DΦs and +D(ΓΦ) and D(∆Φ) for the differentials of Φu and Φs and ΓΦ and ∆Φ at (m1, V1) and +with respect to all variables but V . +Lemma 5.6. The following three statements are equivalent. +32 + +(A) The image of DΦ contains a supplementary subspace of the diagonal W of (R2d)2. +(B) The map D(ΓΦ) is surjective. +(C) The map D(∆Φ) is surjective. +Proof of Lemma 5.6. If statement (A) holds, then, for every (α, β) in (R2d)2, there exist +γ in R2d and δm in Tm1M such that +(5.7) +(γ, γ) + DΦ · δm = (α, β) , +so that +(5.8) +D(∆Φ) · δm = α − β , +and statement (C) holds. Conversely, if statement (C) holds, then, for every (α, β) +in (R2d)2, there exists δm in Tm1M such that (5.8) holds, and as a consequence, if +(δbu, δbs, δξ, δc) denotes the components of δm, the vector α − DΦu(δbu, δξ, δc) is equal +to β − DΦs(δbs, δc), and if this vector is denoted by γ, then equality (5.7) holds, and this +shows that statement (A) holds. Thus statements (A) and (C) are equivalent. +Now, if statement (B) holds, then, for every (δc, δU) in R×R2d, there exist (δbu, δξ, δcu) +in Tbu +1Bu × R2 and (δbs, δcs) in Tbs +1Bs × R such that +(5.9) +(δc, δU) = +�δcu, DΦu · (δbu, δξ, δcu) +� + +�δcs, DΦs · (δbs, δcs) +� , +so that δc is equal to δcu + δcs and so that +δU = DΦu · (δbu, δξ, δcu) + DΦs · (δbs, δc − δcu) += DΦu · (δbu, δξ, δcu) + DΦs · (0, δc) − DΦs · (−δbs, δcu) , +so that finally, if (δbu, −δbs, δξ, δcu) is denoted by δm, then +(5.10) +δU = D(∆Φ) · δm + DΦs · (0, δc) . +By choosing δc equal to 0, this shows that every δU in R2d is in the image of D(∆Φ), +which is statement (C). Conversely, if statement (C) holds, then for every (δc, δU) in +R × R2d, there exists δm in Tm1M such that (5.10) holds, and if δcu denotes the last +component of δm and δcs is the difference δc − δcu, then equality (5.9) holds, and this +shows that statement (B) holds. Thus statements (B) and (C) are equivalent. +Continuation of the proof of Proposition 5.5. To conclude, let us see how both transver- +sality statements 1 and 2 can be expressed in terms of the ingredients of Lemma 5.6. +On the one hand, according to Definition 1.3, the travelling front with profile u1(·) and +speed c1 is transverse if and only if the intersection +(5.11) +� � +cu>0 +{cu} × W u +cu,V +�E−(V ) +� +� +∩ +� � +cs>0 +{cs} × W s +cs,V +�E+(V ) +� +� +33 + +is transverse, in R2d+1, along the set {c1} × U1(R). This transversality can be considered +at a single point, no matter which, of the trajectory U0(R), thus in particular at the point +Φu(bu +1, ξ1, c1, V1), which is equal to Φs(bs +1, c1, V1). By definition, the sum of the tangent +spaces associated to the manifolds intersected in (5.11) is the image of D(ΓΦ) and the +transversality stated in statement 2 is therefore equivalent to the surjectivity of the map +D(ΓΦ) (statement (B) in Lemma 5.6). +On the other hand, the image of the function M → (R2d)2, m �→ Φ(m, V1) is transverse +at Φ(m1, V1) to the diagonal W of (R2d)2 as stated in 1 if and only if the image of DΦ +contains a supplementary subspace of the diagonal (statement (A) in Lemma 5.6). Thus +Proposition 5.5 follows from Lemma 5.6. +According to Proposition 5.5, Proposition 5.2 follows from the conclusion of Theorem 4.2 +applied to the function Φ (see subsection 5.3.6). The next two subsections are devoted +to checking that this function Φ fulfils hypotheses 1 and 2 of this theorem. +5.3.4 Checking hypothesis 1 of Theorem 4.2 +Since the vector field (1.6) defining the flow (2.1) is of class Ck, so is the function Φ. It +follows from subsection 2.1 that +dim(Bu) = d − m(e−,0) − 1 and dim(Bs) = d − 1 , thus dim(M) = 2d − m(e−,0) , +and since the codimension of W in N is equal to 2d, +dim(M) − codim(W) = −m(e−,0) ≤ 0 , +thus +k > dim(M) − codim(W) ; +in other words, hypothesis 1 of Theorem 4.2 is fulfilled. +5.3.5 Checking hypothesis 2 of Theorem 4.2 +Take (m1, V1) in the set Φ−1(W). Let (bu +1, bs +1, ξ1, c1) denote the components of m1, and, +for every real quantity ξ, let us write +U1(ξ) = +�u1(ξ), v1(ξ) +� = Sc1,V1 +�ξ, ˆwu +loc, c1, V1(bu +1) +� . +The function ξ �→ u1(ξ) is the profile of a front travelling at speed c1 and connecting +e−(V1) to e+(V1) for the potential V1; and, according to the empty inclusion (5.5), the +quantity ξ1 is positive. Let us write +DΦ , +DΦu +and +DΦs +for the full differentials (with respect to arguments m in M and V in Λ) of the three +functions Φ and Φu and Φs respectively at the points +�bu +1, bs +1, ξ1, c1, V1 +�, +�bu +1, ξ1, c1, V1 +� +and +�bs +1, c1, V1 +�. Checking hypothesis 2 of Theorem 4.2 amounts to prove that +(5.12) +im(DΦ) + TW = TN . +34 + +To this end, since the subspace R2d × {0R2d} of N is transverse to the diagonal W, it is +sufficient to prove that, for every γ in R2d, the vector +�γ, 0R2d +� can be reached by DΦ. +Thus, it is sufficient to prove that, for every γ in R2d, there exist a real quantity ζ and a +function W in Ck+1 +b +(Rd, R) with a compact support supp(W) satisfying +(5.13) +supp(W) ⊂ BRd(0, R) , +such that +DΦu · (0, ζ, 0, W) = γ , +(5.14) +and +DΦs · (0, 0, W) = 0R2d . +(5.15) +To fulfil equality (5.15), it is sufficient to assume that W satisfies the following additional +condition: +(5.16) +supp(W) ∩ πpos +� +W s +loc, c, V +�E+(V1) +�� += ∅ , +where πpos : R2d → Rd is the projection on the first component defined in (2.10) (this +condition ensures that the local stable manifold of E+(V1) is not changed by a perturbation +of V1 in the direction of W, see the second remark at the end of subsection 2.2). For +convenience, we will also ensure that the same is true for the local unstable manifold of +E−(V1), that is: +(5.17) +supp(W) ∩ πpos +� +W u +loc, c, V +�E−(V1) +�� += ∅ . +Fulfilling equality (5.14) amounts to prove that the orthogonal complement of the subspace +of the directions of R2d that can be reached by DΦu · (0, ζ, 0, W) is trivial, i.e. reduced +to {0R2d}. Observe that +DΦu · (0, ζ, 0, 0) = ζ ˙U1(ξ1) . +Thus the transversality statement (5.12) is a consequence of the following lemma. +Lemma 5.7 (perturbation of the potential reaching a given direction). For every nonzero +vector (φ1, ψ1) in R2d which is orthogonal to ˙U1(ξ1), there exists W in Ck+1 +b +(Rd, R) +satisfying conditions (5.13), (5.16) and (5.17) and the inequality +(5.18) +⟨DΦu · (0, 0, 0, W) | (φ1, ψ1)⟩ ̸= 0 . +Proof of Lemma 5.7. Let (φ1, ψ1) denote a nonzero vector orthogonal to U1(ξ1) in R2d, +and let W be a function in Ck+1 +b +(Rd, R) satisfying the conditions (5.13), (5.16) and (5.17). +Let us consider the linearization of the differential system (1.7), for the potential V1 and +the speed c1, around the solution ξ �→ U1(ξ): +(5.19) +d +dξ +� +δu(ξ) +δv(ξ) +� += +� +0 +id +D2V1 +�u1(ξ) +� +−c1 +� � +δu(ξ) +δv(ξ) +� +, +and let T(ξ, ξ′) denote the family of evolution operators obtained by integrating this +linearized differential system between times ξ and ξ′. It follows from condition (5.17) that +35 + +W only affects the part of Φu corresponding to the flow (not on the function ˆwu +loc, c1, V1) +and the variation of constants formula yields that +(5.20) +DΦu · (0, 0, 0, W) = +� ξ1 +0 +T(ξ, ξ1) +� +0, ∇W +�u1(ξ) +�� +dξ . +For every time ξ, let T ∗(ξ, ξ1) denote the adjoint operator of T(ξ, ξ1), and let +(5.21) +�φ(ξ), ψ(ξ) +� = T ∗(ξ, ξ1) · (φ1, ψ1) . +According to expression (5.20), inequality (5.18) reads +� ξ1 +0 +�� +0, ∇W +�u1(ξ) +�� ��� T ∗(ξ, ξ1) · (φ1, ψ1) +� +dξ ̸= 0 , +or equivalently +(5.22) +� ξ1 +0 +∇W +�u1(ξ) +� · ψ(ξ) dξ ̸= 0 . +Notice that, due to the expression of the linearized differential system (5.19), (φ, ψ) is a +solution of the adjoint linearized system +(5.23) +� ˙φ(ξ) +˙ψ(ξ) +� += − +� +0 +D2V1 +�u1(ξ) +� +id +−c1 +� � +φ(ξ) +ψ(ξ) +� +. +Our task is thus to construct a function W in Ck+1 +b +(Rd, R) satisfying (5.13), (5.16), +(5.17) and (5.22). There are two difficulties to overcome. +1. First, as shown by Figure 3.1, the function ξ �→ u1(ξ) may reach the same value for +different values of the argument ξ, making it difficult to handle the interactions +of the contributions to the integral (5.22) of the perturbation W +�u1(ξ) +� at these +different values of ξ. +2. Second, the integral (5.22) depends on the gradient ∇W of the perturbation W +and not on W itself, and this gradient cannot be any function. +These difficulties have already been tackled in several contexts, see [37, 38, 45] (ordinary +differential equations) and [8–10, 26, 27] (partial differential equations). Each time, +some specific arguments have to be found, using the peculiarities and constraints of the +considered system. +In the present case, the following trick will do the job. According to Lemma 5.3, there +exists a closed interval with nonempty interior Ionce, included in (0, +∞), such that +(5.24) +u1(Ionce) ∩ πpos +� +W u +loc, c, V +�E−(V1) +� ∪ W s +loc, c, V +�E+(V1) +�� += ∅ , +such that ˙u does not vanish on Ionce, and such that +(5.25) +for all ξ∗ in Ionce and ξ in R , +u1(ξ) = u1(ξ∗) =⇒ ξ = ξ∗ . +36 + +According to the empty intersection (5.24) and since u1(ξ) is in W s +loc, c, V +�E+(V1) +� for ξ +larger than ξ1, the interval Ionce is actually included in (0, ξ1). In view of (5.25), the image +u1(Ionce) of this interval provides a suitable place where the trajectory can be perturbed +without the inconvenience 1 emphasized above. Two cases have to be considered (plus a +third one that will turn out to be empty). +Case 1. +There exists a time ξ∗ in Ionce such that ψ(ξ∗) is not collinear to ˙u1(ξ∗). +In this case, up to an affine conformal change of coordinate system in Rd, we may +assume that +(5.26) +u1(ξ∗) = 0 +and +˙u1(ξ∗) = ϵ1 +and +ϵ2 · ψ(ξ∗) ̸= 0 , +where ϵ1 = (1, 0, . . . , 0) and ϵ2 = (0, 1, 0, . . . , 0) are the two first vectors of the canonical +basis of Rd. Let ρ denote an even function in Ck+1�R, [0, 1] +� satisfying +ρ(0) = 1 +and ρ vanishes on R \ (−1, 1). +Let ε denote a small positive quantity to be chosen later and let us consider the bump +function +ρε : Rd → [0, 1], +u �→ ρ +�|u| +ε +� +. +It follows from this definition that +(5.27) ρε(0Rd) = 1 +and +supp(ρε) ⊂ BRd(0, ε) +and +∥∇ρε∥L∞(Rd,R)d ∈ Oε→0(ε−1) . +Let us define the perturbation W as follows: for +every u in Rd, +W(u) = ρε(u)(ϵ2 · u) , +see Figure 5.2, so that +(5.28) +∇W(u) = ρε(u)ϵ2 + (ϵ2 · u)∇ρε(u) . +ϵ1 +ϵ2 +u1(ξ) +Figure 5.2: Graph of W. +It follows from this definition that, if ε is small enough, then, on the one hand conditions +(5.13) (according to inequality (4.3) of Lemma 4.4), and (5.16) and (5.17) (according to +the empty intersection (5.24)) are fulfilled; and, on the other hand, according to (5.25) +and since ˙u1(ξ∗) is nonzero, there exists an open interval I∗ +ε of R satisfying +(5.29) +ξ∗ ∈ I∗ +ε +and, for every ξ in R, +u1(ξ) ∈ BRd(0, ε) ⇐⇒ ξ ∈ I∗ +ε . +Let us assume that ε is chosen as such. It follows from (5.29) that the integral in (5.22) +reduces to: +(5.30) +� +I∗ε +∇W +�u1(ξ) +� · ψ(ξ) dξ . +37 + +As a consequence, if u1(ξ) follows a straight line in the direction of ϵ1 inside the ball +BRd(0, ε), then, for every ξ in I∗ +ε , +∇W +�u1(ξ) +� = ρε +�u1(ξ) +�ϵ2 , +so that the integral (5.30) reduces to +� +I∗ε +ρε +�u1(ξ) +�ϵ2 · ψ(ξ) dξ , +and according to the last property of (5.26), if ε is sufficiently small then this integral +does not vanish, fulfilling inequality (5.22) — and thus also (5.18). +In the general situation where u1(ξ) does not necessarily follow a straight line in the +direction of ϵ1 inside the ball BRd(0, ε), the quantity ϵ2 · u1(ξ) is in Oε→0(ε2) when ξ is in +I∗ +ε , thus it follows from (5.28) and from the last property of (5.27) that, still for ξ in I∗ +ε , +∇W +�u1(ξ) +� = ρε +�u1(ξ) +�ϵ2 + Oε→0(ε) , +and since ρε(0Rd) equals 1, it follows from the last property of (5.26) that, if ε is sufficiently +small, then inequality (5.22) is fulfilled again — thus so is inequality (5.18). +If case 1 does not occur, then ψ(ξ) is collinear to ˙u1(ξ) for every ξ in Ionce, and since +˙u1(·) does not vanish on Ionce, there exists a C1-function α : Ionce → R such that, for +every ξ in Ionce, +(5.31) +ψ(ξ) = α(ξ) ˙u1(ξ) . +The next cases 2 and 3 differ according to whether the function α(·) is constant or not. +Case 2. +For every ξ in Ionce, equality (5.31) holds for some nonconstant function α(·). +For every perturbation W of the potential, if the support of W is localized enough +around some point of u(Ionce) (so that expression (5.31) holds as soon as ∇W +�u(ξ) +� +is nonzero), then an integration by parts shows that the integral in inequality (5.22) +becomes +(5.32) +� +∇W +�u1(ξ) +� · ψ(ξ) dξ = +� +α(ξ)∇W +�u1(ξ) +� · ˙u1(ξ) dξ = − +� +˙α(ξ)W +�u1(ξ) +� dξ +(with integration domain [0, ξ1] for each of these integrals). +The expression of this last integral shows why the assumption (made in the present +case 2) that α(·) is nonconstant matters. According to this assumption, there exists ξ∗ +in Ionce such that ˙α(ξ∗) is nonzero. Let us assume (up to an affine change of variable +in R2d) that u1(ξ∗) is equal to 0Rd. Let us define BRd(0, ε) and ρε and I∗ +ε as in case 1 +above, and let us simply define the perturbation W as +W = ρε . +38 + +As in case 1, for ε sufficiently small, conditions (5.13), (5.16) and (5.17) are fulfilled, and +the integral in inequality (5.22) reduces to the expression (5.30). In view of (5.32), (5.22) +thus becomes +(5.33) +� +I∗ε +˙α(ξ)W +�u1(ξ) +� dξ ̸= 0 , +which is fulfilled if ε sufficiently small. It follows that inequality (5.22) is fulfilled, and +thus so is inequality (5.18). +Case 3. +For every ξ in Ionce, ψ(ξ) = α ˙u(ξ), for some real (constant) quantity α. +In this case, expression (5.32) shows that inequality (5.22) cannot hold if the support +of W is localized around some point of u(Ionce). Fortunately, this third case will lead to +a contradiction (and does therefore actually not happen). Recall that (φ, ψ) is a solution +of the adjoint linearized system (5.23). Thus, for every ξ in Ionce, it follows from the +assumption made in this case 3 that +(5.34) +φ(ξ) = cψ(ξ) − ˙ψ(ξ) = cα ˙u1(ξ) − α¨u1(ξ) . +Besides, recall that (φ1, ψ1) is orthogonal to ˙U1(ξ1) = T(ξ, ξ1) ˙U1(ξ). Thus, +�φ(ξ), ψ(ξ) +� = +T ∗(ξ, ξ1) (φ1, ψ1) is orthogonal to ˙U1(ξ). According to the expression of ψ and expression +(5.34), this last property reads +(5.35) cα| ˙u1|2(ξ) − α¨u1(ξ) · ˙u1(ξ) + α ˙u1(ξ) · ¨u1(ξ) = 0 , +which yields +cα| ˙u1|2(ξ) = 0 . +Since ˙u1 does not vanish on (−∞, ξonce), the quantity α must be zero. This yields +φ ≡ ψ ≡ 0, and contradicts the assumptions of Lemma 5.7. +In short, case 3 cannot happen and, in both cases 1 and 2, a suitable construction +provides a function W in Ck+1 +b +(Rd, R) fulfilling the conditions (5.13) and (5.16) to (5.18). +Lemma 5.7 is proved. +5.3.6 Conclusion +Proof of Proposition 5.2. As seen in subsection 5.3.4, hypothesis 1 of Theorem 4.2 is +fulfilled for the function Φ defined in (5.4). Since the conclusion of Lemma 5.7 yields +equality (5.12), hypothesis 2 of this theorem is also fulfilled. The conclusion of this +theorem ensures that there exists a generic subset Λgen of Λ such that, for every V in Λgen, +the function Φ(·, V ) is transverse to the diagonal W of N. According to Proposition 5.5, +it follows that, for every V in Λgen, every profile ξ �→ u(ξ) of a front travelling at a speed +c in C and connecting e−(V ) to e+(V ), for the potential V , is transverse. In other words +the conclusions of Proposition 5.2 hold with CV0, e−,0, e+,0, c0 = C, νV0, e−,0, e+,0, c0 = ν = Λ +and νV0, e−,0, e+,0, c0, gen = Λgen. +As shown in subsection 5.2, Proposition 5.1 follows from Proposition 5.2. +39 + +6 Generic elementarity of symmetric standing pulses +This section presents strong similarities with the previous section 5. For that reason, the +presentation aims at emphasizing the main differences, while some details or comments +are omitted when they are identical to some already provided in section 5. +6.1 Notation and statements +Notation. For every potential function V in Vfull, let us recall (subsection 5.1) that +Σcrit(V ) denotes the set of non-degenerate critical points of V , and let us consider the set +(6.1) +PV = +�u ∈ Ck+1(R, Rd) : ξ �→ u(ξ) is a global solution of the system ¨u = ∇V (u) , +and there exists e in Σcrit(V ) such that u(ξ) → e as ξ → ±∞ +� . +In other words, u is in PV if and only if ξ �→ u(ξ) is the profile of a standing pulse +connecting a non-degenerate critical point e to itself, for the potential V . +Let us take and fix a positive quantity R. Let us recall that the elementarity of a +symmetric standing pulse was defined in Definition 1.6. The goal of this section is to +prove the following proposition. +Proposition 6.1. There exists a generic subset of Vquad-R such that, for every potential +V in this subset, every symmetric standing pulse in PV is elementary. +Let V0 denote a potential function in Vquad-R, and let e0 denote a non-degenerate critical +point of V0. According to Proposition 2.4 (or simply to the implicit function theorem), +there exists a small neighbourhood νrobust(V0, e0) of V0 in Vquad-R and a Ck+1-function +e(·) defined on νrobust(V0, e0) and with values in Rd, such that e(V0) equals e0 and, for +every V in νrobust(V0, e0), e(V ) is a critical point of V0 close to e0. +Exactly the same arguments as in subsection 5.2 show that Proposition 6.1 is a +consequence of the following local statement. +Proposition 6.2. There exists a neighbourhood νV0, e0 of V0 in Vquad-R, included in +νrobust(V0, e0), and a generic subset νV0, e0, gen of νV0, e0 such that, for every V in νV0, e0, gen, +every symmetric standing front connecting e(V ) to itself is elementary. +The remaining part of section 6 will thus be devoted to the proof of Proposition 6.2. +Let us keep the notation V0 and e0 and νrobust(V0, e0) introduced above. According to +Proposition 2.4, there exist a neighbourhood ν of V0 in Vquad-R, included in νrobust(V0, e0), +and a positive quantity r such that, for every V in ν, there exist Ck-functions +ˆwu +loc, V : Bu +E0(r) → R2d +and +ˆws +loc, V : Bs +E0(r) → R2d +such that the sets +W u +loc, V +�E(V ) +� = ˆwu +loc, V +�Bu +E0(r) +� +and +W s +loc, V +�E(V ) +� = ˆws +loc, V +�Bs +E0(r) +� +40 + +define a local unstable manifold and a local stable manifold of E(V ), respectively (see +the conclusions of Proposition 2.4 and equalities (2.13)). Observe that the departure sets +Bu +E0(r) of ˆwu +loc, V and Bs +E0(r) of ˆws +loc, V do not depend on V . Let +Bu = ∂Bu +E0(r) +and +Bs = ∂Bs +E0(r) . +According to the expression (2.4) of the eigenvectors of the linear system (2.2), +Eu +V0(E0) ∩ Ssym = {0R2d} +and +Es +V0(E0) ∩ Ssym = {0R2d} . +It follows that, up to replacing ν by a smaller neighbourhood of V0 in Vquad-R and r by a +smaller positive quantity, for every V in ν, +(6.2) +W u +loc, V +�E(V ) +� ∩ Ssym = {E(V )} +and +W s +loc, V +�E(V ) +� ∩ Ssym = {E(V )} . +6.2 Proof of Proposition 6.2 +6.2.1 Application of Theorem 4.2 +The setting to which Theorem 4.2 will be applied is as follows. Let +M = Bu × R , +Λ = ν , +N = R2d +and +W = Ssym , +and let us consider the function +(6.3) +Φ : M × Λ → N, +(bu, ξ, V ) �→ SV +�ξ, ˆwu +loc, V (bu) +� . +If the conclusion of Theorem 4.2 holds within this setting, then there exists a generic +subset Λgen of Λ such that, for every V in Λgen, the image of the function M → N, +m �→ Φ(m, V ) is transverse to W. +For a given potential V , the image of m �→ Φ(m, V ) is nothing but the unstable +manifold of E(V ) (deprived of E), see the proof of Proposition 5.4. According to the +characterizations of the symmetric standing pulses stated in Lemma 3.2, the intersection +of Φ(M, V ) with W = Ssym actually corresponds to the set of symmetric standing pulses. +Moreover, by definition (see Definition 1.6), the elementarity of the symmetric standing +pulses for V is equivalent to the transversality of the intersection of Φ(M, V ) with +W = Ssym. Thus, the conclusion of Theorem 4.2 directly implies Proposition 6.2 with +νV0, e0 = ν = Λ and νV0, e0, gen = Λgen. +It remains to show that, in the setting above, the hypotheses of Theorem 4.2 are +fulfilled. +6.2.2 Checking hypothesis 1 of Theorem 4.2 +It follows from subsection 2.1 that dim(Bu) = d − m(e0) − 1. Hence, +dim(M) − codim(W) = +�d − m(e0) +� − d = −m(e0) , +which is less than the positive integer k (the regularity of Φ). Hypothesis 1 of Theorem 4.2 +is thus fulfilled. +41 + +6.2.3 Checking hypothesis 2 of Theorem 4.2 +Take (m1, V1) in the set Φ−1(W). Let (bu +1, ξturn) denote the components of m1, and, for +every real quantity ξ, let us write +U1(ξ) = +�u1(ξ), v1(ξ) +� = SV1 +�ξ, ˆwu +loc, V1(bu +1) +� . +The function ξ �→ u1(ξ) is the profile of a symmetric standing pulse, connecting e(V1) +to itself for the potential V1, and the quantity ξturn is the turning time of this standing +pulse (see Definition 1.5). Observe that, according to the first equality of (6.2), this +turning time ξturn must be positive. Let DΦ denote the full differential (with respect to +m and V ) of Φ at the point +�(bu +1, ξturn), V1 +�. Hypothesis 2 of Theorem 4.2 follows from +the next Lemma 6.3. +Lemma 6.3 (perturbation of the potential reaching a given direction). For every nonzero +vector ψ1 in Rd, there exists W in Ck+1 +b +(Rd, R) such that +(6.4) +⟨DΦ · (0, W) | (0, ψ1)⟩ ̸= 0 , +and +(6.5) +supp(W) ⊂ BRd(0, R) . +Proof. The proof is similar to that of Lemma 5.7. Let ψ1 be a nonzero vector in Rd. , +and let W denote a function in Ck+1 +b +(Rd, R) with a support satisfying the condition +(6.6) +supp(W) ∩ πpos +� +W u +loc, V +�E(V1) +�� += ∅ . +Let us again use the notation T(ξ, ξ′) to denote the family of evolution operators obtained +by integrating the linearized differential system (5.19) (for c1 equal to 0) between the +times ξ and ξ′. It follows from the empty intersection (6.6) that +(6.7) +DΦ · (0, W) = +� ξturn +0 +T(ξ, ξturn) +� +0, ∇W +�u1(ξ) +�� +dξ . +For every time ξ, let T ∗(ξ, ξturn) denote the adjoint operator of T(ξ, ξturn), and let +�φ(ξ), ψ(ξ) +� = T ∗(ξ, ξturn) · (0, ψ1) . +According to (6.7), condition (6.4) reads +� ξturn +0 +�� +0, ∇W +�u1(ξ) +�� ��� T ∗(ξ, ξturn) · (0, ψ1) +� +dξ ̸= 0 , +or equivalently +(6.8) +� ξturn +0 +∇W +�u1(ξ) +� · ψ(ξ) dξ ̸= 0 . +42 + +According to the first equality of (6.2) and due to the Hamiltonian invariance (1.5), for +every (u, v) in W u +loc, V1 +�E(V1) +� and differing from E(V1), the quantity V1(u) is larger than +V1 +�e(V1) +�. On the other hand, since ˙u1(ξturn) vanishes the quantity V1 +�u1(ξturn) +� is equal +to V1 +�e(V1) +�, so that u1(ξturn) does not belong to the (closed) set πpos +� +W u +loc, V1 +�E(V1) +�� +. +As a consequence, there exists a time ξ−, smaller than (and sufficiently close to) ξturn, +such that +(6.9) +u1 +�[ξ−, ξturn] +� ∩ πpos +� +W u +loc, V1 +�E(V1) +�� += ∅ . +Observe that, according to Lemma 3.2, the function ξ �→ ˙u1(ξ) does not vanish on +(−∞, ξturn). As in subsection 5.3.5, three cases have to be considered for the construction +of the perturbation W. +Case 1. +There exists a time ξ† in (ξ−, ξturn) such that ψ(ξ†) is not collinear to ˙u1(ξ†). +In this case, conclusion 3 of Proposition 3.3 provides an open interval Ionce included in +(ξ−, ξturn) and small enough so that, for every ξ∗ in Ionce, +• the vector ψ(ξ∗) is not collinear to ˙u1(ξ∗), +• and for every ξ in (−∞, ξturn), if u1(ξ) equals u1(ξ∗) then ξ equals ξ∗. +The same construction as in case 1 of the proof of Lemma 5.7 can then be carried out. +It leads to a perturbation W such that supp(W) is localized around a point of u(Ionce) +(so that, according to inequality (4.3), inclusion (6.5) holds and according to the empty +intersection (6.9) the empty intersection (6.6) holds) and such that inequality (6.8) holds +— thus so does inequality (6.4). +Case 2. +For every ξ in (ξ−, ξturn), ψ(ξ) = α(ξ) ˙u1(ξ) with α(·) not constant. +Again, conclusion 3 of Proposition 3.3 provides an open interval Ionce included in +(ξ−, ξturn), small enough so that, for every ξ∗ in Ionce, +• ψ(ξ∗) = α(ξ∗) ˙u1(ξ∗), +• and ˙α(ξ∗) ̸= 0, +• and for every ξ in (−∞, ξturn), if u1(ξ) equals u1(ξ∗) then ξ equals ξ∗. +The same construction as in case 2 of the proof of Lemma 5.7 can then be carried out. +Case 3. +For every ξ in (ξ−, ξturn), ψ(ξ) = α ˙u1(ξ), for some real (constant) quantity α. +In case 3 of the proof of Lemma 5.7, the non-nullity of c was mandatory to take +advantage of (5.35). Thus, a new ad hoc argument is now required to preclude the +possibility of the present case 3. Here it is: since ˙u1(ξturn) = 0, it follows from the +assumption made in this case that ψ(ξ) goes to 0 as ξ goes to ξturn, so that ψ1 vanishes, +contradicting the assumptions of Lemma 5.7. +In short, case 3 cannot occur and in both other cases, a suitable perturbation W of +the potential can be constructed by following the constructions introduced in the proof +of Lemma 5.7. Lemma 6.3 is proved. +43 + +7 Generic transversality of asymmetric standing pulses +As in the previous section, the proofs of this section present strong similarities with the +ones which have been already detailed and the presentation will only emphasize the main +differences. +7.1 Notation and statements +The same notation as in the previous section 6 will be used all along the present section 7. +Let us take and fix a positive quantity R. The goal of this section is to prove the following +proposition (the transversality of a standing pulse was defined in Definition 1.4). +Proposition 7.1. There exists a generic subset of Vquad-R such that, for every potential +V in this subset, every asymmetric standing pulse in PV is transverse. +Let V0 denote a potential function in Vquad-R, and let e0 denote a non-degenerate critical +point of V0. As already stated in subsection 6.1, there exists a small neighbourhood +νrobust(V0, e0) of V0 in Vquad-R and a Ck+1-function e(·) defined on νrobust(V0, e0) and +with values in Rd, such that e(V0) equals e0 and, for every V in νrobust(V0, e0), e(V ) is a +critical point of V0 close to e0. +Exactly the same arguments as in subsection 5.2 show that Proposition 7.1 is a +consequence of the following local statement. +Proposition 7.2. There exists a neighbourhood νV0, e0 of V0 in Vquad-R, included in +νrobust(V0, e0), and a generic subset νV0, e0, gen of νV0, e0 such that, for every V in νV0, e0, gen, +every asymmetric standing front connecting e(V ) to itself is transverse. +The remaining part of section 7 will thus be devoted to the proof of Proposition 7.2. Let +us consider the same setting as in subsection 6.1 for local stable and unstable manifolds +of E(V ), for V in a small enough neighbourhood ν of V0. In particular, let us assume +that local stable and unstable manifolds are small enough so that equalities (6.2) hold. In +addition, according to the expression (2.4) of the eigenvectors of the linear system (2.2), +Eu +V0(E0) ∩ +�{0Rd × Rd} +� = {0R2d} +and +Es +V0(E0) ∩ +�{0Rd × Rd} +� = {0R2d} . +It follows that there exists a positive quantity rexit such that, for every U in W u +V0(E0) +differing from E0, +sup +ξ∈R +��πpos +�SV0(ξ, U) +� − e0 +�� > rexit ; +in other words, if a solution ξ �→ U(ξ) = +�u(ξ), ˙u(ξ) +� (for the potential V0) is homoclinic +to E0 then u(ξ) must leave the ball BRd(e0, rexit) before eventually returning into it. Up +to replacing ν by a smaller neighbourhood of V0 in Vquad-R and rexit by a smaller positive +quantity, we may assume that, for every V in ν and for every U in W u +V +�E(V ) +� differing +from E(V ), +(7.1) +sup +ξ∈R +��πpos +�SV (ξ, U) +� − e(V ) +�� > rexit . +44 + +Finally, up to replacing ν by a smaller neighbourhood of V0 in Vquad-R and r by a smaller +positive quantity, we may assume that, for every V in ν, +(7.2) +πpos +� +W u +loc, V +�E(V ) +� ∪ W s +loc, V +�E(V ) +�� +⊂ BRd +�e(V ), rexit/4 +� . +7.2 Asymmetric standing pulses of bounded length and away from Ssym +By comparison with symmetric standing pulses considered in section 6, dealing with +asymmetric standing pulses is less straightforward for the following reasons. +1. Symmetric and asymmetric standing pulses connecting a given critical point to itself +may coexist for some potentials, and while symmetric standing pulses will be proved +to be generically elementary (Definition 1.6), only asymmetric standing pulses will +be proved to be generically transverse, see subsection 7.5). As a consequence, +applying Theorem 4.2 to prove the generic transversality of asymmetric standing +pulses requires to exclude, by a way or another, symmetric ones. +2. The transversality of a standing pulse stated in Definition 1.4 is a transversality +inside the submanifold corresponding to the level set of the Hamiltonian for the +energy −V (e). This submanifold depends on V and a direct application of Theo- +rem 4.2 is not possible because its transversality is stated inside a fixed manifold +N. A simple solution to skip this dependence is to fix V close to e0, but with +the consequence that the considered set of potentials V will not be open, so that +applying Theorem 4.2 in this framework will provide local density but not local +genericity of the potentials for which asymmetric pulses are transverse. Local +genericity will actually be obtained through a countable intersection of open and +dense sets, with separate proofs for their openness and their density. +For every V in ν and for every non negative quantity ¯ξ, let us consider the set +(7.3) +W u +V +�E(V ), ¯ξ +� = SV +�¯ξ, W u +loc, V +�E(V ) +�� += +� +U∈W u +loc, V +� +E(V ) +� SV (¯ξ, U) += {E(V )} ∪ +� +bu∈Bu, ξ∈(−∞,¯ξ] +SV +�ξ, ˆwu +loc, V (bu) +� . +According to this notation, the set W u +V +�E(V ), 0 +� reduces to W u +loc, V +�E(V ) +� and the set +W u +V +�E(V ), ¯ξ +� increases (for inclusion) with ¯ξ and represents (in some sense) the unstable +manifold of the equilibrium E(V ) “until time ¯ξ”. For all positive quantities ¯ξ and ε, let +us consider the set +ν⋔ asym stand pulses(¯ξ, ε) = +� +V ∈ ν : if U0 ∈ W u +V +�E(V ), ¯ξ +� ∩ ∂W s +loc, V +�E(V ) +� and if +dist +� +SV (R, U0) \ +� +W u +loc, V +�E(V ) +� ∪ W s +loc, V +�E(V ) +�� +, Ssym +� +≥ ε , then the +(7.4) +corresponding standing pulse: R → Rd, ξ �→ πpos +�SV (ξ, U0) +� is transverse +� +. +45 + +In other words, a potential function V belonging to ν is in ν⋔ asym stand pulses(¯ξ, ε) if every +standing pulse connecting W u +loc, V +�E(V ) +� to W s +loc, V +�E(V ) +� in a time not larger than ¯ξ +while remaining at a distance not smaller than ε from Ssym, is transverse. Observe +that, according to equalities (6.2), such a standing pulse is necessarily asymmetric. +Proposition 7.2 follows from the next proposition. +Proposition 7.3. For all positive quantities ¯ξ and ε, the set ν⋔ asym stand pulses(¯ξ, ε) is +open and dense in ν. +Proof that Proposition 7.3 yields Proposition 7.2. It follows from Proposition 7.3 that +the set +(7.5) +� +N∈N +ν⋔ asym stand pulses(N, 1/N) +is a generic subset of ν. And, according to the definition of ν⋔ asym stand pulses(·, ·), for +every potential V in this set, every asymmetric standing pulse connecting e(V ) to itself +is transverse. +The remaining of this section is devoted to the proof of Proposition 7.3. +7.3 Openness of ν⋔ asym stand pulses(¯ξ, ε) +For every potential V in ν and for all positive quantities ¯ξ and ε, the manifolds +W u +V +�E(V ), ¯ξ +� and W s +loc, V +�E(V ) +� are compact, and those manifolds depend smoothly on +V . Let (Vn)n∈N denotes a sequence of potentials belonging to ν\ν⋔ asym stand pulses(¯ξ, ε) and +converging to some potential V∞ of ν, and let us prove that, in this case, V∞ is still outside +of ν⋔ asym stand pulses(¯ξ, ε) (this will prove that ν⋔ asym stand pulses(¯ξ, ε) is open in ν). For +every integer n, there exists a non-transverse standing pulse connecting W u +loc, Vn +�E(Vn) +� +to W s +loc, Vn +�E(Vn) +� in a time not larger than ¯ξ while remaining at a distance not smaller +than ε from Ssym. As emphasized in (7.3), this pulse is characterized by a (unique) bu +n in +Bu such that its trajectory in R2d crosses the boundary of W u +loc, Vn +�E(Vn) +� at the point +ˆwu +loc, Vn(bu +n), and a (unique) time ξn in the interval [0, ¯ξ] such that this trajectory crosses +the boundary of W s +loc, Vn +�E(Vn) +� at the point SVn +�ξn, ˆwu +loc, Vn(bu +n) +�. Then, +(i) by compactness (up to considering a subsequence of (Vn)n∈N), we may assume that +(bu +n, ξn) converges to some couple (bu +∞, ξ∞) of Bu × [0, ¯ξ], which in turn characterizes +a standing pulse for V∞. Notice here the importance of considering homoclinic +orbits of bounded “length”, otherwise the limit trajectory would not necessarily be +homoclinic to E(V∞). +(ii) Moreover, both conditions in (7.4) are closed conditions, so that the limit standing +pulse also satisfies them. +(iii) Thanks to the “margin” ε with respect to the symmetry subspace Ssym, the limit +standing pulse is necessarily asymmetric. +46 + +(iv) Last, the limit standing pulse is non-transverse since this property is closed. +The limit potential V∞ is thus not in ν⋔ asym stand pulses(¯ξ, ε), and this completes the proof +that ν⋔ asym stand pulses(¯ξ, ε) is open in ν. +7.4 Density of ν⋔ asym stand pulses(¯ξ, ε) +7.4.1 Application of Theorem 4.2 +The proof of the density assertion of Proposition 7.3 will again follow from applying +Theorem 4.2 to the following appropriate setting. +Take positive quantities ¯ξ and ε, and a potential V1 in ν. Our goal is to prove that +there exist potentials in ν⋔ asym stand pulses(¯ξ, ε) which are arbitrarily close to V1. Let +(7.6) +M = +� +(bu, ξ) ∈ Bu × (0, ¯ξ + 1) : dist +� +SV1 +�[0, ξ], ˆwu +loc, V1(bu) +�, Ssym +� +> ε/2 +and +πpos +� +SV1 +�ξ, ˆwu +loc, V1(bu) +�� +∈ BRd +�e(V1), rexit/2 +�� +. +and let Λ1 denote a neighbourhood of V1 in the set +(7.7) +� +V ∈ ν : V ≡ V1 on the closed ball BRd +�e(V1), rexit +�� +. +We may assume that this neighbourhood Λ1 is small enough so that, for every V in Λ1 +and (bu, ξ) in Bu × (0, ¯ξ + 1), the following two conclusions hold: +1. if (bu, ξ) is not in M, then +(7.8) +either +dist +� +SV +�[0, ξ], ˆwu +loc, V (bu) +�, Ssym +� +< ε +or +πpos +� +SV +�ξ, ˆwu +loc, V (bu) +�� +̸∈ BRd +�e(V1), rexit/4 +� ; +2. if (bu, ξ) is in M, then +(7.9) +dist +� +SV +�[0, ξ], ˆwu +loc, V (bu) +�, Ssym +� +> 0 , +and +πpos +� +SV +�ξ, ˆwu +loc, V (bu) +�� +∈ BRd +�e(V1), rexit +� . +For V in Λ1, let +(7.10) +N = H−1 +V +� +HV +�E(V ) +�� +∩ +�BRd +�e(V ), rexit +� × Rd� \ {E(V )} +and +W = ∂W s +loc, V +�E(V ) +� = ˆws +loc, V (Bs) . +Observe that M, N, and W are submanifolds of R2d and since Λ1 is included in ν, it +follows from inclusion (7.2) that W is included in N. In addition, according to the +condition (7.7) on V and to the inclusion (7.2), M, N and W do actually not depend +on the potential V in Λ1. As already explained in the second remark of the beginning +47 + +of subsection 7.2, this is mandatory to provide a setting where Theorem 4.2 applies. It +follows that, according to (7.9), we may consider the function +(7.11) +Φ : M × Λ1 → N , +(bu, ξ, V ) �→ SV +�ξ, ˆwu +loc, V (bu) +� , +which is well defined. Notice that, even if M contains only couples (bu, ξ) for which, +for V in Λ1, the position u(ξ) = πpos +� +SV +�ξ, ˆwu +loc, V (bu) +� of the corresponding solution +is inside BRd +�e(V1), rexit +� (second condition of (7.9)), it follows from the property (7.1) +defining rexit that this position u(·) exits BRd +�e(V1), rexit/2 +� at other times, and this will +provide a suitable place to perturb the potential. In other words, it will be possible to +modify Φ(bu, ξ, V ) by perturbing V outside of BRd +�e(V1), rexit +�, even if the arrival set of +Φ and its image are restricted to this ball. +Proposition 7.4. For every potential function V in Λ1, if the image of the function M → +N, V �→ Φ(m, V ) is transverse to W, then V belongs to the set ν⋔ asym stand pulses(¯ξ, ε). +Proof. Let us consider V in Λ1 and U0 in W u +V +�E(V ), ¯ξ +� ∩ ∂W s +loc, V +�E(V ) +� satisfying +inequality (7.4). According to the definition (7.3) of W u +V +�E(V ), ¯ξ +�, the point U0 is of the +form (u, ˙u)(ξ) with u a standing pulse such that (u, ˙u)(0) = ˆwu +loc, V (bu) and ξ in [0, ¯ξ]. +According to the inclusion (7.2) satisfied by the local manifolds and the definition of +ν⋔ asym stand pulses(¯ξ, ε), the implication (7.8) shows that (bu, ξ) belongs to M. Thus, the +image Φ +�(bu, ξ), V +� is well defined, and it remains to notice that the transversality of +Φ with W exactly corresponds to the definition Definition 1.4 of the transversality of a +standing pulse. It thus follows from the definition of the set ν⋔ asym stand pulses(¯ξ, ε) that +V belongs to this set. +The remaining part of the proof follows exactly the same arguments as in sections 5 +and 6, except for the exclusion of “case 3”, which will require a slightly different ad hoc +argument. +7.4.2 Checking hypothesis 1 of Theorem 4.2 +By contrast with the previous sections, the ambient space N is now a level set of dimension +2d − 1 (instead of R2d); however the computation is similar. Indeed, it follows from +subsection 2.1 that, on the one hand, dim(M) = dim +�∂Bu +E0(r) +� + 1 = d − m(e0) and, +on the other hand, dim(W) = d − m(e0) − 1 so that codim(W) = d + m(e0). Thus +hypothesis 1 of Theorem 4.2 is fulfilled. +7.4.3 Checking hypothesis 2 of Theorem 4.2 +Take (m2, V2) in the set Φ−1(W). Let (bu +2, ξ2) denote the components of m2, and, for +every real quantity ξ, let us write +U2(ξ) = +�u2(ξ), v2(ξ) +� = SV2 +�ξ, ˆwu +loc, V2(bu +2) +� . +The function ξ �→ u2(ξ) is the profile of a standing pulse, connecting e(V2) to itself, for +the potential V2, and, according to (6.2) and (7.9), this standing pulse is asymmetric. In +48 + +addition, according to (7.1) and (7.2), the quantity ξ2 is positive. Let DΦ denote the +full differential (with respect to m and V ) of Φ at the point (m2, V2 +�. Hypothesis 2 of +Theorem 4.2 follows from the next Lemma 7.5. +Lemma 7.5 (perturbation of the potential reaching a given direction). For every nonzero +vector (φ0, ψ0) ∈ Rd × Rd belonging to TU2(ξ2)N, there exists W in Ck+1 +b +(Rd, R) such that +(7.12) +⟨DΦ · (0, W) | (φ0, ψ0)⟩ ̸= 0 , +and such that W satisfies the condition +(7.13) +supp(W) ∩ BRd +�e(V2), rexit +� = ∅ . +Proof. The proof is similar to those of Lemmas 5.7 and 6.3. Let (φ2, ψ2) be a nonzero +vector in Rd × Rd belonging to TU2(ξ2)N. Let W be a function in Ck+1 +b +(Rd, R), and let us +assume that condition (7.13) holds. Let us again use the notation T(ξ, ξ′) to denote the +family of evolution operators obtained by integrating the linearized differential system +(5.19) (for the potential function V2, and for a speed equal to 0) between the times ξ and +ξ′. For every time ξ, let T ∗(ξ, ξ2) denote the adjoint operator of T(ξ, ξ2), and let +�φ(ξ), ψ(ξ) +� = T ∗(ξ, ξ2) · (φ2, ψ2) . +Using the same computations as in Lemmas 5.7 and 6.3, it follows from the inclusion +(7.2) and the empty intersection (7.13) that inequality (7.12) reads +(7.14) +� ξ2 +0 +∇W +�u2(ξ) +� · ψ(ξ) dξ ̸= 0 . +Observe that, according to inequality (7.1), there exists a (nonempty) open interval I +included in (0, ξ2) and such that, for every ξ in I, u2(ξ) ̸∈ BRd +�e(V2), rexit +�. According +to Lemma 3.2, the function ξ �→ ˙u2(ξ) does not vanish on R, thus a fortiori neither on I. +As in subsections 5.3.5 and 6.2.3, three cases must be considered for the construction of +the perturbation W. +Case 1. +There exists a time ξ† in I such that ψ(ξ†) is not collinear to ˙u2(ξ†). +The same construction as in the first case of the proof of Lemma 5.7 (or as in the first +case of the proof of Lemma 6.3) can then be carried out. +Case 2. +For every ξ in I, ψ(ξ) = α(ξ) ˙u2(ξ) with α(·) not constant. +Again, the same construction as in the second case of the proof of Lemma 5.7 (or as in +the first case of the proof of Lemma 6.3) can then be carried out. +Case 3. +For every ξ in (ξ−, ξturn), ψ(ξ) = α ˙u2(ξ) for some real (constant) quantity α. +As in subsections 5.3.5 and 6.2.3, this third case has to be precluded by a specific +argument. It follows from the adjoint linearized system (5.23) satisfied by φ and ψ (with +c0 equal to zero) that, for every ξ in I, +(7.15) +φ(ξ) = − ˙ψ(ξ) = −α¨u2(ξ) = −α∇V2(u2(ξ)) . +49 + +Besides, since (φ2, ψ2) was assumed to belong to TU2(ξ2)N, it follows that +�φ(ξ), ψ(ξ) +� +belongs to TU2(ξ)H−1 +V2 +� +HV2 +�E(V2) +�� +for all ξ in R (the level set of the energy is invariant +by the flow). The orthogonal space of the tangent space to the level set N is a line +spanned by the gradient of the Hamiltonian ∇HV2(U2) = (−∇V2(u2(ξ)), ˙u2(ξ)). Thus, +the condition (φ2, ψ2) ∈ TU2(ξ2)N reads +(φ2, ψ2) ⊥ (−∇V2(u2(ξ)), ˙u2(ξ)) , +that is +α +�∇V2(u2)2 + ˙u2 +2 +� = 0 . +This implies α = 0 and thus (φ, ψ) ≡ (0, 0), a contradiction with the assumptions of +Lemma 7.5. +In summary, the third case cannot occur and, in both other cases, the same constructions +as in the proofs of Lemmas 5.7 and 6.3 can be carried out, leading to a perturbation +W satisfying the empty intersection (7.13) and inequality (7.14) (and therefore also +inequality (7.12)). +7.4.4 Conclusion +Proof of Proposition 7.3. To complete the proof of Proposition 7.3 amounts to prove +that the set ν⋔ asym stand pulses(¯ξ, ε) is dense in ν. It follows from Lemma 7.5 that both +hypotheses 1 and 2 of Theorem 4.2 are fulfilled for the function Φ defined in (7.11). The +conclusion of this theorem ensures that there exists a generic subset Λgen of Λ1 such that, +for every V in Λgen, the function Φ(·, V ) is transverse to W. According to Proposition 7.4, +the set ν⋔ asym stand pulses(¯ξ, ε) is a superset of Λgen; in particular, there exists potentials +in ν⋔ asym stand pulses(¯ξ, ε) that are arbitrarily close to V1. Since V1 was any potential in ν, +this proves the intended density. Proposition 7.3 is proved. +As shown at the end of subsection 7.2, Proposition 7.3 implies Proposition 7.2, which +in turn implies Proposition 7.1. +7.5 Transversality of symmetric standing pulses? +As it stands, the proof of the generic transversality of asymmetric standing pulses provided +above does not directly apply to symmetric ones. Indeed, for a symmetric standing pulse +ξ �→ u(ξ), with (say) turning time 0, the condition corresponding to (5.22) or (7.14) reads +� ¯ξ +−¯ξ +∇W +�u(ξ) +� · ψ(ξ) dξ ̸= 0 or equivalently +� 0 +−¯ξ +∇W +�u(ξ) +� · +�ψ(ξ) + ψ(−ξ) +� dξ ̸= 0 , +where ¯ξ is a large enough positive quantity. This condition cannot be fulfilled if the +function ξ �→ ψ(ξ) is odd and, due to the symmetry of the adjoint linear equation +¨ψ(ξ) = D2V +�u(ξ) +� · ψ(ξ) , +this happens as soon as ψ(0) vanishes. This case, corresponding to the degeneracy of the +first order derivative with respect to perturbations of the potential, can therefore not be +50 + +excluded. Possibly, the second order derivative could be investigated but the computation +goes beyond the scope of this paper. For that reason, the generic transversality of +symmetric standing pulses is not established here and remains, to our best knowledge, +an open question. +8 Generic non-existence of standing fronts +Let us take and fix a positive quantity R. Due to the Hamiltonian invariance, precluding +the existence of standing fronts is a simple task. +Proposition 8.1. There exists a dense open subset of Vquad-R such that, for every +potential V in this subset, there is no standing front for this potential. +Proof. Let us consider the dense open subset Vquad-R-Morse of Vquad-R containing the +functions of Vquad-R satisfying the Morse property (this notation was introduced in (5.2)), +and let V denote a potential in Vquad-R-Morse. The number of critical points of such a +potential is finite, and, up to applying to V an arbitrarily small localized perturbation +around each of these critical points, it may be assumed that each of these critical points +belongs to a level set of V containing no other critical point. This property is open and +dense in Vquad-R-Morse, thus in Vquad-R, and, since the Hamiltonian HV defined in (1.4) +is constant along the profile of a standing front, it prevents the existence of a standing +front. Proposition 8.1 is proved. +9 Proof of the main results +Propositions 5.1, 6.1, 7.1 and 8.1 show the genericity of the properties considered in +Theorem 1.7, but only inside the space Vquad-R of the potentials that are quadratic past +some radius R. Working in this last space is easier because it is a second countable +Banach space and the flows associated to its potentials are global. +In this section, +the arguments will be adapted to obtain the genericity of the same properties in the +space Vfull = Ck+1(Rd, R) of all potentials, endowed with the extended topology (see +subsection 1.4). +9.1 Proof of conclusion 1 of Theorem 1.7 +Let us recall the notation FV introduced in (5.1), and, for every positive quantity R, let +us consider the set +(9.1) +FV,R = +� +(c, u) ∈ FV : sup +ξ∈R +|u(ξ)| ≤ R +� +of the travelling fronts of FV (invading a minimum point of V ) with a profile contained +in BRd(0, R). As shown thereafter, the following proposition yields conclusion 1 of +Theorem 1.7. +51 + +Proposition 9.1. For every positive quantity R, there exists a generic subset Vfull-⋔-F-R +of Vfull such that, for every potential function V in this subset, V is a Morse function +and every travelling front (c, u) in FV,R is transverse. +Proof that Proposition 9.1 yields conclusion 1 of Theorem 1.7. The set +� +R∈N∗ +Vfull-⋔-F-R , +is a countable intersection of generic subsets of Vfull and is therefore again a generic +subset of Vfull. For every potential function V in this set, V is a Morse function and every +travelling front in FV belongs to FV,R as soon as R is large enough, and is therefore, +according to the property of the set Vfull-⋔-F-R stated in Proposition 9.1, transverse. +Statement 1 of Theorem 1.7 is proved. +The aim of subsection 9.1 is thus to prove Proposition 9.1. Before doing so, here are +a few preliminary comments. Let R be a positive quantity. Proposition 5.1 states that +there exists a generic subset Vquad-R-⋔-F of Vquad-R such that, for every potential Vquad in +this subset, all travelling fronts in FVquad are transverse. However, due to the constraint +at |u| = R, the extension to Rd of all the truncations of these potentials in BRd(0, R) is +meagre. The idea is to take some margin: consider the generic subset Vquad-(R + 1)-⋔-F of +Vquad-(R+1) and, using the notation introduced in definition (4.6), consider the set +(9.2) +res−1 +R,∞ ◦ resR,(R+1)(Vquad-(R + 1)-⋔-F) . +For every potential Vfull in this set, all travelling fronts in FVfull,R are transverse; indeed, +this property depends only on the values of Vfull inside the ball BRd(0, R), where Vfull +must be identically equal to some potential Vquad of Vquad-(R + 1)-⋔-F. It is tempting to +look for an extension of Corollary 4.6 to generic subsets, which would yield the genericity +of the set (9.2). Unfortunately, this corollary definitely applies to open dense subsets, and +not to generic ones. Pursuing further in this direction, observe that, since Vquad-(R + 1)-⋔-F +is a generic subset of Vquad-(R+1), there exists a countable family (ON)N∈N of dense open +subsets of Vquad-(R+1) such that +(9.3) +� +N∈N +ON ⊂ Vquad-(R + 1)-⋔-F , +leading to +res−1 +R,∞ ◦ resR,(R+1) +� � +N∈N +ON +� +⊂ res−1 +R,∞ ◦ resR,(R+1)(Vquad-R + 1-⋔-F) . +According to general properties of functions, the following inclusion holds: +(9.4) +resR,(R+1) +� � +N∈N +ON +� +⊂ +� +N∈N +resR,(R+1)(ON) . +52 + +If this inclusion was an equality, then, still according to general properties of functions, +the following equality would hold: +res−1 +R,∞ ◦ resR,(R+1) +� � +N∈N +ON +� += +� +N∈N +res−1 +R,∞ ◦ resR,(R+1)(ON) , +and, since according to Corollary 4.6 the right-hand side of this equality is a countable +intersection of dense open subsets of Vfull, the intended conclusion that the set (9.2) is +generic in Vfull would follow. Unfortunately, Proposition 5.1 provides no clue about the +sets ON and a strict inclusion in (9.4) cannot be precluded. However, let us make the +following key observation, which enlightens the remaining of the proof: if the property +“a given potential V belongs to ON” only depends on the values of V inside the ball +BRd(0, R), then inclusion (9.4) is actually an equality. +The main step in the proof is thus to construct dense subsets ON of Vquad-(R+1) such +that: +1. for every potential Vquad in � +n ON, every travelling front in FV,R is transverse, +2. and the property “a given potential V belongs to ON” only depends on the values +of V inside the ball BRd(0, R). +Proof of Proposition 9.1. As above, let R denote a positive quantity. Let V0 denote a +potential function in Vquad-(R+1), let e−,0 and e+,0 denote a non-degenerate critical point +and a non-degenerate minimum point of V0 and let c0 denote a positive speed. Let us +consider the neighbourhoods νV0, e−,0, e+,0, c0 of V0 in Vquad-(R+1) and CV0, e−,0, e+,0, c0 of c0 in +(0, +∞) provided by Proposition 5.2 for these objects. Recall that those neighbourhoods +are the ones from which, for every V in νV0, e−,0, e+,0, c0 and every c in CV0, e−,0, e+,0, c0, the +functions ˆwu +loc, c, V , the sets M and W and the functions Φu and Φs and Φ were defined +in subsection 5.3. Up to replacing the neighbourhood νV0, e−,0, e+,0, c0 by its interior, we +may assume that it is open in Vquad-(R+1). Similarly, we may assume that CV0, e−,0, e+,0, c0 +is compact in R. Let N denote a non negative integer and let us consider the set +(9.5) +MN = Bu × Bs × (−∞, N] × CV0, e−,0, e+,0, c0 = +�(bu, bs, ξ, c) ∈ M : ξ ≤ N +� . +As in subsection 5.3, let us define N as (R2d)2, and let us consider the set +(9.6) +OV0, e−,0, e+,0, c0, N = +� +V ∈ νV0, e−,0, e+,0, c0 : Φ +�MN, V +� is transverse to W in N +� +. +As shown in Proposition 5.5, this set OV0, e−,0, e+,0, c0, N is made of the potential functions +V in νV0, e−,0, e+,0, c0 such that every profile ξ �→ u(ξ) of a front travelling at a speed +c in CV0, e−,0, e+,0, c0 and connecting e−(V ) to e+(V ) for this potential, and connecting +∂W u +loc, c, V +�E−(V ) +� to ∂W s +loc, c, V +�E+(V ) +� in a time not larger than N, is transverse. +Lemma 9.2. The set OV0, e−,0, e+,0, c0, N is a dense open subset of νV0, e−,0, e+,0, c0. +Proof of Lemma 9.2. The density is a direct consequence of Proposition 5.2 which states +that, generically with respect to V in νV0, e−,0, e+,0, c0, the whole image of M by the map +53 + +m �→ Φ(m, V ) is transverse to W. To prove the openness, let us argue as in subsection 7.3. +Let us consider a sequence (Vn)n∈N of potentials in νV0, e−,0, e+,0, c0 converging to a potential +V∞ in νV0, e−,0, e+,0, c0, and such that, for every n in N, there exists mn = (bu +n, bs +n, ξn, cn) +in MN such that the set Φ(MN, Vn) is not transverse to W at Φ(mn, Vn). Observe that, +according to the empty intersection (5.5), ξn must be positive. As a consequence, by +compactness of Bu × Bs × [0, N] × CV0, e−,0, e+,0, c0, we may assume that mn converges, as +n goes to +∞, to a point m∞ of MN. Then, by continuity, the image Φ(MN, V∞) is +not transverse to W at Φ(m∞, V∞). This proves that νV0, e−,0, e+,0, c0 \ OV0, e−,0, e+,0, c0, N +is closed in νV0, e−,0, e+,0, c0, and yields the intended conclusion. +Continuation of the proof of Proposition 9.1. Let us make the additional assumption +that the potential V0 is a Morse function. Then, the set of critical points of V0 is finite +and depends smoothly on V in a neighbourhood νrobust(V0) of V0. Intersecting the +sets νV0, e−,0, e+,0, c0 and CV0, e−,0, e+,0, c0 and OV0, e−,0, e+,0, c0, N above over all the possible +couples (e−,0, e+,0) in Σcrit(V0) × Σmin(V0) provides an open neighbourhood νV0, c0 of V0, +a compact neighbourhood CV0, c0 of c0 and an open dense subset OV0, c0, N of νV0, c0 such +that, for all V ∈ OV0, c0, N, every front travelling at speed c ∈ CV0, c0 and connecting the +local (un)stable manifolds of two points (e−, e+) in Σcrit(V ) × Σmin(V ) within the “time” +N, is transverse. +Denoting by int(A) the interior of a set A and using the notation of definition (4.6), +let us introduce the sets +˜νV0, c0 = res−1 +R,∞ ◦ resR,(R+1)(νV0, c0) , +(9.7) +and +˜OV0, c0, N = res−1 +R,∞ ◦ resR,(R+1) +�OV0, c0, N +� , +(9.8) +and +˜Oext +V0, c0, N = ˜OV0, c0, N ⊔ int +�Vfull \ ˜νV0, c0 +� . +(9.9) +In other words, a potential ˜V of Vfull is in ˜νV0, c0 (in ˜OV0, c0, N) if it coincides, inside the +ball BRd(0, R), with a potential Vquad quadratic past R + 1 and belonging to νV0, c0 (to +OV0, c0, N). The last set ˜Oext +V0, c0, N is an extension of the open dense subset ˜OV0, c0, N of +˜νV0, c0, obtained by adding all potentials outside (the closure of) ˜νV0, c0. +Lemma 9.3. The set ˜Oext +V0, c0, N is a dense open subset of Vfull. +Proof of Lemma 9.3. According to Corollary 4.6, the set ˜νV0, c0 is an open subset of Vfull, +and the set ˜OV0, c0, N is a dense open subset of ˜νV0, c0. Thus, according to its definition +(9.9), the set ˜Oext +V0, c0, N is a dense open subset of Vfull. +Continuation of the proof of Proposition 9.1. Since Vquad-(R+1) is a separable space, it is +second-countable. Thus Vquad-(R+1)-Morse × (0, +∞) is also second-countable and can be +covered by a countable number of products νV0, c0 × CV0, c0. With symbols, there exists a +countable family (V0,i, c0,i)i∈N of elements of Vquad-(R+1)-Morse × (0, +∞) so that +(9.10) +Vquad-(R+1)-Morse × (0, +∞) = +� +i∈N +νV0,i, c0,i × CV0,i, c0,i . +54 + +Notice here the importance of first working with Vquad-(R+1), which is second-countable, +instead of the full space Vfull, which is not. Let us consider the set +(9.11) +Vfull-⋔-F-R = Vfull-Morse ∩ +� +� +� +(i,N)∈N2 +˜Oext +V0,i, c0,i, N +� +� , +where Vfull-Morse is the set of potentials in Vfull which are Morse functions. +Lemma 9.4. For every potential ˜V in the set Vfull-⋔-F-R, every travelling front (u, c) in +F ˜V ,R is transverse. +Proof of Lemma 9.4. Let ˜V be a potential function in the set Vfull-⋔-F-R and (c, u) be +a travelling front in F ˜V ,R. +According to Lemma 4.5, the map resR,(R+1) is surjec- +tive, thus there exists a potential function V in Vquad-(R+1) such that V belongs to +res−1 +R,(R+1) ◦ resR,∞( ˜V ) (in other words V coincides with ˜V on BRd(0, R)). Since ˜V is a +Morse function, the critical points of V in BRd(0, R) are degenerate, and up to applying +to V a small perturbation in BRd(0, R + 1) \ BRd(0, R), we may assume that its critical +point in this set are also nondegenerate, so that V is actually also a Morse function. +Since ˜V coincides with V inside BRd(0, R) and since the travelling front u is contained +in this ball, it is also a travelling front of V and it is sufficient to show that (u, c) is a +transverse travelling front for V . +According to equality (9.10), there exists an integer i such that V belongs to νV0,i, c0,i +and c belongs to CV0,i, c0,i. Then, since V and ˜V coincide on BRd(0, R), ˜V belongs to +˜νV0,i, c0,i (definition (9.7)). Besides, it follows from definition (9.11) that, for every integer +N, ˜V belongs to ˜Oext +V0,i, c0,i, N; and since V is also in ˜νV0,i, c0,i, it follows from definition +(9.9) that ˜V actually belongs to ˜OV0,i, c0,i, N. +Let us denote by e− and e+ the critical points of V (and ˜V ) approached by u(ξ) as +ξ goes to −∞ and +∞ respectively. According to the definition of the neighbourhood +νV0,i, c0,i of V0,i, there exists a (unique) critical point e−,0,i and a (unique) minimum +point e+,0,i of V0,i such that, if W �→ e−,i(W) and W �→ e+,i(W) denote the functions +which “follow” these critical points for W in νrobust(V0,i), then e− equals e−,i(V ) and e+ +equals e+,i(V ). Let us keep the notation M and Φ to denote the objects defined as in +subsection 5.3 for the neighbourhoods νV0,i, e−,0,i, e+,0,i, c0,i of V0,i and CV0,i, e−,0,i, e+,0,i, c0,i of +c0,i. The travelling front (c, u) therefore corresponds to an intersection between Φ(M, V ) +and W, which occurs at a certain point m of M and thus for a certain (positive) time ξ +which is the time that the profile of this travelling front takes to go from the border of +the local unstable manifold of e− to the border of the local stable manifold of e+. +Let N denote an integer not smaller than ξ. Since ˜V belongs to ˜OV0,i, c0,i, N, there +must exist (according to definition (9.8)) a potential VN in νV0,i, c0,i identically equal to ˜V +(and V ) on the ball BRd(0, R) and belonging to OV0,i, c0,i, N. Again, (c, u) is a travelling +front for VN and the previous correspondence between this front and an intersection +between Φ(M, VN) and W still holds. Since VN belongs to OV0,i, c0,i, N, the aforementioned +intersection must be transverse, leading to the transversality of the front (u, c) for VN. +55 + +Again, the three potentials ˜V and V and VN considered here have the same values +along the profile of the travelling front (u, c). Thus, this front is also transverse for ˜V . +End of the proof of Proposition 9.1. The set Vfull-⋔-F-R defined in (9.11) is a countable +intersection of dense open subsets of Vfull, and is therefore a generic subset of Vfull. In +view of Lemma 9.4, Proposition 9.1 is proved. +9.2 Proof of conclusions 2 and 3 of Theorem 1.7 +The proof of conclusions 2 and 3 of Theorem 1.7 is similar to the proof of conclusion +1 provided in the previous subsection. As a consequence, only the core arguments will +be reproduced here. Let us recall the notation PV introduced in (6.1), and, for every +positive quantity R, let us consider the set +PV,R = +� +u ∈ PV : sup +ξ∈R +|u(ξ)| ≤ R +� +. +As shown in the previous subsection for Proposition 9.1 and conclusion 1 of Theorem 1.7, +the following proposition yields conclusions 2 and 3 of of Theorem 1.7. +Proposition 9.5. For every positive quantity R, there exists a generic subset Vfull-⋔-P-R +of Vfull, included in Vfull-Morse, such that, for every potential function V in Vfull-⋔-P-R, +every standing pulse u in PV,R is: elementary if this standing pulse is symmetric, and +transverse if this standing pulse is asymmetric. +Proof. Let R denote a positive quantity and let V0 denote a Morse potential function in +Vquad-(R+1). Let e0 denote a non-degenerate critical point of V0 and let us consider an +open neighbourhood νV0, e0 of V0 in Vquad-(R+1) included in both neighbourhoods provided +by Propositions 6.2 and 7.2. For every N in N∗ and for every V in νV0, e0, let us consider +the subset OV0, e0, N of νV0, e0 defined as the set of potentials V in νV0, e0 satisfying the +following two conditions: +1. every symmetric standing pulse of V , connecting ∂W u +loc, V +�E(V ) +� to the symmetric +subspace Ssym in a time not larger than N, is elementary; +2. and every asymmetric standing pulse of V , connecting ∂W u +loc, V +�E(V ) +� to +∂W s +loc, V +�E(V ) +� in a time not larger than N while remaining at a distance not +smaller than 1/N of Ssym, is transverse. +The same arguments as in the proof of Lemma 9.2 show that the set OV0, e0, N is a dense +open subset of νV0, e0: the density follows from Propositions 6.2 and 7.2 and, regarding the +openness, the key new ingredient is the condition that every asymmetric standing pulse +remains at a distance at least 1/N of Ssym. Indeed, a sequence of asymmetric standing +pulses (as considered in the proof) may (generally speaking) approach a symmetric +standing pulse which may be non-transverse even if it is elementary. Staying away from +Ssym precludes this possibility. +56 + +As on page 54, let us consider the intersections of the previous sets over all the critical +points of V0: +νV0 = +� +e0∈Σcrit(V0) +νV0, e0 +and +OV0, N = +� +e0∈Σcrit(V0) +OV0, e0, N . +The set νV0 is still open in Vquad-(R+1) and the set OV0, N is still a dense open subset of +νV0. As in definitions (9.7) to (9.9), these sets can be extended as follows: +˜νV0 = res−1 +R,∞ ◦ resR,(R+1)(νV0) , +˜OV0, N = res−1 +R,∞ ◦ resR,(R+1) +�OV0, N +� , +and +˜Oext +V0, N = ˜OV0, N ⊔ int +�Vquad-(R+1) \ ˜νV0 +� . +The end of the proof follows the same arguments as the ones of subsection 9.1. The set +Vquad-(R+1)-Morse can be covered by a countable number of subsets ˜νV0,i and the set +Vfull-⋔-P-R = Vfull-Morse ∩ +� +� +� +(i,N)∈N2 +˜Oext +V0,i, N +� +� +is the generic subset the existence of which was stated in Proposition 9.5. +9.3 Proof of conclusion 4 of Theorem 1.7 +Let us consider the set OR of potentials V of Vfull such that all the critical points of +V in BRd(0, R) are non-degenerate and have different values. The same arguments as +in Proposition 8.1 show that this set OR is an open dense subset of Vfull, so that the +intersection ∩R∈N∗OR is generic in Vfull. Since the critical points connected by a standing +front must belong to the same level set of the potential, no standing front can exist for a +potential in this intersection. +9.4 Proof of conclusions 1 to 4 of Corollary 1.1 +Let V be a potential function belonging to the generic subset provided by Theorem 1.7, +let (c, u) be a travelling front in FV , and let e− and e+ denote the critical point and the +minimum point of V connected by this travelling front. According to Table 2.1, +dim +� +� � +c′>0 +{c′} × W u +c′,V (E−) +� +� = d − m(e−) + 1 , +and +dim +� +� � +c′>0 +{c′} × W s +c′,V (E+) +� +� = d + 1 . +The intersection between these two manifolds contains at least the curve {c} × U(R) +corresponding to the travelling front. Thus, the dimension of the sum of the tangent +spaces to these two manifolds is not larger than the quantity +�d − m(e−) + 1 +� + (d + 1) − 1 = 2d + 1 − m(e−) . +57 + +Since according to Definition 1.3 and Theorem 1.7 the intersection between these two +manifolds is transverse in R2d+1, along the set {c} × U(R), this quantity is not smaller +than 2d + 1, so that the Morse index m(e−) must be zero. This proves conclusion 1 of +Corollary 1.1. +Now let us assume that u is the profile of a standing pulse and let e denote the critical +point of V such that this pulse connects e to itself. According to Table 2.1, +dim (W u +V (E)) = d − m(e) +and +dim (W s +V (E)) = d − m(e) . +According to Definition 1.6 and Theorem 1.7, if u is symmetric then the intersection +between W u +V (E) and the d−dimensional manifold Ssym is transverse in R2d, at the point +U(ξturn) and this can happen only if m(e) = 0. If u is asymmetric then the intersection +between W u +V (E) and W s +V (E) is transverse, in H−1 +V +�V (E) +�, along the trajectory U(R). +The intersection of W u +V (E) and W s +V (E) is at least one-dimensional and the dimension of +H−1 +V +�V (E) +� is equal to 2d − 1. Again, the transversality can happen only if m(e) = 0. +This proves conclusion 2 of Corollary 1.1. +In all the cases considered above, the counting of the dimensions and the transversality +imply that the intersections of the stable and unstable manifolds reduce to the smallest +possible set, that is: the one-dimensional curve drawn by the trajectory U for travelling +fronts or asymmetric pulses, and the singleton +�U(ξturn) +� defined by the turning point for +symmetric pulses. By local compactness of the unstable manifolds, this implies that the +trajectories of a given class are isolated from each other (even if a family of asymmetric +standing pulses may accumulate on a non-degenerate — and in this case non-transverse +— symmetric pulse). In particular, there is only a countable number of such trajectories. +Conclusion 3 of Corollary 1.1 is proved. +Finally, conclusion 4 about the robustness of travelling fronts and standing pulses +(the fact that they persist under small perturbations of the potential) follows from their +transversality (that, is, the transversality of the intersections considered above). +10 Generic asymptotic behaviour for the profiles of bistable +travelling fronts and of standing pulses stable at infinity +The goal of this section is to prove Theorem 1.8 (and thus also conclusion 5 of Corol- +lary 1.1). +10.1 Asymptotic behaviour of profiles +Let V0 denote a potential in Vfull, let e0 denote a nondegenerate minimum point of V , +and let c denote a nonnegative quantity (speed). As in subsection 2.1, let (u1, . . . , ud) +denote an orthonormal basis of Rd made of eigenvectors of D2V (e0), and let µ1, . . . , µd +denote the corresponding (positive) eigenvalues, with µ1 ≤ · · · ≤ µd. The statement “the +smallest eigenvalue of D2V (e0) is simple”, in conclusion 1 of Theorem 1.8, just mean +that µ1 is smaller than µ2 (and thus also than all the other eigenvalues of D2V (e0)). Let +58 + +us make this assumption. With the notation of subsection 2.1, it follows that, for every j +in {2, . . . , d}, +λj,− < λ1,− < 0 < λ1,+ < λj,+ ; +in other words, λ1,− and λ1,+ are, among all the eigenvalues of DFc,V (E0) (which are +real), the closest ones to 0 (here E0 = (e0, 0Rd) is the equilibrium point of the flow Sc,V +corresponding to e0). If a solution ξ �→ u(ξ) of the differential system (1.7) goes to e0 +as ξ goes to −∞ (+∞), then one among the following two possible cases occurs (see +Proposition 10.1 below for a more precise statement): +1. there exists a real quantity K such that +u(ξ) − e0 = Keλ1,+ξu1 + oξ→−∞(eλ1,+ξ) +(and u(ξ) − e0 = Keλ1,−ξu1 + oξ→+∞(eλ1,−ξ) , respectively); +2. u(ξ) − e0 = oξ→−∞(eλ1,+ξ) (and u(ξ) − e0 = oξ→+∞(eλ1,−ξ), respectively). +The words “u(ξ) approaches its limit (at ±∞) tangentially to the eigenspace corresponding +to the smallest eigenvalue of D2V at this point”, used in conclusion 5 of Corollary 1.1 +and in conclusion 2 of Theorem 1.8, mean that case 1 above occurs. As illustrated on +Figure 10.1 (see also Figure 10.2), approach of equilibria “at the slowest possible rate” +(case 1 above) is a generic feature among solutions of differential systems. The main goal +of this section is thus to provide a formal proof that this feature is indeed generic (with +respect to the potential V ) for bistable travelling fronts and standing pulses stable at +infinity of the parabolic system (1.1). +Figure 10.1: Attractive node of a two-dimensional vector field. +In the language of +subsection 10.2, the vertical axis is the “strongly stable subspace” of the equilibrium. +10.2 Local strongly stable and unstable manifolds when the speed c is +positive +Let us keep the notation and assumptions of the previous subsection and let us assume +that c is positive. The aim of this subsection is to provide a variant of Proposition 2.2 +59 + +→+ +N +→ ++ ++ +1 ++devoted to the “strongly” local stable and unstable manifolds, which are characterized +by a “fast” convergence (case 2 above). Concerning the references, the same comments +as in subsection 2.2 apply. +Calling upon the notation of subsection 2.1, let +Esu(E0) = span +�{U2,+, . . . , Ud,+} +� +and +Ess(E0) = span +�{U2,−, . . . , Ud,−} +� , +and +Em(E0) = span +�{U1,−, U1,+} +� +(the superscripts “su”, “ss”, and “m” stand for “strongly unstable”, “strongly stable”, +and “mild”, respectively), and +βsu = λ2,+ +and +βss = λ2,− . +As in subsection 2.2, there exist norms ∥·∥su on Esu(E0) and ∥·∥ss on Ess(E0) such that +inequalities (2.7) (with “su” instead of “u” and “ss” instead of “s” everywhere) hold. For +every positive quantity r, let us define the balls Bsu +E0(r) and Bss +E0(r) as in (2.8) (with the +same substitutions “u”←“su” and “s”←“ss”), let Bm +E0(r) denote the closed ball centred +at E0 and with radius r, in the subspace Em(E0), for the usual euclidean norm on these +subspace, and let +BE0(r) = +�Usu + Uss + Um : Usu ∈ Bsu +E0(r) and Uss ∈ Bss +E0(r) and Um ∈ Bm +E0(r) +� . +Let λ3/2,− and λ3/2,+ denote real quantities satisfying +λ2,− < λ3/2,− < λ1,− +and +λ1,+ < λ3/2,+ < λ2,+ . +Proposition 10.1 (local strong stable and unstable manifolds). There exist a neigh- +bourhood ν of V0 in Vfull, a neighbourhood C of c0 in (0, +∞) and a positive quantity r +such that, for every (c, V ) in C × ν, in addition to the conclusions of Proposition 2.2, the +following statements hold. +There exist Ck-functions +wsu +loc, c, V : Bsu +E0(r) → Bm +E0(r) + Bss +E0(r) +and +wss +loc, c, V : Bss +E0(r) → Bm +E0(r) + Bsu +E0(r) +such that, if we consider the sets +W su +loc, c, V +�E(V ) +� = +� +E(V ) + Usu + wsu +loc, c, V (Usu) : Usu ∈ Bsu +E0(r) +� +and +W ss +loc, c, V +�E(V ) +� = +� +E(V ) + Uss + wss +loc, c, V (Uss) : Uss ∈ Bss +E0(r) +� +, +then, for every U in BE0(r) the following two assertions are equivalent: +1. U is in W su +loc, c, V +�E(V ) +�; +2. Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in (−∞, 0] and +|Sc,V (ξ, U) − E(V )| = oξ→−∞(eλ3/2,+ξ) ; +60 + +and for every U in BE0(r) the following two assertions are equivalent: +1. U ∈ W ss +loc, c, V +�E(V ) +�; +2. Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in [0, +∞) and +|Sc,V (ξ, U) − E(V )| = oξ→+∞(eλ3/2,−ξ) . +Both differentials Dwsu +loc, c0, V0(0) and Dwss +loc, c0, V0(0) vanish, and both maps +C × ν × Bsu +E0(r) → Bm +E0(r) + Bss +E0(r), +(c, V, Usu) �→ wsu +loc, c, V (Usu) +and +C × ν × Bss +E0(r) → Bm +E0(r) + Bsu +E0(r), +(c, V, Uss) �→ wss +loc, c, V (Uss) +are of class Ck. +10.3 Idea of the proof of conclusion 2 of Theorem 1.8 +The goal of this subsection is to provide a rough idea of the proof of Theorem 1.8, more +precisely of the main conclusion of this theorem which is conclusion 2 (the proof of +conclusion 1, carried out in the next subsection, is straightforward). +The proof of conclusion 2 is actually almost identical to the proof of Theorem 1.7. +Observe that, by contrast with the proof of Theorem 1.7, only bistable travelling fronts +and standing pulses that are stable at infinity need to be considered. In each case +(bistable travelling fronts, symmetric and asymmetric standing pulses stable at infinity), +the proof relies on applying Sard–Smale Theorem 4.2 to the same settings as in the proof +of Theorem 1.7, both for potentials that are quadratic past a certain radius and for the +extension to general potentials, except for the following change: +1. either the unstable manifold of the left end equilibrium E−(V ) is replaced by its +strongly unstable manifold, +2. or the stable manifold of the right end equilibrium E+(V ) is replaced by its strongly +stable manifold. +More precisely, both replacements have to be (separately) considered both for travelling +fronts and asymmetric standing pulses, while only the first replacement is relevant for +symmetric standing pulses. +Let us see why such change (replacement) in the setting does not affect the validity of the +two assumptions of Theorem 4.2, and how its conclusions can be interpreted. Concerning +assumption 1 of Theorem 4.2, this replacement leads to the following consequences: +1. either the dimension of the manifold denoted by M is decreased by 1 (this is what +happens for travelling fronts, be it with replacement 1 or 2, for symmetric standing +pulses with replacement 1, and for asymmetric standing pulses with replacement 1), +2. or the dimension of the manifold denoted by W is decreased by 1 (this is what +happens for asymmetric standing pulses with replacement 2). +61 + +In each of these cases, the dimension of the arrival manifold N is unchanged, and as +a consequence, the difference dim(M) − codim(W) is exactly decreased by 1. More +precisely, since only bistable travelling fronts and standing pulses stable at infinity are +considered, this difference is actually exactly equal to −1. Assumption 1 of Theorem 4.2 +is therefore still satisfied. +Concerning assumption 2 of Theorem 4.2, it is also fulfilled in each of these cases, due +to the key following observation: in the proof of each of the three lemmas proving that +this assumption holds (Lemmas 5.7, 6.3 and 7.5), the freedom provided by the variables +bu and bs is not used — only the freedom provided by the time variable ξ and by the +potential V are. As a consequence, the fact that the unstable manifold of E−(V ) is +replaced by its strongly unstable manifold does not affect the validity of the conclusion +of the lemma, and neither does the fact that the stable manifold of E+(V ) is replaced by +its strongly stable manifold. In other words, the key assumption 2 of Theorem 4.2 still +holds. +In each case and for each of the two replacements 1 and 2, the conclusions of Theorem 4.2 +thus still hold, and ensure that, locally generically with respect to V , the profiles of +travelling fronts or of (a)symmetric standing pulses locally correspond to transverse +intersections between the image of m �→ Φ(m, V ) and W in N. +But the fact that +dim(M) − codim(W) is now equal to −1 actually precludes the very existence of such +transverse intersections. In other words, locally generically with respect to V , profiles +of bistable travelling fronts or of (a)symmetric pulses stable at infinity approaching +their limit at −∞ through its strongly stable manifold or their limit at +∞ through its +strongly stable manifold do simply (locally) not exist, which is the intended conclusion. +The emptiness of such a transverse intersection due to the value −1 of the difference +dim(M) − codim(W) is illustrated by Figure 10.2. +Figure 10.2: Whereas the sum of the dimensions of W u(E−) and W s(E+) has the minimal +value for a nonempty transverse intersection between these two manifolds to exist, for +W su(E−) and W s(E+) (or for W u(E−) and W ss(E+)) this sum is smaller, so that a +transverse intersection between such manifolds must be empty. This figure actually +depicts the intersection defining a transverse asymmetric bistable standing front, but +the same principle applies for bistable travelling fronts, and elementary symmetric (or +transverse asymmetric) standing pulses that are stable at infinity. +The remaining arguments, ensuring the first extension to global statements for poten- +tials quadratic past a certain radius (subsections 5.2, 6.1 and 7.1), and then the second +extension to general potentials (subsections 9.1 and 9.2), are unchanged. +62 + +Wu(E-) n Ws(E+ +E +Wsu(E_) +-)nM +Ws(E+To complete these arguments, a few milestones of the proof for travelling fronts are +detailed in subsection 10.5 below. +10.4 Proof of conclusion 1 of Theorem 1.8 +Let R denote a positive quantity, let us recall the notation Vquad-R introduced in (4.2) +and Vquad-R-Morse introduced in (5.2), and let us consider the set +Vquad-R-Morse-ss-eig = +�V ∈ Vquad-R-Morse : at every minimum point of V , +the smallest eigenvalue of D2V is simple +� +(the subscript “ss-eig” stands for “simple smallest eigenvalue”). +Proposition 10.2. The set Vquad-R-Morse-ss-eig is a dense open subset of Vquad-R-Morse +(and thus of Vquad-R). +Proof. Openness follows from the continuity of the roots (eigenvalues of D2V at a +minimum point) of a polynomial with respect to its coefficients. To prove the density, +let V be in Vquad-R-Morse, and let us assume that there exists a minimum point e of V +such that the smallest eigenvalue µ1 of D2V (e) is not simple. Let δ denote a positive +quantity, small enough so that the closed ball BRd(e, δ) contains no critical point of V +but e. Let ρ denote a smooth function [0, +∞) → R satisfying +ρ(r) = 1 +for +r in [0, 1/2] +and +ρ(r) = 0 +for +r in [1, +∞) , +and let ε denote a small positive quantity to be chosen below. Let u1 denote an unit +eigenvector of D2V (e) associated to µ1, and let us consider the perturbed potential Vpert +defined as: +Vpert(u) = V (u) − ε +2 +�(u − e) · u1 +�2ρ +�|u − e| /δ +� . +Then, e is still a critical point of Vpert and, for every v in Rd, +D2Vpert(e)(v, v) = D2V (e)(v, v) − ε(v · u1)2 . +As a consequence, u1 is still an eigenvector of D2Vpert(e), the corresponding eigenvalue +µ1 − ε is simple, and the other eigenvalues of D2Vpert(e) are the same as those of D2V (e) +(the difference D2Vpert(e) − D2V (e) vanishes on the orthogonal subspace to u1 in Rd), +these other eigenvalues are therefore larger than µ1 − ε. In addition, if ε is small enough, +then µ1 − ε is positive (so that e is still a minimum point of Vpert) and the closed ball +BRd(e, δ) contains no critical point of Vpert but e. The same procedure, repeated for each +minimum point of V such that the smallest eigenvalue of D2V at this minimum point is +not simple, provides an arbitrarily small perturbation of V belonging to Vquad-R-Morse-ss-eig, +and therefore proves the intended density. +Let VMorse-ss-eig denote the subset of Vfull containing Morse potentials V such that, at +every minimum point point of V , the smallest eigenvalue of the Hessian D2V at this +minimum point is simple. Proceeding as in subsection 9.3, the same arguments as in the +proof of Proposition 10.2 above show that this set VMorse-ss-eig is a generic subset of Vfull, +which proves conclusion 1 of Theorem 1.8. +63 + +10.5 Proof of conclusion 2 of Theorem 1.8 for bistable travelling fronts +The aim of this subsection is to complete the idea of the proof of conclusion 2 of +Theorem 1.8 provided in subsection 10.3 with a few milestones of this proof, in the case +of travelling fronts (only). +As for conclusion 1 of Theorem 1.7, the first goal is to prove the intended conclusion +among potentials that are quadratic past a certain (positive) radius R. This is stated +by the following proposition, which is an extension of Proposition 5.1. It calls upon the +notation FV introduced in (5.1). +Proposition 10.3. There exists a generic subset of Vquad-R, included in Vquad-R-Morse-ss-eig +such that, for every potential V in this subset, every travelling front (c, u) in FV is trans- +verse, bistable, and its profile u approaches its limit at +∞ (−∞) tangentially to the +eigenspace corresponding to the smallest eigenvalue of D2V at this point. +10.5.1 Reduction to a local statement +Let V0 denote a potential function in Vquad-R-Morse-ss-eig, and let e−,0 and e+,0 denote non- +degenerate minimum points of V0. Let us consider the neighbourhood νrobust(V0, e−,0, e+,0) +of V0 introduced in subsection 5.2, and let us denote by ˜νrobust(V0, e−,0, e+,0) the inter- +section νrobust(V0, e−,0, e+,0) ∩ Vquad-R-Morse-ss-eig. The following proposition is a variant +(extension in the case of bistable travelling fronts) of Proposition 5.2. The notation +is similar, except for the “tilde” added to the symbols of the various sets, in order to +differentiate them for the corresponding sets introduced in Proposition 5.2. +Proposition 10.4. For every positive speed c0, there exist a neighbourhood ˜νV0, e−,0, e+,0, c0 +of V0 in Vquad-R, included in ˜νrobust(V0, e−,0, e+,0), a neighbourhood ˜CV0, e−,0, e+,0, c0 of c0 in +(0, +∞), and a generic subset ˜νV0, e−,0, e+,0, c0, gen of ˜νV0, e−,0, e+,0, c0 such that, for every V +in ˜νV0, e−,0, e+,0, c0, gen, every front travelling at a speed c in ˜CV0, e−,0, e+,0, c0 and connecting +e−(V ) to e+(V ), for the potential V , is transverse and its profile u approaches its limit +at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of +D2V at this point. +Proof that Proposition 10.4 yields Proposition 10.3. Proposition 5.1 already ensures the +existence of a generic subset Vquad-R-⋔-F of Vquad-R such that, for every potential function +V in this subset, every travelling front (c, u) in FV is transverse. According to the +arguments of subsection 9.4, such a front is necessarily bistable. Thus, only the conclusion +of Proposition 10.4 relative to the asymptotic behaviour of the profile remains to be +proved. +To this end, the arguments are the same as in subsection 5.2. We may introduce the sets +˜νV0, c0 and CV0, c0 and νV0, c0, gen, defined exactly as the corresponding sets (without tilde) +in (5.3) (with νrobust(V0, e−,0, e+,0) replaced with ˜νrobust(V0, e−,0, e+,0)), and the same +remaining arguments (replacing Vquad-R-Morse with Vquad-R-Morse-ss-eig) show the existence +of a generic subset of Vquad-R, included in Vquad-R-Morse-ss-eig, such that, for every potential +V in this subset, every bistable travelling front (c, u) in FV is transverse and its profile u +approaches its limit at both ends of R according to the intended conclusion. Intersecting +64 + +this generic subset with the one provided by Proposition 5.1 provides a generic subset of +Vquad-R for which all conclusions of Proposition 10.3 hold. +10.5.2 Proof of the local statement +The proof of Proposition 10.4 may be derived from the proof of Proposition 5.2, up to a +few changes and thanks to some key arguments, all of which are exposed in subsection 10.3 +above. +10.5.3 Extension to all potentials +The extension to all potentials is obtained by applying the same strategy as in subsec- +tion 9.1. Let us recall the notation FV,R introduced in (9.1). The same arguments as +in subsection 9.1 show that the intended extension is a consequence of the following +extension of Proposition 9.1. +Proposition 10.5. For every positive quantity R, there exists a generic subset +Vfull-⋔-F-min-rate-R of Vfull, included in VMorse-ss-eig, such that, for every potential V in this +subset, every travelling front (c, u) in FV,R is transverse, bistable, and approaches its limit +at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of +D2V at this point. +Proof. Proposition 9.1 already provides a generic subset Vfull-⋔-F-R of Vfull such that, +for every potential V in this subset, every travelling front (c, u) in FV,R is transverse, +and therefore bistable (subsection 9.4). Therefore, only the conclusion relative to the +asymptotic behaviour of the profiles remains to be proved. +The proof of this conclusion is a variation of the proof of Proposition 9.1 and follows +the ideas exposed in subsection 10.3: for some potential V0 in Vquad-(R+1) and for some +non-degenerate minimum points e−,0 and e+,0 of V , and for every nonnegative integer N, +two variants of the set MN defined in (9.5) (and of the open subset OV0, e−,0, e+,0, c0, N +defined in (9.6)) can be introduced: one where Bu is replaced by Bsu, and one where Bs +is replaced by Bss. In each of theses two cases, the condition “Φ +�MN, V +� is transverse +to W in N” can be read as “the intersection between Φ +�MN, V +� and W is empty”, +due to the missing dimension induced by the change in each of theses variants. Then, +replacing the open subset OV0, e−,0, e+,0, c0, N by the intersection of its two variants, the +remaining arguments are exactly the same. 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Soc. 198.925 (2009). doi: 10.1090/memo/0925 +(cit. on p. 11). +Romain Joly +Université Grenoble Alpes, CNRS UMR 5582, Institut Fourier, +38000 Grenoble, France. +romain.joly@univ-grenoble-alpes.fr +Emmanuel Risler +Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, +F-69621 Villeurbanne, France. +emmanuel.risler@insa-lyon.fr +69 + diff --git a/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/load_file.txt b/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a1f979d0ae321210c4335d123a06cad56109352 --- /dev/null +++ b/ndA0T4oBgHgl3EQfJv-v/content/tmp_files/load_file.txt @@ -0,0 +1,3915 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf,len=3914 +page_content='Generic transversality of travelling fronts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' standing fronts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and standing pulses for parabolic gradient systems Romain Joly and Emmanuel Risler January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2023 For nonlinear parabolic systems of the form ∂tw(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' t) = ∂2 xw(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' t) − ∇V �w(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' t) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the following conclusions are proved to hold generically with respect to the potential V : every travelling front invading a minimum point of V is bistable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there is no standing front,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every standing pulse is stable at infinity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the profiles of these fronts and pulses approach their limits at ±∞ tangentially to the eigenspaces corresponding to the smallest eigenvalues of D2V at these points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' these fronts and pulses are robust with respect to small perturbations of the potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and the set of their profiles is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' These conclusions are obtained as consequences of generic transversality results for heteroclinic and homoclinic solutions of the differential systems governing the profiles of such fronts and pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Among these results, it is proved that, for a generic Hamiltonian system of the form ¨u = ∇V (u) , every asymmetric homoclinic orbit is transverse and every symmetric homo- clinic orbit is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: 35K57, 37C20, 37C29, 37J46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Key words and phrases: parabolic gradient systems, travelling fronts, standing fronts and pulses, homoclinic and heteroclinic orbits of Hamiltonian systems, generic transversality, Morse–Smale theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='02095v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='AP] 5 Jan 2023 Contents 1 Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Travelling fronts, standing fronts and standing pulses .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Differential system governing the profiles of fronts and pulses .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Transversality of fronts and pulses .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 The space of potentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Main results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 Short historical review .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10 2 Stable and unstable manifolds of equilibria 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Linearization around an equilibrium point .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Local stable and unstable manifolds when the speed c is positive .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Local stable and unstable manifolds when the speed c equals 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 16 3 Preliminary properties of travelling fronts and standing fronts and pulses 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Proof of Proposition 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 19 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Additional conditions on ν and r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Equivalent characterizations of transversality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 39 6 Generic elementarity of symmetric standing pulses 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Notation and statements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Checking hypothesis 1 of Theorem 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 42 7 Generic transversality of asymmetric standing pulses 44 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Notation and statements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} 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+page_content='3 Openness of ν⋔ asym stand pulses(¯ξ, ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 48 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Conclusion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 50 8 Generic non-existence of standing fronts 51 9 Proof of the main results 51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Proof of conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Proof of conclusions 2 and 3 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Proof of conclusion 4 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Proof of conclusions 1 to 4 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 57 10 Generic asymptotic behaviour for the profiles of bistable travelling fronts and of standing pulses stable at infinity 58 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Asymptotic behaviour of profiles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 58 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Local strongly stable and unstable manifolds when the speed c is positive 59 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Idea of the proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 61 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Proof of conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 for bistable travelling fronts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 64 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Reduction to a local statement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 65 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Extension to all potentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 65 1 Introduction The purpose of this paper is to address the generic properties of travelling fronts and standing fronts/pulses of nonlinear parabolic systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) ∂tw(x, t) = ∂2 xw(x, t) − ∇V �w(x, t) � , where time variable t and space variable x are real, the spatial domain is the whole real line, the function (x, t) �→ w(x, t) takes its values in Rd with d a positive integer, and the nonlinearity is the gradient of a potential function V : Rd → R, which is assumed to be regular (of class at least C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Travelling fronts and standing fronts/pulses are the solutions of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) of the form w(x, t) = u(x−ct) that are stationary in a travelling (c > 0) or standing (c = 0) frame and that approach critical points of V at the two ends of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' An insight into the main results of this paper (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, completed with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) is provided by the following corollary, illustrated by Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Its terms are precisely defined in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For a generic potential V the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every travelling front invading a minimum point of V is bistable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there is no standing front, and every standing pulse is stable at infinity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the set of all bistable travelling fronts and all standing pulses is discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every travelling front and every standing pulse (considered individually) is robust with respect to small perturbations of V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the profile of every bistable travelling front or standing pulse stable at infinity approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Travelling fronts, standing fronts and standing pulses Let c be a real quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A function u : R → Rd, ξ �→ u(ξ) is the profile of a wave travelling at speed c (if c is nonzero), respectively a stationary solution (if c equals 0), for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) if the function w : (x, t) �→ u(x − ct) is a solution of this system, that is if u is a solution of the second order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) ¨u = −c ˙u + ∇V (u) , where ˙u and ¨u denote the first and second derivatives of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Up to applying the transform (ξ, c) �→ (−ξ, −c), which leaves system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) unchanged, we may assume that that the speed c is nonnegative (and will always do so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall that a critical point of the 1 c1 c2 e3,− e2,− e1,− c = 0 c3 x Rd e+ e+ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: Illustration of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The coloured lines represent the profiles of travelling fronts or standing fronts/pulses wi(x, t) = ui(x − cit) approaching a minimum point e+ of a given potential at the right end of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If this potential is generic, the critical point ei,− approached at the left end of space by every such profile must be a minimum point, and for the standing profile (speed c = 0) this minimum point must be e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, these profiles (up to translation of the argument) are isolated from each other, so that the set of such profiles (up to translation of the argument) is countable with respect to both speed and profile, and robust with respect to small perturbations of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Additionally (conclusion 5), these profiles approach their limits e+ (ei,−) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V (e+) (D2V (ei,−)), but this last feature is not displayed on the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' potential V is a point e of Rd such that ∇V (e) = 0, and that a non-degenerate local minimum point of V is a critical point m of V such that D2V (m) is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If e− and e+ are two critical points of V , and if u is a non-constant global solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) such that the following limits hold (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) u(ξ) −−−−→ ξ→−∞ e− and u(ξ) −−−−→ ξ→−∞ e+ , then the solution (x, t) �→ u(x − ct) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) is said to connect e− to e+ and is called: a travelling front if c ̸= 0 and e− ̸= e+, a standing front if c = 0 and e− ̸= e+, a standing pulse if c = 0 and e− = e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, a travelling or standing front connecting a critical point e− to a critical point e+ is said to be bistable if both these critical points are non-degenerate (local or global) minimum points of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Accordingly, a standing pulse connecting a critical point to itself is said to be stable at infinity if this critical point is a non-degenerate (local or global) minimum point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Among standing pulses, it is relevant to distinguish symmetric pulses, which are even with respect to some time (the solution goes back and forth following the same path), from asymmetric pulses which are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Travelling fronts and standing fronts and pulses can be interpreted in terms of energy as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us denote by ˜V the opposite potential −V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, in system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) (where the argument ξ plays the role of a time), the speed plays the role of a damping coefficient, and 2 the nonlinear conservative force derives from the potential ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, the system governs the motion of a ball rolling on the graph of ˜V , submitted to the gravitational force and to a friction force −c ˙u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Its Hamiltonian energy is the function HV defined as: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) HV : R2d → R , (u, v) �→ 1 2|v|2 − V (u) = 1 2|v|2 + ˜V (u) , and, for every solution ξ �→ u(ξ) of this system and every time ξ where this solution is defined, the time derivative of HV along this solution reads (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) d dξ HV �u(ξ), ˙u(ξ) � = −c| ˙u(ξ)|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, if such a solution satisfies the limits (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3), if c is positive then e− and e+ must differ and V (e−) must be smaller than V (e+);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' then, – from the point of view of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), the travelling front will be said to invade the “higher” (with respect to V ) critical point e+ (which is “replaced” with the “lower” one e−);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' – from the point of view of the Hamiltonian system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) the damping “absorbs” the positive lag ˜V (e−) − ˜V (e+) (the “higher” critical point with respect to ˜V is e− and the “lower” one is e+);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and if c is zero then e− and e+ must belong to the same level set of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, as explained on Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, the mechanical interpretation provides an intuitive explanation of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Differential system governing the profiles of fronts and pulses Keeping the previous notation, let us consider the vector field (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) Fc,V : R2d → R2d, � u v � �→ � v ∇V (u) − cv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The second order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) is equivalent to the first order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) � ˙u = v ˙v = ∇V (u) − cv or equivalently ˙U = Fc,V (U) with U = (u, v) ∈ R2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A point E of R2d is an equilibrium point of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) if and only if there exists a critical point e of V such that E equals (e, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that e is non-degenerate, or in other words that 0 is not in the spectrum of the symmetric matrix D2V (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let W s c,V (E) and W u c,V (E) denote the stable and unstable manifolds of E for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Recall that each of these manifolds is defined as the union of the images of the 3 e1 e2 ˜V = −V e3 u(ξ) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2: Heteroclinic connections between critical points belonging to different level sets of V for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) (dimension d equals 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This system governs the motion of a ball rolling on the surface u �→ ˜V (u) = −V (u), submitted to the gravitational force and to a friction force −c ˙u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The minimum points e1 and e3 of V are maximum points for −V , whereas e2 denotes a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A travelling front connecting e1 or e2 to e3 corresponds to the ball leaving e1 or e2 with speed zero at time −∞, and rolling towards e3 with the suitable damping c such that is reaches e3 at rest when time goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Roughly speaking, this asymptotic behaviour in the future requires two conditions: the right direction (towards e3) and the right damping (to reach e3 and stop there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As can be intuitively seen on the figure, starting from e1 provides two degrees of freedom (direction and damping), whereas starting from e2 provides only one (damping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For that reason, connections between e1 and e3 (bistable travelling fronts invading e3) are expected to occur generically and to be a robust feature, by contrast with connections between e2 and e3 (non bistable travelling fronts invading e3), which should occur only for rare potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conclusions 1 and 3 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 above and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 below formally confirm these expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4 e Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3: A symmetric standing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A ball is dropped with speed zero at the same level of V as the critical point e, and there is no damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Because the Hamiltonian (energy) is conserved, reaching e as time goes to +∞ only requires to adjust the “direction” towards e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If e is a minimum point of V (a maximum point of −V ) as on the figure, this condition can be fulfilled by choosing the adequate dropping point on the one-dimensional level set V −1�{e} �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If by contrast e was a saddle point, the dropping point should also lie on the one-dimensional stable manifold of e, adding an additional condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For that reason, symmetric standing pulses stable at infinity are expected to be a generic and robust feature, whereas those not stable at infinity should not occur but for rare potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conclusions 2 and 3 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 above and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 below confirm these expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' u(ξ) e Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4: An asymmetric standing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A ball “leaves” the critical point e with speed zero at time −∞, and there is no damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Because the Hamiltonian (energy) is conserved, going back to e as time goes to +∞ only requires to adjust the “direction” towards e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If e is a minimum point of V (a maximum point of −V ) as on the figure, this condition can be fulfilled by leaving e in the adequate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If by contrast e was a saddle point, there would be no choice for the direction of leaving (and in addition, going back to e would require to do so through a particular direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For that reason, asymmetric standing pulses stable at infinity are expected to be a generic and robust feature, whereas those not stable at infinity should not occur but for rare potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conclusions 2 and 3 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 above and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 below confirm these expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5 solutions ξ �→ U(ξ) that converge to E at an exponential rate as ξ goes to +∞/−∞, tangentially to the stable/unstable linear space of this equilibrium (see section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following statement, proved in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, formalizes the correspondence between the profiles of travelling fronts and standing fronts/pulses and the intersections between such manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let e− and e+ be two (possibly equal) non-degenerate critical points of V , let E− and E+ denote the corresponding equilibria for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), and let c denote a real (zero or nonzero) quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every profile ξ �→ u(ξ) of a front/pulse connecting e− to e+ and travelling at speed c (or standing if c equals zero), the image of the corresponding solution ξ �→ �u(ξ), ˙u(ξ) � of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) belongs to W u c,V (E−) ∩ W s c,V (E+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The meaning of this proposition is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' First, it states that the convergence of u(ξ) towards e± at ±∞ yields the convergence of �u(ξ), ˙u(ξ) � towards (e±, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, every profile of a travelling or standing front of the partial differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) corresponds to a heteroclinic orbit of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), and every profile of a standing pulse corresponds to a homoclinic orbit of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Second, those convergences occur at an exponential rate, thus not along a centre manifold (which exists for a non-degenerate critical point which is not a minimum point when c vanishes, see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Transversality of fronts and pulses Usually, the transversality of a heteroclinic orbit connecting two equilibria E− and E+ is defined as the transversality of the intersection between the unstable manifold of E− and the stable manifold of E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For travelling fronts, however, the freedom of moving the speed c must be taken into account, and leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 (transversality of a travelling front).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let e− and e+ be two non-degenerate critical points of V and let E− and E+ denote the corresponding equilibria for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A front with profile ξ �→ u(ξ) travelling at a positive speed c and connecting e− to e+ is said to be transverse if the intersection � � � c′>0 {c′} × W u c′,V (E−) � � ∩ � � � c′>0 {c′} × W s c′,V (E+) � � is transverse, in R2d+1, along the set {c} × U(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For a standing pulse (connecting a critical point e of V to itself) the speed c equals 0, so that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) preserves the Hamiltonian HV defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, the stable and unstable manifolds of the equilibrium E corresponding to e belong to the same level set of HV , so that the transversality between those manifolds cannot hold in R2d, but only in this level set (which is a 2d − 1-manifold of class Ck+1 outside of the set of equilibria).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 6 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 (transversality of a standing pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let e denote a non-degenerate critical point of V and let E = (e, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A standing pulse with profile ξ �→ u(ξ) and connecting e to itself is said to be transverse if the intersection W u 0,V (E) ∩ W s 0,V (E) is transverse, inside the level set H−1 V �−V (e) � deprived of E, along the trajectory U(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As mentioned above, standing pulses divide into two classes (symmetric and asymmetric, see Figures 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4), which will require separate treatments in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Here is a more precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 (symmetric standing pulse, turning time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let e denote a non-degenerate critical point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A standing pulse with profile ξ �→ u(ξ) connecting e to itself is said to be symmetric if there exists a time ξturn, called the turning time of the pulse, such that ˙u(ξturn) vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' or equivalently, such that U(ξturn) belongs to Rd × {0Rd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This subspace Rd × {0Rd}, often called the reversibility or symmetry subspace, will be denoted by Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If such a turning time exists then it is unique and the profile of the pulse is indeed symmetric with respect to this turning time, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Note that in the scalar case d = 1 every standing pulse is symmetric (the derivative ˙u must vanish if the solution approaches the same limits at both ends of R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For symmetric standing pulses (for any value of the dimension d), instead of the transversality defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, the following weaker transversality property ([17, 23, 51]) will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 (elementary symmetric standing pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that the standing pulse ξ �→ u(ξ) is symmetric with turning time ξturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This pulse is said to be elementary if the intersection W u 0,V (E) ∩ Ssym is transverse, in R2d, at the point U(ξturn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The feature of being elementary, for a standing pulse, will be called elementarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Note that every transverse symmetric standing pulse is elementary: due to the time reversibility when c is zero, a non-transverse intersection between W u 0,V (E) and Ssym induces a non-transverse intersection between W u 0,V (E) and W s 0,V (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' But the converse is false: for a symmetric standing pulse, the intersection W u 0,V (E) ∩ W s 0,V (E) may be non- transverse in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 while this intersection still crosses transversally the reversibility subspace Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This may occur, for instance, if a symmetric standing pulse is the limit of a one-parameter family of asymmetric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 The space of potentials For the remaining of the paper, let us take and fix an integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us denote by Ck+1 b (Rd, R) the space of functions Rd → R of class Ck+1 which are bounded, as well as their derivatives of order not larger than k + 1, equipped with the norm ∥W∥Ck+1 b = max α multi-index, |α|≤k+1 ∥∂|α| uαW∥L∞(Rd,R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7 Let us embed the larger space Ck+1(Rd, R) with the following topology: for V in this space, a basis of neighbourhoods of V is given by the sets V + O, where O is an open subset of Ck+1 b (Rd, R) embedded with the topology defined by ∥·∥Ck+1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This topology (which can be viewed as the one of an extended metric) is convenient, since local properties may be studied in Banach spaces of the form V + � Ck+1 b (Rd, R), ∥·∥Ck+1 b � , with ∥·∥Ck+1 b viewed as a classical norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this paper, the space Ck+1(Rd, R) will always be embedded with this topology (if a topology is needed) and � Ck+1(Rd, R), ∥·∥Ck+1 b � will be denoted simply by Ck+1(Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall that a subset A of a topological set B is said to be a generic subset of B if it contains a countable intersection of dense open subsets of B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' accordingly, a property is said to hold for a generic potential if it holds for every potential in a generic subset of Ck+1(Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It is important to notice that Ck+1(Rd, R) is a Baire space because it is locally equal to the Baire space Ck+1 b (Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, the notion of genericity provides relevant definitions of “large” subsets and “almost everywhere satisfied” properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Other definitions exist and the results stated in this paper presumably still hold for those (the interested reader may consider [2, 3, 25, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Actually, the results stated in this paper also hold with other natural topologies, such as Whitney’s topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' However the space Ck+1(Rd, R) is not locally a metric space for Whitney’s topology (which is not characterized by sequences) and this leads to technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The framework chosen above is thus convenient to state the main arguments while avoiding unessential technicalities, but the choice of the topology is not a key issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To finish, let us recall that a function having only non-degenerate critical points is commonly called a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to a classical result (see for instance [24]), the set of Morse functions is a generic subset of Ck+1(Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the intersection of two generic subsets is still a generic subset, and since our purpose is to state results which hold generically, assuming that the potential V under consideration is a Morse function does not present any inconvenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, only nondegenerate critical points will be considered in the following, and the potential V will often be assumed to be a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Main results The following generic transversality statement is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (generic transversality of fronts and pulses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Ck+1(Rd, R) such that, for every potential function V in this subset, V is a Morse function and the following conclusions hold for the fronts and pulses defined by V : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every travelling front invading a minimum point of V is transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every symmetric standing pulse is elementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every asymmetric standing pulse is transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there is no standing front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The core of this paper (sections 5 to 8) is devoted to the proof of this result among potentials which are quadratic past some radius (see their definition in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For such potentials, conclusion 1 is proved by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, conclusion 2, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, conclusion 3 by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, and conclusion 4 by Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Sections 5 and 6 are devoted, respectively, to the proofs of these propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In section 9, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 is completed by extending these conclusions to general potentials of Ck+1(Rd, R) (not necessarily quadratic past some radius), and the qualitative conclusions 1 to 4 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 are derived from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Using the same techniques, the following extension of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (and of conclusions 1 to 4 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) is proved in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The second conclusion of this extension is nothing but the last conclusion 5 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Ck+1(Rd, R) such that, for every potential function V in this subset, in addition to the conclusions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (and to the conclusions 1 to 4 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), the following two additional conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every minimum point of V , the smallest eigenvalue of the Hessian D2V at this minimum point is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the profile of every bistable travelling front or standing pulse stable at infinity approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As explained in subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, conclusions 2 to 4 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 can be interpreted in terms of homoclinic and heteroclinic orbits of the Hamiltonian system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) � ˙u = ∂vH(u, v) ˙v = −∂uH(u, v) where H(u, v) = 1 2|v|2 + ˜V (u) and ˜V = −V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following statement explicitly provides this interpretation (for conclusions 2 and 3 only, since conclusion 4 is actually elementary and well known, see section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' No proof is given since it is exactly a translation of these conclusions, with obvious meanings for (a)symmetry and elementarity of homoclinic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9 (the Hamiltonian point of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Ck+1(Rd, R) such that, for every potential function ˜V in this subset, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every asymmetric homoclinic orbit of the Hamiltonian system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) is transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every symmetric homoclinic orbit of the Hamiltonian system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 Short historical review Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 and its proof rely on transversality theorems, also known as Sard–Smale or Thom’s theorems, and are closely related to classical transversality results for differential systems, see for instance [1, 31, 36, 45, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Significant differences with respect to previous works still deserve to be mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' First, genericity in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 holds with respect to the sole potential function V , not general vector fields in R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, perturbations of a given potential only provide a partial control on the dynamics (in other words, differential systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) do not generate all possible flows in R2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This constraint is balanced by the peculiarities of the systems considered, which will have to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To our best knowledge, the first genericity result about the dynamics of a special class of differential systems goes back to [45], and deals with polynomial flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning Hamiltonian flows, homoclinic orbits play an important role, both from theoretical and physical points of view, see for instance the reviews [12, 16] and articles [4, 15, 23, 32, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The transversality/elementarity of such orbits has important dynamical consequences, as the presence of Smale horseshoes associated to complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In [30, 50], the genericity of these properties is considered in a general abstract framework, and obtained only under sufficient conditions corresponding to the assumptions of the transversality Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In [32], this genericity is proved, but in the case of non-autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Other references dealing with the generic transversality of connecting orbits include [33, 37, 38, 46] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In [46], genericity holds with respect to all Hamiltonian flows, and not only second order conservative systems as (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In [33, 37, 38], genericity holds with respect to the potential ˜V only, in a more general setting where the “kinetic energy” |v|2 /2 of the Hamiltonian in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) is replaced by a more general expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' But the transversality of homoclinic orbits is not considered in these papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In [33], the transversality of heteroclinic orbits is derived from a perturbation result of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The others results of [33, 37, 38] are concerned with closed orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, to the best of our knowledge, even Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9 (the results concerning standing pulses, in the language of Hamiltonian systems) is new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning the nonzero dissipation case (conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), the statement differs from usual genericity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If c is fixed (and nonzero), heteroclinic connections corresponding to travelling fronts invading a minimum point of V do generically not exist for the flow of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' But the freedom provided by the parameter c ensures the generic existence, transversality, and robustness of heteroclinic connections corresponding to bistable travelling fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This parameter c will thus have to be taken into account in the setting where transversality theorems will be applied, a significant difference with classical genericity results about the flows of differential systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The initial motivation for this paper actually relates to parabolic systems of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For such systems, the global dynamics of bistable solutions, that is solutions close at both ends of space to local minimum points of the potential V , has been described under rather general (assumed to be generic) hypotheses on V by the second author in [39, 40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Every such solutions must approach, as time goes to +∞, far to the left in space a stacked family of bistable fronts travelling to the left, far to the right 10 in space a stacked family of bistable fronts travelling to the right, and in between a pattern of standing pulses/fronts going slowly away from one another (this extends to gradient systems the program initiated in the late seventies by Fife and McLeod for scalar equations [19–21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The present paper provides a rigorous proof of the genericity of the hypotheses made on the potential V in [39, 40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same hypotheses yield similar conclusions for hyperbolic gradient systems [41] and for radially symmetric solutions of parabolic gradient systems in higher space dimension [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The results obtained in this last reference rely on an additional hypothesis, which is the higher space dimension analogue of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (elementarity of symmetric standing pulses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The genericity of this hypothesis is proved in the companion paper [44], using the same approach as in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The extension Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (comprising the last conclusion 5 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) is motivated by the study of the long-range interaction between fronts and pulses of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The long-range interaction between such “localized structures” is the object of a large body of literature, both in Mathematics and Physics, see for instance [6, 11, 14, 18, 28, 52] among many other possible references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The conclusions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 are especially relevant in conjunction with this topic, for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Consider a solution of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) close to, say, two standing fronts or pulses or two fronts travelling at the same speed, far away from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us denote by uleft(·) and uright(·) their profiles, so that the solution is close to a translate of uleft on R− and to a translate of uright on R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, the (large) distance between these two translates is expected to vary slowly, according to a (long-range) interaction law that can be computed at first order, and which is related to the asymptotics of uleft at +∞ and of uright at −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Basically, when (as in the present context) the tails of uleft and uright are not oscillating, this first order long-range interaction can be attractive or repulsive or neutral, depending on the sign of a scalar product between the (oriented) directions through which uleft and uright approach their (common) limit (at +∞ and at −∞ respectively), see for instance the conjecture at the bottom of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 59 of [5], or expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In the present context, according to the conclusions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 and for a generic potential, these two oriented directions are aligned with the one-dimensional eigenspace associated with the smallest eigenvalue of the Hessian D2V of the potential at the minimum point which is the common limit mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Among the consequences, the first order long-range interaction is thus either attractive or repulsive, but not neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2 Stable and unstable manifolds of equilibria Throughout all this section V denotes a potential function in Ck+1(Rd, R) and c denotes a non-negative quantity (speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As stated in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the travelling fronts and standing fronts/pulses of the parabolic equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) correspond to heteroclinic and homoclinic connections for the flow in R2d generated by the first order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let Ω be the maximal (open) subset of R × R2d where this flow is defined and let 11 us consider its flow Sc,V defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) Sc,V : Ω → R2d , (ξ, U0) �→ U(ξ) , where U(ξ) is the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) with U(0) = U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By definition, for every (ξ, U0) in Ω, ∂ ∂ξ Sc,V (ξ, U0) = Fc,V �Sc,V (ξ, U0) � where Fc,V : � u v � �→ � v ∇V (u) − cv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Although the variable ξ denotes primarily the space variable in a frame travelling at speed c for the initial partial differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), this variable also plays the role of a time in the differential systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) prescribing the profiles of travelling and standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In the following, this variable will thus often be referred to as a “time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Linearization around an equilibrium point Let e denote a non-degenerate critical point of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , ud) denote an orthonormal basis of Rd made of eigenvectors of the Hessian D2V (e) and let µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , µd denote the corresponding (real) eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us call Morse index of e, denoted by m(e), the number of negative eigenvalues of D2V (e), counted with their algebraic multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the critical point e is assumed to be non-degenerate, it is: a minimum point if m(e) equals 0, a saddle point if m(e) is between 1 and d − 1, and a maximum point if m(e) equals d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, none of the quantities µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , µd vanishes, and we may assume that µ1 ≤ · · · ≤ µm(e) < 0 < µm(e)+1 ≤ · · · ≤ µd if m(e) > 0 , and 0 < µ1 ≤ · · · ≤ µd if m(e) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, let us consider the equilibrium point E = (e, 0Rd) of Sc,V corresponding to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The linearized differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) at E reads: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) ˙U = DFc,V (E)U , or equivalently � ˙u = v ˙v = D2V (e)u − cv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that a complex quantity λ is an eigenvalue for the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) if and only if the quantity λ(λ + c) is an eigenvalue for the Hessian D2V (e), that is if λ(λ + c) is equal to one of the quantities µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For j in {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , d}, let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) λj,+ = −c 2 + � c2 4 + µj and λj,− = −c 2 − � c2 4 + µj denote the two (real or complex) eigenvalues of the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) corresponding to µj, and let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) Uj,+ = � uj λj,+uj � and Uj,− = � uj λj,−uj � 12 denote the corresponding eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) Es c,V (E) and Ec c,V (E) and Eu c,V (E) denote the stable, centre, and unstable subspaces of R2d for the linear operator DFc,V defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), that is the eigenspaces corresponding to eigenvalues with negative, zero and positive real parts respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The dimensions of those spaces and of the corresponding invariant manifolds (defined below) derive from expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3), and are as summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The case of a negative speed c can be derived by the c = 0 c > 0 Dimension of Eu c,V (E) and W u c,V (E) d − m(e) d − m(e) Dimension of Es c,V (E) and W s c,V (E) d − m(e) d + m(e) Dimension of Ec c,V (E) and W c loc,c,V(E) 2m(e) 0 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: Dimensions of stable, unstable, and centre manifolds for an equilibrium point E = (e, 0) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), corresponding to a critical point e of the potential with Morse index m(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' transformation (c, ξ) �→ (−c, −ξ) which leaves the systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) unchanged (and exchanges the stable and unstable dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The dimension of Eu c,V (E) is also commonly called the Morse index of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To avoid any confusion, the denomination Morse index will only be used for critical points of the potential, not for the corresponding equilibria in R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Local stable and unstable manifolds when the speed c is positive The construction of the local stable (unstable) manifold of an equilibrium of a differential system is classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A historical reference is Kelley’s article [29], comprising the construc- tion and the dependence on the parameters, however with a slightly non-optimal regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A complete construction can be found in many textbooks, for example Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 of [22] or Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 of [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The goal of this subsection and of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 below is to provide precise statements (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 below and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 when the speed c equals 0) concerning these manifolds (for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7)), and the associated notation (without proofs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' those statements and notation will be called upon in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Take V0 in Ck+1(Rd, R), let e0 denote a non-degenerate critical point of V0, and let c0 denote a positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, the point (e0, 0), which will be denoted by E0, is a hyperbolic equilibrium point and the subspaces Eu c0,V0(E0) and Es c0,V0(E0) introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) generate the whole space R2d (or in other words the central part Ec c0,V0(E0) reduces to {0R2d}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) βu = min {Re(λ) : λ eigenvalue of DFc0,V0(E0) with Re(λ) > 0} and βs = max {Re(λ) : λ eigenvalue of DFc0,V0(E0) with Re(λ) < 0} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 13 There exist norms ∥·∥u on the unstable subspace Eu c0,V0(E0) and ∥·∥s on the stable subspace Es c0,V0(E0) such that, for every non negative quantity ξ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) ���exp � −ξDFc0,V0(E0)|Eu c0,V0(E−) ���� u ≤ exp � −βu 2 ξ � , and ���exp � ξDFc0,V0(E0)|Es c0,V0(E+) ���� s ≤ exp �βs 2 ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity r, let (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) Bu E0(r) = �Uu ∈ Eu c0,V0(E0) : ∥Uu∥u ≤ r � , and Bs E0(r) = �Us ∈ Es c0,V0(E0) : ∥Us∥s ≤ r � , and BE0(r) = �Uu + Us : Uu ∈ Bu E0(r) and Us ∈ Bs E0(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (local stable and unstable manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist a neighbourhood ν of V0 in Ck+1(Rd, R), a neighbourhood C of c0 in (0, +∞) and a positive quantity r such that, for every (c, V ) in C × ν, the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a unique critical point e(V ) of V such that E(V ) = (e(V ), 0) belongs to E0 + BE0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, e(V ) has the same Morse index as e0 and the map ν → Rd, V �→ e(V ) is of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist Ck-functions wu loc, c, V : Bu E0(r) → Bs E0(r) and ws loc, c, V : Bs E0(r) → Bu E0(r) such that, if we consider the sets W u loc, c, V �E(V ) � = � E(V ) + Uu + wu loc, c, V (Uu) : Uu ∈ Bu E0(r) � and W s loc, c, V �E(V ) � = � E(V ) + Us + ws loc, c, V (Us) : Us ∈ Bs E0(r) � , then, for every U in BE0(r) the following two assertions are equivalent: a) U is in W u loc, c, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' b) Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in (−∞, 0] and Sc,V (ξ, U) → E(V ) as ξ → −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and for every U in BE0(r) the following two assertions are equivalent: c) U ∈ W s loc, c, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' d) Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in [0, +∞) and Sc,V (ξ, U) → E(V ) as ξ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Both differentials Dwu loc, c0, V0(0) and Dws loc, c0, V0(0) vanish, and both maps C × ν × Bu E0(r) → Bs E0(r), (c, V, Uu) �→ wu loc, c, V (Uu) and C × ν × Bs E0(r) → Bu E0(r), (c, V, Us) �→ ws loc, c, V (Us) are of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 14 With the notation provided by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, for every (c, V ) in C×ν, let us introduce the maps ˆwu loc, c, V : Bu E0(r) → R2d, Uu �→ E(V ) + Uu + wu loc, c, V (Uu) , and ˆws loc, c, V : Bs E0(r) → R2d, Us �→ E(V ) + Us + ws loc, c, V (Us) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Local unstable and stable manifolds of E(V ) can be defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) W u loc, c, V �E(V ) � = ˆwu loc, c, V �Bu E0(r) � , and W s loc, c, V �E(V ) � = ˆws loc, c, V �Bs E0(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Those manifolds depend smoothly of c and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The global unstable and stable manifolds W u c,V �E(V ) � = {U ∈ R2d : Sc,V (ξ, U) → E(V ) when ξ → −∞} and W s c,V �E(V ) � = {U ∈ R2d : Sc,V (ξ, U) → E(V ) when ξ → +∞} can then be derived from those local manifolds through the flow Sc,V as follows: W u c,V �E(V ) � = Sc,V � R × W u loc, c, V �E(V ) �� , and W s c,V �E(V ) � = Sc,V � R × W s loc, c, V �E(V ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Here are two observations that will turn out to play some role in the forthcoming proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the characterization provided by this proposition (equivalence between 2a and 2b and between 2c and 2d), for every solution ξ �→ U(ξ) of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), if this solution belongs to the stable (unstable) manifold of E(V ) then it crosses exactly once the border ∂W s loc, c, V �E(V ) � of the local stable manifold (the border ∂W u loc, c, V �E(V ) � of the local unstable manifold) of E(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, according to the the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) satisfied by the norms ∥·∥u and ∥·∥s, up to replacing the radius r by a smaller quantity, this intersection between the trajectory of ξ �→ U(ξ) and the border of the local stable (unstable) manifold of E(V ) is transverse inside the full stable (unstable) manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Although the transversality of this intersection is not formally required in the following, assuming that it holds helps figuring out the broad scheme, see for instance Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The functions wu loc, c, V and ws loc, c, V are uniquely defined by the characterization provided by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 once the radius r and the departure sets of these two functions are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, those two functions and the local stable and unstable manifolds W u loc, c, V �E(V ) � and W s loc, c, V �E(V ) � remain unchanged if the potential function V is modified outside a neighbourhood of the set πpos � W u loc, c, V �E(V ) � ∪ W s loc, c, V �E(V ) �� , where πpos is the projection onto the position coordinates: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) πpos : R2d → Rd, (u, v) �→ u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Local stable and unstable manifolds when the speed c equals 0 As Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 shows, an equilibrium E is hyperbolic except if c vanishes and m(e) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this case, there exists, in addition to the stable and unstable manifolds of E, a centre manifold with dimension 2m(e) (corresponding to the central part of the spectrum of the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) at E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' However, as shown by the following lemma, a solution ξ �→ U(ξ) of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) cannot asymptotically approach E through such a centre manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 below and the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, provided in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, rely on this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 (approach of critical points through stable/unstable manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that c equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every critical point e of V such that the Morse index m(e) is positive, and for every (maximal) solution ξ �→ U(ξ) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), if E denotes the point (e, 0), the following conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' if U(ξ) goes to E as ξ goes to +∞, then the trajectory of ξ �→ U(ξ) converges to E tangentially to the stable space Es V (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' if U(ξ) goes to E as ξ goes to −∞, then the trajectory of ξ �→ U(ξ) converges to E tangentially to the unstable space Eu V (E), Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ξ �→ U(ξ) = (u, v)(ξ) denote a solution of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) for a speed c equal to 0, and let us assume that U(ξ) goes to E as ξ goes to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from the invariance of the Hamiltonian function HV (defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4)) along U(·) that HV (U(ξ)) = HV (E), or in other words that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) 1 2 |v(ξ)|2 − V �u(ξ) � = −V (e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us proceed by contradiction and assume that this solution does not belong to the stable manifold of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, it follows that, as ξ goes to +∞, the component of U(ξ) − E along the centre subspace Ec V (E) is dominant compared to its component along the hyperbolic subspace Es V (E) + Eu V (E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' with symbols, if πcent denotes the projection along Es V (E) + Eu V (E) onto Ec V (E) in R2d, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) U(ξ) − E = πcent �U(ξ) − E � + oξ→+∞ � πcent �U(ξ) − E �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from the expressions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) of the eigenvalues and eigenvectors of DF0,V (E) that Ec V (E) = span �U1,+, U1,−, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , Um(e),+, Um(e),− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, applying the projection πpos (projection onto the position coordinates, defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10)) to equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12), it follows that, if we denote by πm(e) the orthogonal projection onto span{u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , um(e)} in Rd, u(ξ) − e = πm(e) �u(ξ) − e � + oξ→+∞ � πm(e) �u(ξ) − e �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the restriction of D2V (e) to the image of πm(e) is negative definite, it follows that, for ξ sufficiently large, V �u(ξ) � is smaller than V (e), thus −V �u(ξ) � is larger than −V (e), contradicting equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 16 As for Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 in the case c > 0, the aim of the next Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 is to provide (in the case c = 0) a precise statement and the associated notation concerning the local stable and unstable manifolds of an equilibrium for the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this case c = 0, the conclusions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 show that centre manifolds are not relevant for homoclinic and heteroclinic solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for that reason, those centre manifolds are ignored in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning the construction and properties of the local stable and unstable manifolds, there is no difference with respect to the positive speed case considered in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, see again [22, 29, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, by contrast with the statements that can be found in textbooks, the characterization of these local stable and unstable manifolds does not require an exponential rate of convergence towards E, again due to the conclusions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 (see the equivalence between assertions 2a and 2b and between assertions 2c and 2d in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For the remaining of this paper, when the speed c vanishes, it will be omitted in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, concerning the previously introduced notation, FV SV Es V Ec V Eu V W s V W u V stand for: F0,V S0,V Es 0,V Ec 0,V Eu 0,V W s 0,V W u 0,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Take V0 in Ck+1(Rd, R) and let e0 denote a non-degenerate critical point of V0 and let E0 = (e0, 0) (which is not necessarily hyperbolic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let βu and βs be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in the case c > 0, there exist norms ∥·∥u on the unstable subspace Eu V0(E0) and ∥·∥s on the stable subspace Es V0(E0) such that inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) hold for every non negative quantity ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ∥·∥c denote any norm on the centre subspace Ec V0(E0) (for instance the euclidean norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity r, let Bu E0(r) = {Uu ∈ Eu V0(E0) : ∥Uu∥u ≤ r} , Bs E0(r) = {Us ∈ Es V0(E0) : ∥Us∥s ≤ r} , Bc E0(r) = {Uc ∈ Ec V0(E0) : ∥Uc∥c ≤ r} , and BE0(r) = {Uu + Us + Uc : Uu ∈ Bu E0(r) and Us ∈ Bs E0(r) and Uc ∈ Bc E0(r)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 (local stable and unstable manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist a neighbourhood ν of V0 in Ck+1(Rd, R) and a positive quantity r such that, for every V in ν, the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a unique critical point e(V ) of V such that E(V ) = (e(V ), 0) belongs to E0 + BE0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, e(V ) has the same Morse index as e0 and the map ν → Rd, V �→ e(V ) is of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist Ck-functions wu loc, V : Bu E0(r) → Bs E0(r) + Bc E0(r) and ws loc, V : Bs E0(r) → Bu E0(r) + Bc E0(r) such that, if we consider the sets W u loc, V �E(V ) � = � E(V ) + Uu + wu loc, V (Uu) : Uu ∈ Bu E0(r) � and W s loc, V �E(V ) � = � E(V ) + Us + ws loc, V (Us) : Us ∈ Bs E0(r) � , 17 then, for every U in BE0(r), the following two assertions are equivalent: a) U is in W u loc, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' b) SV (ξ, U)−E(V ) remains in BE0(r) for all ξ in (−∞, 0] and SV (ξ, U) → E(V ) as ξ → −∞, and for every U in BE0(r), the following two assertions are equivalent: c) U ∈ W s loc, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' d) SV (ξ, U)−E(V ) remains in BE0(r) for all ξ in [0, +∞) and SV (ξ, U) → E(V ) as ξ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Both differentials Dwu loc, V0(0) and Dws loc, V0(0) vanish, and both maps ν × Bu E0(r) → Bs E0(r), (V, Uu) �→ W u loc, V (Uu) and ν × Bs E0(r) → Bu E0(r), (V, Us) �→ W s loc, V (Us) are of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation provided by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, for every V in ν, let us introduce the maps ˆwu loc, V : Bu E0(r) → R2d, Uu �→ E(V ) + Uu + wu loc, V (Uu) , and ˆws loc, V : Bs E0(r) → R2d, Us �→ E(V ) + Us + ws loc, V (Us) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Local unstable and stable manifolds of E(V ) can be defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) W u loc, V �E(V ) � = ˆwu loc, V �Bu E0(r) � , and W s loc, V �E(V ) � = ˆws loc, V �Bs E0(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Those manifolds depend smoothly of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the global unstable/stable manifolds of E(V ), denoted by W u V (E(V )) and W s V (E(V )) can be expressed in terms of those local manifolds and of the flow SV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Both observations made in the remark ending the previous subsection are still valid in the present case of zero speed and potential existence of a centre manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3 Preliminary properties of travelling fronts and standing fronts and pulses Let us take and fix, for this whole section, a potential function V in Ck+1(Rd, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Let e− and e+ be two (possibly equal) non-degenerate critical points of V , let c denote a non negative quantity (speed), and let ξ �→ u(ξ) denote the profile of a front or pulse connecting e− to e+ and travelling at speed c (or standing if c equals zero) for the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 18 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The derivative ˙u(ξ) goes to 0 as ξ goes to ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If the speed c is positive, then ξ �→ u(ξ) is the profile of a travelling front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from integrating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) lim ξ→+∞ HV (u(ξ)) − lim ξ→−∞ HV (u(ξ)) = −c � R | ˙u(ξ)|2 dξ and thus that ˙u(·) is in L2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus 0 is an adherent value of the kinetic part of the Hamiltonian function ξ �→ HV �U(ξ) � as ξ goes to ±∞, meaning that V (e±) are adherent values of HV (U(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since according to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) this last function decreases with ξ, it follows that HV �U(ξ) � goes to V (e±) as ξ goes to ±∞, and the intended conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If the speed c equals 0, it follows from the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and the convergence of u(·) to critical points that ¨u(ξ) goes to 0 as ξ goes to ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus ˙u(·) is uniformly continuous and the convergence of u yields the intended conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us use the notation of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If c is non zero or if c equals 0 and both Morse indices m(e−) and m(e+) of e− and e+ vanish, then E− and E+ are hyperbolic equilibria of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) and the conclusions of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 follow from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If c equals 0 and the Morse indices m(e−) and m(e+) are any, then the equilibria E− and E+ are not necessarily hyperbolic, but again in this case it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 that U(ξ) goes to E± as ξ → ±∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 that the values of ξ �→ U(ξ) belong to the unstable manifold of E− and to the stable manifold of E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Equivalent definitions of a symmetric standing pulse Let e denote a non-degenerate critical point of V , and let ξ �→ u(ξ) denote the profile of a standing pulse connecting e to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, the symmetry of such a pulse was defined by the existence of a “turning time” where ˙u vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following standard result (see for instance [17]) completes this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (equivalent definitions of a symmetric standing pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every real quantity ξturn, the following properties are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' ξturn is a turning time in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, that is ˙u(ξturn) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every ξ in R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) u(ξturn − ξ) = u(ξturn + ξ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there exists ξ in R such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) u(ξturn − ξ) = u(ξturn + ξ) and ˙u(ξturn − ξ) = − ˙u(ξturn + ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, these statements hold for at most one real quantity ξturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 19 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Differentiating equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) with respect to ξ yields equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) for all ξ, so that property 2 implies property 3, and property 3 for ξ equal to 0 is equivalent to property 1, so that property 2 implies property 1 and property 1 implies property 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It remains to prove that property 3 implies property 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that property 3 holds, and, for every real quantity ξ, let us write u1(ξ) = u(ξturn + ξ) and u2(ξ) = u(ξturn − ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since ξ �→ u(ξ) is a solution of the second order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) with c equal to zero, both ξ �→ U1(ξ) and ξ �→ U2(ξ) are solutions of the first order differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) (again with c equal to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to property 3, there exists ξ such that U1(ξ) is equal to U2(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus U1(ξ) must be equal to U2(ξ) for every real time ξ, and property 2 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus the three properties of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, if property 2 holds for two different turning times ξturn and ξ′ turn, then ξ �→ u(ξ) is periodic with a period equal to 2(ξ′ turn − ξturn), a contradiction with the assumption that u is a standing pulse connecting e to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Values reached only once by profiles of travelling fronts / standing pulses The proofs carried on in the sections 5 to 7 below rely on the construction of suitable perturbations of the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Whereas the uniqueness of the solutions of differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) ensures that the function ξ �→ �u(ξ), ˙u(ξ) � defined by such a solution is one-to-one, this is not necessarily true for the function ξ �→ u(ξ) (as shown by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, a perturbation of the potential V may affect this solution at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The goal of the following proposition is to avoid this inconvenience, by providing in each case under consideration a time interval where u(ξ) is reached only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every profile ξ �→ u(ξ) of a front travelling at a positive speed c and connecting two non-degenerate critical points, there exists a time ξonce such that, for all times ξ∗ in (−∞, ξonce] and ξ in R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) u(ξ) = u(ξ∗) =⇒ ξ = ξ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every profile ξ �→ u(ξ) of an asymmetric standing pulse and for every nonempty open interval I of R, there exists a nonempty open interval Ionce, included in I, such that, for all times ξ∗ in Ionce and ξ in R, implication (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every profile ξ �→ u(ξ) of a symmetric standing pulse, if ξturn denotes the turning time of this pulse (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), then, for every nonempty open interval I included in (−∞, ξturn], there exists a nonempty open interval Ionce, included in I, such that, for all times ξ∗ in Ionce and ξ in (−∞, ξturn], implication (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 20 u 0 −V e− e+ u(ξ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: The one-dimensional example of this figure shows that property 1 of Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 may not hold outside a small neighbourhood of the critical point e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of statement 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ξ �→ u(ξ) denote the profile of a front trav- elling at a positive speed c for the potential V , and let e− denote the critical point, assumed to be non-degenerate, approached by u(ξ) as ξ goes to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since all eigenval- ues of DFc,V (E−)|Eu c,V (E−) are real and positive (see subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), the corresponding solution U(ξ) of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) must approach E− tangentially to some (real, unstable) eigenvector Ueig of DFc,V (E−) as ξ goes to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If λ denotes the corresponding (positive) eigenvalue, then Ueig is of the form (ueig, λueig), where ueig is an eigenvector of D2V (e−), see expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus there must exist a nonzero scalar function ξ �→ α(ξ) so that, as ξ goes to −∞, U(ξ) = E− + α(ξ)Ueig + o �α(ξ) � , that is � u(ξ) = e− + α(ξ)ueig + o �α(ξ) � , ˙u(ξ) = α(ξ)λueig + o �α(ξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows that there exists a large negative time ξ0 such that, for every ξ in (−∞, ξ0], d dξ |u(ξ) − e−|2 = 2(u(ξ) − e−) · ˙u(ξ) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In particular, the function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) (−∞, ξ0] → Rd, ξ �→ u(ξ) is a C1-diffeomorphism onto its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the decrease (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) of the Hamiltonian, the quantity HV �U(ξ0) � is smaller than −V (e−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, there exists a time ξonce in (−∞, ξ0) such that, for every ξ∗ in (−∞, ξonce], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) HV �U(ξ0) � < −V �u(ξ∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Take a time ξ∗ in (−∞, ξonce] and a time ξ in R and let us assume that u(ξ) equals u(ξ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If ξ was larger than ξ0 then it would follow from the expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) of the Hamiltonian, its decrease (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) and inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) that −V �u(ξ) � ≤ HV �U(ξ) � ≤ HV �U(ξ0) � < −V �u(ξ∗) � , a contradiction with the equality of u(ξ) and u(ξ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus ξ is not larger than ξ0, and it follows from the one-to-one property of the function (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) that ξ must be equal to ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Statement 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 21 Proof of statement 2 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ξ �→ u(ξ) be the profile of an asymmetric standing pulse for the potential V , let e denote the critical point approached by u(ξ) as ξ goes to ±∞, and let I be a nonempty open interval of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In view of the intended conclusion (statement 2), we may assume that I is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the invariance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) of the Hamiltonian HV , for every ξ in R, the difference V �u(ξ) � − V (e) is equal to | ˙u(ξ)|2/2 and is therefore nonzero, so that the critical point e is never reached by the function ξ �→ u(ξ) on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence there exists a (small) positive quantity r such that |u(ξ) − e| is larger than r for all ξ in I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and since u(ξ) approaches e as ξ goes to ±∞, there exists a (large) positive quantity M such that |u(ξ) − e| is smaller than r outside of [−M, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that there exist two different times ξ and ξ′ in R such that u(ξ) equals u(ξ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, again according to the invariance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) of the Hamiltonian HV , the time derivatives ˙u(ξ) and ˙u(ξ′) must have the same norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Besides, these two vectors cannot be equal (or else the profile u would be periodic) nor opposite (or else according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 the pulse would be symmetric), thus they are not proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus the couples (ξ, ξ′) such that u(ξ) is equal to u(ξ′) are isolated in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, if (ξ, ξ′) is such a couple and ξ is in I then ξ′ must belong to [−M, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This shows by compactness that there exists only a finite number of couples (ξ, ξ′) in I × R such that u(ξ) equals u(ξ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Statement 2 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of statement 3 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The arguments are the same as in the proof of statement 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ξ �→ u(ξ) be the profile of a symmetric pulse with turning time ξturn for the potential V , let I be a nonempty open interval of (−∞, ξturn], assumed to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If there exist two different times ξ and ξ′ in (−∞, ξturn] such that u(ξ) equals u(ξ′), again the time derivatives ˙u(ξ) and ˙u(ξ′) have the same norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' These two vectors cannot be equal (or else the profile u would be periodic) nor opposite (or else, according to statement 3 of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, (ξ + ξ′)/2 would be a second turning time — smaller than ξturn — for u, a contradiction with the conclusion of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus again, ˙u(ξ) and ˙u(ξ′) cannot be proportional, and the same arguments as in the proof of statement 2 above show that there exists only a finite number of couples (ξ, ξ′) in I × (−∞, ξturn] such that u(ξ) = u(ξ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4 Tools for genericity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 An instance of the Sard–Smale transversality theorem To prove that a given property generically holds, a standard method is to express this property as a transversality problem and to use one instance among the family of theorems known as Sard–Smale theorem (or Thom’s theorems, or transversality theorems), see [1, 7, 24, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this paper the following instance will be used (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider a function Φ : M × Λ → N , where M and N are two finite-dimensional manifolds and Λ (“parameter space”) is a Banach manifold, together with a submanifold W of N (see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us assume 22 that these four manifolds and the function Φ are of class Ck (as everywhere in the paper k denotes an integer which is not smaller than 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Finally, let codim(W) denote the codimension of W in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation above, the image of Φ is said to be transverse to W, if, for every (m, λ) in M × Λ such that Φ(m, λ) is in W, the following equality holds: DΦ(TmM × TλΛ) + TΦ(m,λ)W = TΦ(m,λ)N (here DΦ denotes the differential of Φ at (m, λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Accordingly, for every λ in Λ, if Φλ denotes the function: M → N, m �→ Φ(m, λ) , then the image of Φλ is said to be transverse to W if, for every m in M such that Φ(m, λ) is in W, denoting DΦλ the differential of Φλ at m, DΦλ(TmM) + TΦ(m,λ)W = TΦ(m,λ)N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (Sard–Smale transversality theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation above, if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' k > dim(M) − codim(W), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and the image of Φ is transverse to W, then there exists a generic subset Λgen of Λ such that, for every λ in Λgen, the image of Φλ is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The proof of this result can be found in [1] or in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The key hypothesis, which is often difficult to check, is the transversality hypothesis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notice that the conclusion is stronger than this hypothesis since it states that the transversality holds for a fixed generic parameter λ, whereas hypothesis 2 uses the freedom of moving λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' λ moves Φ M Φ(M, λ) W Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: Geometric interpretation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that for a given parameter λ0, DΦλ0(TM) + TW is not the whole tangent space TN, but that the dependence of Φ on λ provides the missing directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then for almost every λ close to λ0, the image Φ(M, λ) intersects W transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Extending local genericity to global genericity Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (under the form above or another) is the standard tool to prove that a property generically holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' However, it turns out that is is often difficult, in practice, to express a property using a single function Φ as above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' thus one is often led to patch together several conclusions provided by this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following lemma provides a way to carry out this patching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This lemma is identical to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 of Chapter 3 of [35], where a proof can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 (local genericity implies global genericity in a separable Baire space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V be a separable Baire space and Vdense be a dense subset of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every subset Vgen of V, the following two assertions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the subset Vgen is generic in V;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every V0 in Vdense, there exists a neighbourhood ν of V0 in V such that Vgen ∩ ν is generic in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Potentials that are quadratic past a given radius The whole space Ck+1(Rd, R) of potentials is somewhat difficult to handle, for various reasons: it is not separable, even locally, and the flow of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) is not globally well-defined for some of the potentials V in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To get around these difficulties, the proofs of the next sections 5 and 6 will be carried out on a more restricted class Vquad-R of potentials, after what the results will be extended to the full set Ck+1(Rd, R) in the final section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) Vfull = Ck+1(Rd, R) , and, for a positive quantity R, let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) Vquad-R = � V ∈ Vfull : for all u in Rd, |u| ≥ R =⇒ V (u) = |u|2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By contrast with Vfull, the affine subspace Vquad-R of Vfull is separable, and therefore provides a framework where Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The next lemma states some (nice) properties of the flow of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) for a potential V in Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It is followed by another one (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 below) providing the adequate tools to proceed with the extension mentioned above and carried out in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every non negative quantity r, let BRd(0, r) and BRd(0, r) denote the open ball and the closed ball centred at the origin and of radius r in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity R and for every potential V in Vquad-R, the following conclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every speed c, the flow defined by the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) is global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Every profile ξ �→ u(ξ) of a travelling front or a standing front or a standing pulse, for this potential, satisfies the following bound: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) sup ξ∈R |u(ξ)| < R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V be in Vquad-R and let c be a real quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) of Vquad-R, there exists a positive quantity K such that, for every u in Rd, |∇V (u)| ≤ K + |u| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, it follows from the expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) of Fc,V that, for every solution U = (u, v) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) in R2d, ��� ˙U(ξ) ��� = |Fc,V (u, v)| = |(v, ∇V (u) − cv)| = O|U|→∞ � |U(ξ)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This bound prevents solutions from blowing up in finite time, proving conclusion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now let ξ �→ u(ξ) denote a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) approaching critical points of V at both ends of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us write q = |u|2/2, so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) ˙q = u · ˙u and ¨q = −c ˙q + ˙u2 + u · ∇V (u) , and so that, since V is in Vquad-R, for every real quantity ξ, |u(ξ)| ≥ R =⇒ d dξ |ξ �ecξ ˙q(ξ) � = ecξ� ˙u2(ξ) + u2(ξ) � > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since V is quadratic outside the ball BRd(0, R), its critical points must belong to the interior of BRd(0, R), and the same must be true for u(ξ) when |ξ| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, if |u(·)| were to reach the value R at some (finite) time ξ0, then (if ξ0 is the first time when this happens) ˙q(ξ0) would be nonnegative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the implications above show that, from this time on, the quantity ecξ ˙q(ξ) (and thus also the quantity ˙q(ξ)) would remain positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' so that q(ξ) and |u(ξ)| would keep increasing with ξ, a contradiction with the fact that u(ξ) must be back inside BRd(0, R) for ξ large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conclusion 2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Topological properties of restriction maps Let R denote a positive quantity and let us consider the set (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) Vres-R = Ck+1�BRd(0, R), R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The next Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 will be used to carry out, in section 9, the extension mentioned at the beginning of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To ease its formulation, let us adopt Vquad-∞ as an alternative notation for the space Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let R′ denote either a quantity larger than R or ∞, and let us consider the restriction operator: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) resR,R′ : Vquad-R′ → Vres-R , V �→ V|BRd(0,R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 25 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The restriction map resR,R′ is continuous, surjective and open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If two potentials of Vquad-R′ are Ck-close, then their restrictions to the closed ball BRd(0, R) are still Ck-close on this ball, so that the map resR,R′ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To prove that the map resR,R′ is surjective and open, it is sufficient to construct a continuous right inverse for this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For this purpose we may consider Seeley’s extension ext∞,R : Vres-R → Vfull , which is a right inverse for resR,∞ (that is resR,∞ ◦ ext∞,R is the identity map of Vres-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The map defined in Seeley’s original paper [47] extends to the whole space Rd a function initially defined on a half space, but using spherical coordinates the same definition leads to this extension ext∞,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This map ext∞,R is linear and continuous for the usual topology for the departure set Vres-R and the topology of uniform convergence of derivatives up to order k on compact subsets of Rd for the arrival set Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, if χ : [0, +∞) → R denotes a smooth truncation function satisfying χ ≡ 1 on [0, R] and χ ≡ 0 on �min(R + 1, R′), +∞ � , then the map extR′,R : Vres-R → Vquad-R defined, for every V in Vres-R, by extR′,R(V )(u) = χ(|u|) ext∞,R(V )(u) + �1 − χ(|u|) �|u|2 2 , is a right inverse of resR,R′ and is continuous (for the topologies of uniform convergence of derivatives up to order k for the departure and arrival sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every couple (A, B) of subsets of Vres-R, let A′ = res−1 R,R′(A) and B′ = res−1 R,R′(B) denote the sets of the potentials of Vquad-R′ whose restrictions to BRd(0, R) belong to A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then the following equivalences hold: A is open in Vres-R ⇐⇒ A′ is open in Vquad-R′ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) A is dense in B ⇐⇒ A′ is dense in B′ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) A is dense in Vres-R ⇐⇒ A′ is dense in Vquad-R′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) follows from the continuity and the openness of resR,R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the surjectivity of resR,R′, the set resR,R′(A′) is equal to A and the set resR,R′(B′) is equal to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the image of a dense set by a continuous map is dense in its image, if A′ is dense in B′ then A is dense in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Implication “ ⇐= ” of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' On the other hand, if A is dense in B, then, for every open subset Ω′ of B′, its image Ω := resR,R′(Ω′) is, according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, open in B so that the intersection A ∩ Ω is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the surjectivity of resR,R′, the set res−1 R,R′(A∩Ω) is also nonempty and it is by construction included in A′ ∩ Ω′, which is a fortiori nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This proves that A′ is dense in B′ and completes the proof of equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Finally, equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) by setting B′ equal to Vquad-R′ and B equal to Vres-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 26 5 Generic transversality of travelling fronts 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Notation and statement Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall the notation Vfull and Vquad-R introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential function V in Vfull, let Σcrit(V ) and Σmin(V ) denote the set of non- degenerate critical points and of non-degenerate minimum points of V , respectively, and let us consider the set (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) FV = �(c, u) ∈ (0, +∞) × Ck+1(R, Rd) : ξ �→ u(ξ) is a global solution of the system ¨u = −c ˙u + ∇V (u) and there exists (e−, e+) in Σcrit(V ) × Σmin(V ) such that lim ξ→−∞ u(ξ) = e− and lim ξ→+∞ u(ξ) = e+ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, (c, u) is in FV if and only if c is a positive quantity and ξ �→ u(ξ) is the profile of a front travelling at speed c and connecting a non-degenerate critical point (at the left end) to a non-degenerate minimum point (at the right end), for the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us take and fix a positive quantity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The goal of this section is to prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 below, which is a weaker version of statement 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 since the potentials under consideration belong to the subspace Vquad-R and not to the full space Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The reasons for first proving the intended genericity result in this restricted setting are explained at the beginning of subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, and the extension from Vquad-R to Vfull will be carried out in the last section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a reminder, the transversality of a travelling front was defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset of Vquad-R such that, for every potential V in this subset, every travelling front (c, u) in FV is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Reduction to a local statement Let V0 denote a potential function in Vquad-R, and let e−,0 and e+,0 denote a non-degenerate critical point and a non-degenerate minimum point of V0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (or simply to the implicit function theorem), there exists a small neighbourhood νrobust(V0, e−,0, e+,0) of V0 in Vquad-R and two Ck+1-functions e−(·) and e+(·), defined on νrobust(V0, e−,0, e+,0) and with values in Rd, such that e−(V0) equals e−,0 and e+(V0) equals e+,0 and, for every V in νrobust(V0, e−,0, e+,0), both e−(V ) and e+(V ) are critical point of V close to e+,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following local generic transversality statement, which calls upon this notation, yields Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 (as shown below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive speed c0, there exist a neighbourhood νV0, e−,0, e+,0, c0 of V0 in Vquad-R, included in νrobust(V0, e−,0, e+,0), a neighbourhood CV0, e−,0, e+,0, c0 of c0 in (0, +∞), and a generic subset νV0, e−,0, e+,0, c0, gen of νV0, e−,0, e+,0, c0 such that, for every V in νV0, e−,0, e+,0, c0, gen, every front travelling at a speed c in CV0, e−,0, e+,0, c0 and connecting e−(V ) to e+(V ), for the potential V , is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 27 Proof that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 yields Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us denote by Vquad-R-Morse the dense open subset of Vquad-R defined by the Morse property (see [24]): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) Vquad-R-Morse = {V ∈ Vquad-R : all critical points of V are non-degenerate} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V0 denote a potential function in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Its critical points are non-degenerate and thus isolated and, since V0 is in Vquad-R, they belong to the open ball BRd(0, R), so that those critical points are in finite number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assume that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation of this proposition, let us consider the following three intersections, at each time over all couples (e−,0, e+,0) with e−,0 a non-degenerate critical point and e+,0 a non-degenerate minimum point of V0: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) νV0, c0 = νrobust(V0) ∩ �� νV0, e−,0, e+,0, c0 � , CV0, c0 = � CV0, e−,0, e+,0, c0 and νV0, c0, gen = νrobust(V0) ∩ �� νV0, e−,0, e+,0, c0, gen � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Those are finite intersections, so that νV0, c0 is still a neighbourhood of V0 in Vquad-R, CV0, c0 is still a neighbourhood of c0 in (0, +∞) and the set νV0, c0, gen is still a generic subset of νV0, c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let I denote a compact sub-interval of (0, +∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the three sets defined above in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) can be constructed likewise for every c0 in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since I is compact, it can be covered by a finite union of sets CV0,c0,i, corresponding to a finite set {c0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , c0,p} of speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Again the intersections νV0, I = � 1≤i≤p νV0, c0,i and νV0, I, gen = � 1≤i≤p νV0, c0,i, gen .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' are finite and thus νV0, I, gen is still a generic subset of νV0, I, which is a neighbourhood of V0 in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By construction, for every potential function V in νV0, I, gen, all fronts travelling at a speed belonging to I and connecting a critical point of V to a minimum point of V are transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, the set Vquad-R-Morse-⋔-F-I = �V ∈ Vquad-R-Morse : for every travelling front (c, u) in FV , if c is in I then (c, u) is transverse � , is locally generic in the sense that Vquad-R-Morse-⋔-F-I ∩ νV0, I is generic in νV0, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since Vquad-R is separable, applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 with V = Vquad-R, Vdense = Vquad-R-Morse, Vgen = Vquad-R-Morse-⋔-F-I and ν = νV0, I shows that the set Vquad-R-Morse-⋔-F-I is generic in the whole set Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, the set � q∈N∗ Vquad-R-Morse-⋔-F-[−1/q,q] is still generic in Vquad-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential V in this set, all travelling fronts belonging to FV are transverse, so that this set fulfils the conclusions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining part of section 5 will thus be devoted to the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Proof of the local statement (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Setting For the remaining part of this section, let us fix a potential function V0 in Vquad-R, a non-degenerate critical point e−,0 of V0 and a non-degenerate minimum point e+,0 of V0, differing from e−,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there exist a neighbourhood ν of V0 in Vquad-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' included in νrobust(V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' a neighbourhood C of c0 in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' +∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and a positive quantity r such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every (c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V ) in C × ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there exist Ck+1-functions ˆwu loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V : Bu E−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0(r) → R2d and ˆws loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V : Bs E+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0(r) → R2d such that the sets W u loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V �E−(V ) � = ˆwu loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V �Bu E−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0(r) � and W s loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V �E+(V ) � = ˆws loc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' V �Bs E+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0(r) � define a local unstable manifold of E−(V ) and a local stable manifold of E+(V ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' respec- tively (see the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 and equalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Here is the setting where Sard–Smale theorem (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) will be applied (see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let W u loc, c, V (E−(V )) freedom when bu moves freedom when ξ moves ˆwu loc, c, V (bu) Bu bu Φu(bu, ξ, c, V ) freedom when c moves ˆws loc, c, V (bs) E+(V ) Φs(bs, c, V ) bs E−(V ) flow Sc,V (ξ, ·) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: The function ˆwu loc, c, V (·) maps Bu onto the boundary of the local unstable manifold W u loc, c, V �E−(V ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A point ˆwu loc, c, V (bu) of this boundary is pushed forward during a time ξ by the flow Sc,V (ξ, ·) to give the image Φu(bu), which still belongs to the global unstable manifold of E−(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' On the other hand, Φs maps Bs onto the boundary of the local stable manifold W s loc, c, V (E+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The dependence of Φu on the time ξ and the point bu provides a number of degrees of freedom equal to the dimension of the unstable manifold, while an additional degree of freedom is provided by the speed c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This additional dependence makes the difference between the transversality of a travelling front as defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 and the classical transversality of stable and unstable manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Bu = ∂Bu E−,0(r) , Bs = ∂Bs E+,0(r) , M = Bu × Bs × R × C , Λ = ν , N = (R2d)2 , and W = {(A, B) ∈ N : A = B} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notice that W is the diagonal of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the functions Φu : Bu × R × C × Λ → R2d , (bu, ξ, c, V ) �−→ Sc,V �ξ, ˆwu loc, c, V (bu) � and Φs : Bs × C × Λ → R2d , (bs, c, V ) �−→ ˆws loc, c, V (bs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 29 For every V in Λ and c in C, the image of Φu(·, ·, c, V ) is the global unstable manifold W u c,V �E−(V ) � (except the point E−(V ) itself), whereas the image of Φs(·, c, V ) is the boundary of the local stable manifold W s loc, c, V �E+(V ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Finally, let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) Φ : M × Λ → N , (m, V ) = �(bu, bs, ξ, c), V � �→ �Φu(bu, ξ, c, V ), Φs(bs, c, V ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Additional conditions on ν and r The main step in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is the construction of a suitable perturbation W of V (carried out in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This construction requires more accurate conditions on the setting above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' First, since e−,0 and e+,0 differ, we may assume that ν and C and r are small enough so that, for every V in ν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) πpos � W u loc, c, V �E−(V ) �� ∩ πpos � W s loc, c, V �E+(V ) �� = ∅ , where πpos is the projection on the first component defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Next, the following lemma is a more uniform version of assertion 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, the key difference being that r can be chosen small enough such that Ionce contains positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Up to replacing ν by a smaller neighbourhood of V0 in Vquad-R, and C by a smaller neighbourhood of c0 in (0, +∞), and r by a smaller radius, we may assume that the following conclusions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every V in ν, every c in C, and every solution ξ �→ U(ξ) = �u(ξ), ˙u(ξ) � of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) such that U(0) belongs to the boundary of W u loc, c, V �E−(V ) � (in other words there exists bu in Bu such that U(0) equals ˆwu loc, c, V (bu)), there exists a a compact interval with nonempty interior Ionce, included in (0, +∞), such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' the function ξ �→ |u(ξ) − e−(V )| is increasing on Ionce (so that u|Ionce is a diffeo- morphism onto its image), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and for all ξ∗ in Ionce and ξ in R, u(ξ) = u(ξ∗) implies ξ = ξ∗, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) holds, and in addition, u(Ionce) ∩ πpos � W u loc, c, V �E−(V ) � ∪ W s loc, c, V �E+(V ) �� = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Consider for now that ν and C and r are as in the previous subsection, and, for some bu in Bu, let us consider the solution ξ �→ U(ξ) = �u(ξ), ˙u(ξ) � of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) defined as: U(ξ) = Sc0,V0 �ξ, ˆwu loc, c0, V0(bu) � (so that U(0) = ˆwu loc, c0, V0(bu)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same arguments as in the proof of statement 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 yield the following conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' First, there exists a (large, negative) time ξ0(bu) such that the function ξ �→ 30 |u(ξ) − e−,0| is increasing on �−∞, ξ0(bu) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, there exists ξonce(bu) in �−∞, ξ0(bu) � such that, for every ξ∗ in �−∞, ξonce(bu) �, HV � U �ξ0(bu) �� < −V �u(ξ∗) � (which is nothing but inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, it follows from statement 1 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 that, for the interval Ionce equal to �ξonce(bu) − 2, ξonce(bu) − 1 �, conclusions 1 and 2 of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 hold for the solution U (and they still hold if ξonce(bu) is replaced by a smaller quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, observe that, due to the smooth dependence of the map (−∞, 0] → R2d, ξ �→ Sc,V �ξ, ˆwu loc, c, V (bu) � on V and c and bu, this construction can be made uniform with respect to bu in a (small) open subset Ω of Bu and V in a (small) neighbourhood νΩ (included in ν) of V0, and to c in a (small) neighbourhood CΩ (included in C) of c0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' in other words, there exists a (sufficiently large negative) quantity ξonce(Ω) such that the conclusions above hold for all such V and c and bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since Bu is compact, it can be covered by a finite number Ω1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Ωn of such open subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, replacing ν by n � i=1 νΩi and C by n � i=1 CΩi , and choosing ξonce = min i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=',n} ξonce(Ωi) and Ionce = [ξonce − 2, ξonce − 1] , conclusions 1 and 2 of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Up to replacing r by a smaller positive quantity, we may assume in addition that Ionce belongs to (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Finally, again up to replacing r by a smaller positive quantity, we may assume that conclusion 3 also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Equivalent characterizations of transversality Let us consider the set FΛ, C = �(V, c, u) : V ∈ Λ and c ∈ C and u is the profile of a front travelling at speed c and connecting e−(V ) to e+(V ), for the potential V � , and let us denote by �FΛ, C the set of equivalence classes of FΛ, C for the equivalence relation: (V, c, u) ∼ (V †, c†, u†) if and only if V = V † and c = c† and u = u† up to a translation of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The aim of this subsection is to prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 below, relating the transversality of the intersection Φ(M × Λ) ∩ W to the transversality of travelling fronts belonging to FΛ, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To begin with, the next Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 formalizes the correspondence between the intersection of the image of Φ with the diagonal W and the profiles of such travelling fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) Φ−1(W) → FΛ, C , (bu, bs, ξ, c, V ) �→ � V, c, ξ′ �→ πpos � Sc,V �ξ′, ˆwu loc, c, V (bu) ��� defines a a one-to-one correspondence between Φ−1(W) and the quotient set �FΛ, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 31 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The image by Φ of a point (bu, bs, ξ, c, V ) of M × Λ belongs to the diagonal W of N if and only if Φu(bu, ξ, c, V ) = Φs(bs, c, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If this last equality holds, the function u : ξ′ �→ Φu(bu, ξ′, c, V ) is a solution belonging to the unstable manifold W u c,V �E−(V ) � such that u(ξ) = Φs(bs, c, V ) belongs to the local stable manifold of E+(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus u defines the profile of a front travelling at speed c and connecting e−(V ) to e+(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) is thus well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, if ξ �→ u(ξ) is the profile of a front travelling at a speed c in C for the potential V and connecting e−(V ) to e+(V ), then, according to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the image of ξ �→ �u(ξ), ˙u(ξ) � belongs to the intersection W u c,V �E−(V ) � ∩ W s c,V �E+(V ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a con- sequence, this image must cross the boundary of W u loc, c, V �E−(V ) � at a time ξ− and the boundary of W s loc, c, V �E+(V ) � at a time ξ+: there exists bu in Bu and bs in Bs such that �u(ξ−), ˙u(ξ−) � = ˆwu loc, c, V (bu) and �u(ξ+), ˙u(ξ+) � = ˆws loc, c, V (bs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By construction, Φu(bu, ξ+ − ξ−, c, V ) = Φs(bs, c, V ) and thus Φ(bu, bs, ξ+ − ξ−, c, V )) is in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, according to the remark at the end of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the times ξ− and ξ+ at which these intersections occur are unique (for a given profile ξ �→ u(ξ)), thus so are the points bu in Bu and bs in Bs and the time lag ξ+ − ξ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This completes the proof of this one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Both corresponding notions of transversality are related as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential function V in Λ, the following two statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The image of the function M → N, m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Every profile ξ �→ u(ξ) of a front travelling at a speed c in C and connecting e−(V ) to e+(V ), for the potential V , is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us take (m1, V1) in M × Λ such that Φ(m1, V1) is in W, let (bu 1, bs 1, ξ1, c1) denote the point m1 and let ξ �→ u1(ξ) denote the profile of the corresponding travelling front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, for all ξ in R , U1(ξ) = Φu(bu 1, ξ, c1, V1) , where U1(ξ) = �u1(ξ), ˙u1(ξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the maps ΓΦ : (Bu × R × C) × (Bs × C) → R × R2d �(bu, ξ, cu), (bs, cs) � �→ �cu, Φu(bu, ξ, cu, V1) � + �cs, Φs(bs, cs, V1) � , and ∆Φ : M → R2d , (bu, bs, ξ, c) �→ Φu(bu, ξ, c, V1) − Φs(bs, c, V1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and let us write, only for this proof, DΦ for DTm1MΦ, and similarly DΦu and DΦs and D(ΓΦ) and D(∆Φ) for the differentials of Φu and Φs and ΓΦ and ∆Φ at (m1, V1) and with respect to all variables but V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following three statements are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 32 (A) The image of DΦ contains a supplementary subspace of the diagonal W of (R2d)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (B) The map D(ΓΦ) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (C) The map D(∆Φ) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If statement (A) holds, then, for every (α, β) in (R2d)2, there exist γ in R2d and δm in Tm1M such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) (γ, γ) + DΦ · δm = (α, β) , so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) D(∆Φ) · δm = α − β , and statement (C) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conversely, if statement (C) holds, then, for every (α, β) in (R2d)2, there exists δm in Tm1M such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) holds, and as a consequence, if (δbu, δbs, δξ, δc) denotes the components of δm, the vector α − DΦu(δbu, δξ, δc) is equal to β − DΦs(δbs, δc), and if this vector is denoted by γ, then equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) holds, and this shows that statement (A) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus statements (A) and (C) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now, if statement (B) holds, then, for every (δc, δU) in R×R2d, there exist (δbu, δξ, δcu) in Tbu 1Bu × R2 and (δbs, δcs) in Tbs 1Bs × R such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) (δc, δU) = �δcu, DΦu · (δbu, δξ, δcu) � + �δcs, DΦs · (δbs, δcs) � , so that δc is equal to δcu + δcs and so that δU = DΦu · (δbu, δξ, δcu) + DΦs · (δbs, δc − δcu) = DΦu · (δbu, δξ, δcu) + DΦs · (0, δc) − DΦs · (−δbs, δcu) , so that finally, if (δbu, −δbs, δξ, δcu) is denoted by δm, then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) δU = D(∆Φ) · δm + DΦs · (0, δc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By choosing δc equal to 0, this shows that every δU in R2d is in the image of D(∆Φ), which is statement (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conversely, if statement (C) holds, then for every (δc, δU) in R × R2d, there exists δm in Tm1M such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) holds, and if δcu denotes the last component of δm and δcs is the difference δc − δcu, then equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) holds, and this shows that statement (B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus statements (B) and (C) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Continuation of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To conclude, let us see how both transver- sality statements 1 and 2 can be expressed in terms of the ingredients of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' On the one hand, according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, the travelling front with profile u1(·) and speed c1 is transverse if and only if the intersection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) � � cu>0 {cu} × W u cu,V �E−(V ) � � ∩ � � cs>0 {cs} × W s cs,V �E+(V ) � � 33 is transverse, in R2d+1, along the set {c1} × U1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This transversality can be considered at a single point, no matter which, of the trajectory U0(R), thus in particular at the point Φu(bu 1, ξ1, c1, V1), which is equal to Φs(bs 1, c1, V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By definition, the sum of the tangent spaces associated to the manifolds intersected in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) is the image of D(ΓΦ) and the transversality stated in statement 2 is therefore equivalent to the surjectivity of the map D(ΓΦ) (statement (B) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' On the other hand, the image of the function M → (R2d)2, m �→ Φ(m, V1) is transverse at Φ(m1, V1) to the diagonal W of (R2d)2 as stated in 1 if and only if the image of DΦ contains a supplementary subspace of the diagonal (statement (A) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 follows from the conclusion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 applied to the function Φ (see subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The next two subsections are devoted to checking that this function Φ fulfils hypotheses 1 and 2 of this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Since the vector field (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) defining the flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) is of class Ck, so is the function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 that dim(Bu) = d − m(e−,0) − 1 and dim(Bs) = d − 1 , thus dim(M) = 2d − m(e−,0) , and since the codimension of W in N is equal to 2d, dim(M) − codim(W) = −m(e−,0) ≤ 0 , thus k > dim(M) − codim(W) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' in other words, hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Take (m1, V1) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (bu 1, bs 1, ξ1, c1) denote the components of m1, and, for every real quantity ξ, let us write U1(ξ) = �u1(ξ), v1(ξ) � = Sc1,V1 �ξ, ˆwu loc, c1, V1(bu 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The function ξ �→ u1(ξ) is the profile of a front travelling at speed c1 and connecting e−(V1) to e+(V1) for the potential V1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and, according to the empty inclusion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5), the quantity ξ1 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us write DΦ , DΦu and DΦs for the full differentials (with respect to arguments m in M and V in Λ) of the three functions Φ and Φu and Φs respectively at the points �bu 1, bs 1, ξ1, c1, V1 �, �bu 1, ξ1, c1, V1 � and �bs 1, c1, V1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 amounts to prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) im(DΦ) + TW = TN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 34 To this end, since the subspace R2d × {0R2d} of N is transverse to the diagonal W, it is sufficient to prove that, for every γ in R2d, the vector �γ, 0R2d � can be reached by DΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, it is sufficient to prove that, for every γ in R2d, there exist a real quantity ζ and a function W in Ck+1 b (Rd, R) with a compact support supp(W) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) supp(W) ⊂ BRd(0, R) , such that DΦu · (0, ζ, 0, W) = γ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='14) and DΦs · (0, 0, W) = 0R2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='15) To fulfil equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='15), it is sufficient to assume that W satisfies the following additional condition: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) supp(W) ∩ πpos � W s loc, c, V �E+(V1) �� = ∅ , where πpos : R2d → Rd is the projection on the first component defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) (this condition ensures that the local stable manifold of E+(V1) is not changed by a perturbation of V1 in the direction of W, see the second remark at the end of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For convenience, we will also ensure that the same is true for the local unstable manifold of E−(V1), that is: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) supp(W) ∩ πpos � W u loc, c, V �E−(V1) �� = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Fulfilling equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='14) amounts to prove that the orthogonal complement of the subspace of the directions of R2d that can be reached by DΦu · (0, ζ, 0, W) is trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' reduced to {0R2d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that DΦu · (0, ζ, 0, 0) = ζ ˙U1(ξ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus the transversality statement (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) is a consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (perturbation of the potential reaching a given direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every nonzero vector (φ1, ψ1) in R2d which is orthogonal to ˙U1(ξ1), there exists W in Ck+1 b (Rd, R) satisfying conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) and the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18) ⟨DΦu · (0, 0, 0, W) | (φ1, ψ1)⟩ ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (φ1, ψ1) denote a nonzero vector orthogonal to U1(ξ1) in R2d, and let W be a function in Ck+1 b (Rd, R) satisfying the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the linearization of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), for the potential V1 and the speed c1, around the solution ξ �→ U1(ξ): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='19) d dξ � δu(ξ) δv(ξ) � = � 0 id D2V1 �u1(ξ) � −c1 � � δu(ξ) δv(ξ) � , and let T(ξ, ξ′) denote the family of evolution operators obtained by integrating this linearized differential system between times ξ and ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) that 35 W only affects the part of Φu corresponding to the flow (not on the function ˆwu loc, c1, V1) and the variation of constants formula yields that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='20) DΦu · (0, 0, 0, W) = � ξ1 0 T(ξ, ξ1) � 0, ∇W �u1(ξ) �� dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every time ξ, let T ∗(ξ, ξ1) denote the adjoint operator of T(ξ, ξ1), and let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='21) �φ(ξ), ψ(ξ) � = T ∗(ξ, ξ1) · (φ1, ψ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='20), inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18) reads � ξ1 0 �� 0, ∇W �u1(ξ) �� ��� T ∗(ξ, ξ1) · (φ1, ψ1) � dξ ̸= 0 , or equivalently (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) � ξ1 0 ∇W �u1(ξ) � · ψ(ξ) dξ ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notice that, due to the expression of the linearized differential system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='19), (φ, ψ) is a solution of the adjoint linearized system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='23) � ˙φ(ξ) ˙ψ(ξ) � = − � 0 D2V1 �u1(ξ) � id −c1 � � φ(ξ) ψ(ξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Our task is thus to construct a function W in Ck+1 b (Rd, R) satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There are two difficulties to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' First, as shown by Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, the function ξ �→ u1(ξ) may reach the same value for different values of the argument ξ, making it difficult to handle the interactions of the contributions to the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) of the perturbation W �u1(ξ) � at these different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Second, the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) depends on the gradient ∇W of the perturbation W and not on W itself, and this gradient cannot be any function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' These difficulties have already been tackled in several contexts, see [37, 38, 45] (ordinary differential equations) and [8–10, 26, 27] (partial differential equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Each time, some specific arguments have to be found, using the peculiarities and constraints of the considered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In the present case, the following trick will do the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, there exists a closed interval with nonempty interior Ionce, included in (0, +∞), such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='24) u1(Ionce) ∩ πpos � W u loc, c, V �E−(V1) � ∪ W s loc, c, V �E+(V1) �� = ∅ , such that ˙u does not vanish on Ionce, and such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='25) for all ξ∗ in Ionce and ξ in R , u1(ξ) = u1(ξ∗) =⇒ ξ = ξ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 36 According to the empty intersection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='24) and since u1(ξ) is in W s loc, c, V �E+(V1) � for ξ larger than ξ1, the interval Ionce is actually included in (0, ξ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='25), the image u1(Ionce) of this interval provides a suitable place where the trajectory can be perturbed without the inconvenience 1 emphasized above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Two cases have to be considered (plus a third one that will turn out to be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a time ξ∗ in Ionce such that ψ(ξ∗) is not collinear to ˙u1(ξ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this case, up to an affine conformal change of coordinate system in Rd, we may assume that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='26) u1(ξ∗) = 0 and ˙u1(ξ∗) = ϵ1 and ϵ2 · ψ(ξ∗) ̸= 0 , where ϵ1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , 0) and ϵ2 = (0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , 0) are the two first vectors of the canonical basis of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ρ denote an even function in Ck+1�R, [0, 1] � satisfying ρ(0) = 1 and ρ vanishes on R \\ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ε denote a small positive quantity to be chosen later and let us consider the bump function ρε : Rd → [0, 1], u �→ ρ �|u| ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from this definition that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='27) ρε(0Rd) = 1 and supp(ρε) ⊂ BRd(0, ε) and ∥∇ρε∥L∞(Rd,R)d ∈ Oε→0(ε−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us define the perturbation W as follows: for every u in Rd, W(u) = ρε(u)(ϵ2 · u) , see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, so that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='28) ∇W(u) = ρε(u)ϵ2 + (ϵ2 · u)∇ρε(u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' ϵ1 ϵ2 u1(ξ) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2: Graph of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from this definition that, if ε is small enough, then, on the one hand conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) (according to inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) (according to the empty intersection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='24)) are fulfilled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and, on the other hand, according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='25) and since ˙u1(ξ∗) is nonzero, there exists an open interval I∗ ε of R satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='29) ξ∗ ∈ I∗ ε and, for every ξ in R, u1(ξ) ∈ BRd(0, ε) ⇐⇒ ξ ∈ I∗ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us assume that ε is chosen as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='29) that the integral in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) reduces to: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='30) � I∗ε ∇W �u1(ξ) � · ψ(ξ) dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 37 As a consequence, if u1(ξ) follows a straight line in the direction of ϵ1 inside the ball BRd(0, ε), then, for every ξ in I∗ ε , ∇W �u1(ξ) � = ρε �u1(ξ) �ϵ2 , so that the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='30) reduces to � I∗ε ρε �u1(ξ) �ϵ2 · ψ(ξ) dξ , and according to the last property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='26), if ε is sufficiently small then this integral does not vanish, fulfilling inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) — and thus also (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In the general situation where u1(ξ) does not necessarily follow a straight line in the direction of ϵ1 inside the ball BRd(0, ε), the quantity ϵ2 · u1(ξ) is in Oε→0(ε2) when ξ is in I∗ ε , thus it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='28) and from the last property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='27) that, still for ξ in I∗ ε , ∇W �u1(ξ) � = ρε �u1(ξ) �ϵ2 + Oε→0(ε) , and since ρε(0Rd) equals 1, it follows from the last property of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='26) that, if ε is sufficiently small, then inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) is fulfilled again — thus so is inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If case 1 does not occur, then ψ(ξ) is collinear to ˙u1(ξ) for every ξ in Ionce, and since ˙u1(·) does not vanish on Ionce, there exists a C1-function α : Ionce → R such that, for every ξ in Ionce, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='31) ψ(ξ) = α(ξ) ˙u1(ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The next cases 2 and 3 differ according to whether the function α(·) is constant or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in Ionce, equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='31) holds for some nonconstant function α(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every perturbation W of the potential, if the support of W is localized enough around some point of u(Ionce) (so that expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='31) holds as soon as ∇W �u(ξ) � is nonzero), then an integration by parts shows that the integral in inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) becomes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='32) � ∇W �u1(ξ) � · ψ(ξ) dξ = � α(ξ)∇W �u1(ξ) � · ˙u1(ξ) dξ = − � ˙α(ξ)W �u1(ξ) � dξ (with integration domain [0, ξ1] for each of these integrals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The expression of this last integral shows why the assumption (made in the present case 2) that α(·) is nonconstant matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to this assumption, there exists ξ∗ in Ionce such that ˙α(ξ∗) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us assume (up to an affine change of variable in R2d) that u1(ξ∗) is equal to 0Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us define BRd(0, ε) and ρε and I∗ ε as in case 1 above, and let us simply define the perturbation W as W = ρε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 38 As in case 1, for ε sufficiently small, conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='17) are fulfilled, and the integral in inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) reduces to the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='32), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) thus becomes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='33) � I∗ε ˙α(ξ)W �u1(ξ) � dξ ̸= 0 , which is fulfilled if ε sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows that inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) is fulfilled, and thus so is inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in Ionce, ψ(ξ) = α ˙u(ξ), for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this case, expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='32) shows that inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) cannot hold if the support of W is localized around some point of u(Ionce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Fortunately, this third case will lead to a contradiction (and does therefore actually not happen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Recall that (φ, ψ) is a solution of the adjoint linearized system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, for every ξ in Ionce, it follows from the assumption made in this case 3 that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='34) φ(ξ) = cψ(ξ) − ˙ψ(ξ) = cα ˙u1(ξ) − α¨u1(ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Besides, recall that (φ1, ψ1) is orthogonal to ˙U1(ξ1) = T(ξ, ξ1) ˙U1(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, �φ(ξ), ψ(ξ) � = T ∗(ξ, ξ1) (φ1, ψ1) is orthogonal to ˙U1(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the expression of ψ and expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='34), this last property reads (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='35) cα| ˙u1|2(ξ) − α¨u1(ξ) · ˙u1(ξ) + α ˙u1(ξ) · ¨u1(ξ) = 0 , which yields cα| ˙u1|2(ξ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since ˙u1 does not vanish on (−∞, ξonce), the quantity α must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This yields φ ≡ ψ ≡ 0, and contradicts the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In short, case 3 cannot happen and, in both cases 1 and 2, a suitable construction provides a function W in Ck+1 b (Rd, R) fulfilling the conditions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='16) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 Conclusion Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As seen in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is fulfilled for the function Φ defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the conclusion of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 yields equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12), hypothesis 2 of this theorem is also fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The conclusion of this theorem ensures that there exists a generic subset Λgen of Λ such that, for every V in Λgen, the function Φ(·, V ) is transverse to the diagonal W of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, it follows that, for every V in Λgen, every profile ξ �→ u(ξ) of a front travelling at a speed c in C and connecting e−(V ) to e+(V ), for the potential V , is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words the conclusions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 hold with CV0, e−,0, e+,0, c0 = C, νV0, e−,0, e+,0, c0 = ν = Λ and νV0, e−,0, e+,0, c0, gen = Λgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As shown in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 39 6 Generic elementarity of symmetric standing pulses This section presents strong similarities with the previous section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For that reason, the presentation aims at emphasizing the main differences, while some details or comments are omitted when they are identical to some already provided in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Notation and statements Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential function V in Vfull, let us recall (subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) that Σcrit(V ) denotes the set of non-degenerate critical points of V , and let us consider the set (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) PV = �u ∈ Ck+1(R, Rd) : ξ �→ u(ξ) is a global solution of the system ¨u = ∇V (u) , and there exists e in Σcrit(V ) such that u(ξ) → e as ξ → ±∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, u is in PV if and only if ξ �→ u(ξ) is the profile of a standing pulse connecting a non-degenerate critical point e to itself, for the potential V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us take and fix a positive quantity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall that the elementarity of a symmetric standing pulse was defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The goal of this section is to prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Vquad-R such that, for every potential V in this subset, every symmetric standing pulse in PV is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V0 denote a potential function in Vquad-R, and let e0 denote a non-degenerate critical point of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 (or simply to the implicit function theorem), there exists a small neighbourhood νrobust(V0, e0) of V0 in Vquad-R and a Ck+1-function e(·) defined on νrobust(V0, e0) and with values in Rd, such that e(V0) equals e0 and, for every V in νrobust(V0, e0), e(V ) is a critical point of V0 close to e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Exactly the same arguments as in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 show that Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is a consequence of the following local statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a neighbourhood νV0, e0 of V0 in Vquad-R, included in νrobust(V0, e0), and a generic subset νV0, e0, gen of νV0, e0 such that, for every V in νV0, e0, gen, every symmetric standing front connecting e(V ) to itself is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining part of section 6 will thus be devoted to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us keep the notation V0 and e0 and νrobust(V0, e0) introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, there exist a neighbourhood ν of V0 in Vquad-R, included in νrobust(V0, e0), and a positive quantity r such that, for every V in ν, there exist Ck-functions ˆwu loc, V : Bu E0(r) → R2d and ˆws loc, V : Bs E0(r) → R2d such that the sets W u loc, V �E(V ) � = ˆwu loc, V �Bu E0(r) � and W s loc, V �E(V ) � = ˆws loc, V �Bs E0(r) � 40 define a local unstable manifold and a local stable manifold of E(V ), respectively (see the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 and equalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that the departure sets Bu E0(r) of ˆwu loc, V and Bs E0(r) of ˆws loc, V do not depend on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let Bu = ∂Bu E0(r) and Bs = ∂Bs E0(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) of the eigenvectors of the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), Eu V0(E0) ∩ Ssym = {0R2d} and Es V0(E0) ∩ Ssym = {0R2d} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows that, up to replacing ν by a smaller neighbourhood of V0 in Vquad-R and r by a smaller positive quantity, for every V in ν, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) W u loc, V �E(V ) � ∩ Ssym = {E(V )} and W s loc, V �E(V ) � ∩ Ssym = {E(V )} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 The setting to which Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 will be applied is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let M = Bu × R , Λ = ν , N = R2d and W = Ssym , and let us consider the function (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) Φ : M × Λ → N, (bu, ξ, V ) �→ SV �ξ, ˆwu loc, V (bu) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If the conclusion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 holds within this setting, then there exists a generic subset Λgen of Λ such that, for every V in Λgen, the image of the function M → N, m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For a given potential V , the image of m �→ Φ(m, V ) is nothing but the unstable manifold of E(V ) (deprived of E), see the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the characterizations of the symmetric standing pulses stated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the intersection of Φ(M, V ) with W = Ssym actually corresponds to the set of symmetric standing pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Moreover, by definition (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6), the elementarity of the symmetric standing pulses for V is equivalent to the transversality of the intersection of Φ(M, V ) with W = Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, the conclusion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 directly implies Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 with νV0, e0 = ν = Λ and νV0, e0, gen = Λgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It remains to show that, in the setting above, the hypotheses of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 It follows from subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 that dim(Bu) = d − m(e0) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Hence, dim(M) − codim(W) = �d − m(e0) � − d = −m(e0) , which is less than the positive integer k (the regularity of Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is thus fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Take (m1, V1) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (bu 1, ξturn) denote the components of m1, and, for every real quantity ξ, let us write U1(ξ) = �u1(ξ), v1(ξ) � = SV1 �ξ, ˆwu loc, V1(bu 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The function ξ �→ u1(ξ) is the profile of a symmetric standing pulse, connecting e(V1) to itself for the potential V1, and the quantity ξturn is the turning time of this standing pulse (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, according to the first equality of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), this turning time ξturn must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let DΦ denote the full differential (with respect to m and V ) of Φ at the point �(bu 1, ξturn), V1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 follows from the next Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 (perturbation of the potential reaching a given direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every nonzero vector ψ1 in Rd, there exists W in Ck+1 b (Rd, R) such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) ⟨DΦ · (0, W) | (0, ψ1)⟩ ̸= 0 , and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) supp(W) ⊂ BRd(0, R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The proof is similar to that of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ψ1 be a nonzero vector in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , and let W denote a function in Ck+1 b (Rd, R) with a support satisfying the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) supp(W) ∩ πpos � W u loc, V �E(V1) �� = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us again use the notation T(ξ, ξ′) to denote the family of evolution operators obtained by integrating the linearized differential system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='19) (for c1 equal to 0) between the times ξ and ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from the empty intersection (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) DΦ · (0, W) = � ξturn 0 T(ξ, ξturn) � 0, ∇W �u1(ξ) �� dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every time ξ, let T ∗(ξ, ξturn) denote the adjoint operator of T(ξ, ξturn), and let �φ(ξ), ψ(ξ) � = T ∗(ξ, ξturn) · (0, ψ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7), condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) reads � ξturn 0 �� 0, ∇W �u1(ξ) �� ��� T ∗(ξ, ξturn) · (0, ψ1) � dξ ̸= 0 , or equivalently (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) � ξturn 0 ∇W �u1(ξ) � · ψ(ξ) dξ ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 42 According to the first equality of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and due to the Hamiltonian invariance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5), for every (u, v) in W u loc, V1 �E(V1) � and differing from E(V1), the quantity V1(u) is larger than V1 �e(V1) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' On the other hand, since ˙u1(ξturn) vanishes the quantity V1 �u1(ξturn) � is equal to V1 �e(V1) �, so that u1(ξturn) does not belong to the (closed) set πpos � W u loc, V1 �E(V1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, there exists a time ξ−, smaller than (and sufficiently close to) ξturn, such that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) u1 �[ξ−, ξturn] � ∩ πpos � W u loc, V1 �E(V1) �� = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the function ξ �→ ˙u1(ξ) does not vanish on (−∞, ξturn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, three cases have to be considered for the construction of the perturbation W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a time ξ† in (ξ−, ξturn) such that ψ(ξ†) is not collinear to ˙u1(ξ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this case, conclusion 3 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 provides an open interval Ionce included in (ξ−, ξturn) and small enough so that, for every ξ∗ in Ionce, the vector ψ(ξ∗) is not collinear to ˙u1(ξ∗), and for every ξ in (−∞, ξturn), if u1(ξ) equals u1(ξ∗) then ξ equals ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same construction as in case 1 of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 can then be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It leads to a perturbation W such that supp(W) is localized around a point of u(Ionce) (so that, according to inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3), inclusion (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) holds and according to the empty intersection (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) the empty intersection (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) holds) and such that inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) holds — thus so does inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in (ξ−, ξturn), ψ(ξ) = α(ξ) ˙u1(ξ) with α(·) not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Again, conclusion 3 of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 provides an open interval Ionce included in (ξ−, ξturn), small enough so that, for every ξ∗ in Ionce, ψ(ξ∗) = α(ξ∗) ˙u1(ξ∗), and ˙α(ξ∗) ̸= 0, and for every ξ in (−∞, ξturn), if u1(ξ) equals u1(ξ∗) then ξ equals ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same construction as in case 2 of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 can then be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in (ξ−, ξturn), ψ(ξ) = α ˙u1(ξ), for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In case 3 of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, the non-nullity of c was mandatory to take advantage of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, a new ad hoc argument is now required to preclude the possibility of the present case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Here it is: since ˙u1(ξturn) = 0, it follows from the assumption made in this case that ψ(ξ) goes to 0 as ξ goes to ξturn, so that ψ1 vanishes, contradicting the assumptions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In short, case 3 cannot occur and in both other cases, a suitable perturbation W of the potential can be constructed by following the constructions introduced in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 43 7 Generic transversality of asymmetric standing pulses As in the previous section, the proofs of this section present strong similarities with the ones which have been already detailed and the presentation will only emphasize the main differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Notation and statements The same notation as in the previous section 6 will be used all along the present section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us take and fix a positive quantity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The goal of this section is to prove the following proposition (the transversality of a standing pulse was defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Vquad-R such that, for every potential V in this subset, every asymmetric standing pulse in PV is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V0 denote a potential function in Vquad-R, and let e0 denote a non-degenerate critical point of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As already stated in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, there exists a small neighbourhood νrobust(V0, e0) of V0 in Vquad-R and a Ck+1-function e(·) defined on νrobust(V0, e0) and with values in Rd, such that e(V0) equals e0 and, for every V in νrobust(V0, e0), e(V ) is a critical point of V0 close to e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Exactly the same arguments as in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 show that Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is a consequence of the following local statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a neighbourhood νV0, e0 of V0 in Vquad-R, included in νrobust(V0, e0), and a generic subset νV0, e0, gen of νV0, e0 such that, for every V in νV0, e0, gen, every asymmetric standing front connecting e(V ) to itself is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining part of section 7 will thus be devoted to the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the same setting as in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 for local stable and unstable manifolds of E(V ), for V in a small enough neighbourhood ν of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In particular, let us assume that local stable and unstable manifolds are small enough so that equalities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, according to the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) of the eigenvectors of the linear system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), Eu V0(E0) ∩ �{0Rd × Rd} � = {0R2d} and Es V0(E0) ∩ �{0Rd × Rd} � = {0R2d} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows that there exists a positive quantity rexit such that, for every U in W u V0(E0) differing from E0, sup ξ∈R ��πpos �SV0(ξ, U) � − e0 �� > rexit ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' in other words, if a solution ξ �→ U(ξ) = �u(ξ), ˙u(ξ) � (for the potential V0) is homoclinic to E0 then u(ξ) must leave the ball BRd(e0, rexit) before eventually returning into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Up to replacing ν by a smaller neighbourhood of V0 in Vquad-R and rexit by a smaller positive quantity, we may assume that, for every V in ν and for every U in W u V �E(V ) � differing from E(V ), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) sup ξ∈R ��πpos �SV (ξ, U) � − e(V ) �� > rexit .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 44 Finally, up to replacing ν by a smaller neighbourhood of V0 in Vquad-R and r by a smaller positive quantity, we may assume that, for every V in ν, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) πpos � W u loc, V �E(V ) � ∪ W s loc, V �E(V ) �� ⊂ BRd �e(V ), rexit/4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Asymmetric standing pulses of bounded length and away from Ssym By comparison with symmetric standing pulses considered in section 6, dealing with asymmetric standing pulses is less straightforward for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Symmetric and asymmetric standing pulses connecting a given critical point to itself may coexist for some potentials, and while symmetric standing pulses will be proved to be generically elementary (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6), only asymmetric standing pulses will be proved to be generically transverse, see subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 to prove the generic transversality of asymmetric standing pulses requires to exclude, by a way or another, symmetric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The transversality of a standing pulse stated in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 is a transversality inside the submanifold corresponding to the level set of the Hamiltonian for the energy −V (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This submanifold depends on V and a direct application of Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is not possible because its transversality is stated inside a fixed manifold N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A simple solution to skip this dependence is to fix V close to e0, but with the consequence that the considered set of potentials V will not be open, so that applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 in this framework will provide local density but not local genericity of the potentials for which asymmetric pulses are transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Local genericity will actually be obtained through a countable intersection of open and dense sets, with separate proofs for their openness and their density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every V in ν and for every non negative quantity ¯ξ, let us consider the set (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) W u V �E(V ), ¯ξ � = SV �¯ξ, W u loc, V �E(V ) �� = � U∈W u loc, V � E(V ) � SV (¯ξ, U) = {E(V )} ∪ � bu∈Bu, ξ∈(−∞,¯ξ] SV �ξ, ˆwu loc, V (bu) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to this notation, the set W u V �E(V ), 0 � reduces to W u loc, V �E(V ) � and the set W u V �E(V ), ¯ξ � increases (for inclusion) with ¯ξ and represents (in some sense) the unstable manifold of the equilibrium E(V ) “until time ¯ξ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For all positive quantities ¯ξ and ε, let us consider the set ν⋔ asym stand pulses(¯ξ, ε) = � V ∈ ν : if U0 ∈ W u V �E(V ), ¯ξ � ∩ ∂W s loc, V �E(V ) � and if dist � SV (R, U0) \\ � W u loc, V �E(V ) � ∪ W s loc, V �E(V ) �� , Ssym � ≥ ε , then the (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) corresponding standing pulse: R → Rd, ξ �→ πpos �SV (ξ, U0) � is transverse � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 45 In other words, a potential function V belonging to ν is in ν⋔ asym stand pulses(¯ξ, ε) if every standing pulse connecting W u loc, V �E(V ) � to W s loc, V �E(V ) � in a time not larger than ¯ξ while remaining at a distance not smaller than ε from Ssym, is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, according to equalities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), such a standing pulse is necessarily asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 follows from the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For all positive quantities ¯ξ and ε, the set ν⋔ asym stand pulses(¯ξ, ε) is open and dense in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof that Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 yields Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 that the set (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) � N∈N ν⋔ asym stand pulses(N, 1/N) is a generic subset of ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' And, according to the definition of ν⋔ asym stand pulses(·, ·), for every potential V in this set, every asymmetric standing pulse connecting e(V ) to itself is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining of this section is devoted to the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Openness of ν⋔ asym stand pulses(¯ξ, ε) For every potential V in ν and for all positive quantities ¯ξ and ε, the manifolds W u V �E(V ), ¯ξ � and W s loc, V �E(V ) � are compact, and those manifolds depend smoothly on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (Vn)n∈N denotes a sequence of potentials belonging to ν\\ν⋔ asym stand pulses(¯ξ, ε) and converging to some potential V∞ of ν, and let us prove that, in this case, V∞ is still outside of ν⋔ asym stand pulses(¯ξ, ε) (this will prove that ν⋔ asym stand pulses(¯ξ, ε) is open in ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every integer n, there exists a non-transverse standing pulse connecting W u loc, Vn �E(Vn) � to W s loc, Vn �E(Vn) � in a time not larger than ¯ξ while remaining at a distance not smaller than ε from Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As emphasized in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3), this pulse is characterized by a (unique) bu n in Bu such that its trajectory in R2d crosses the boundary of W u loc, Vn �E(Vn) � at the point ˆwu loc, Vn(bu n), and a (unique) time ξn in the interval [0, ¯ξ] such that this trajectory crosses the boundary of W s loc, Vn �E(Vn) � at the point SVn �ξn, ˆwu loc, Vn(bu n) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, (i) by compactness (up to considering a subsequence of (Vn)n∈N), we may assume that (bu n, ξn) converges to some couple (bu ∞, ξ∞) of Bu × [0, ¯ξ], which in turn characterizes a standing pulse for V∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notice here the importance of considering homoclinic orbits of bounded “length”, otherwise the limit trajectory would not necessarily be homoclinic to E(V∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (ii) Moreover, both conditions in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) are closed conditions, so that the limit standing pulse also satisfies them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (iii) Thanks to the “margin” ε with respect to the symmetry subspace Ssym, the limit standing pulse is necessarily asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 46 (iv) Last, the limit standing pulse is non-transverse since this property is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The limit potential V∞ is thus not in ν⋔ asym stand pulses(¯ξ, ε), and this completes the proof that ν⋔ asym stand pulses(¯ξ, ε) is open in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Density of ν⋔ asym stand pulses(¯ξ, ε) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 The proof of the density assertion of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 will again follow from applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 to the following appropriate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Take positive quantities ¯ξ and ε, and a potential V1 in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Our goal is to prove that there exist potentials in ν⋔ asym stand pulses(¯ξ, ε) which are arbitrarily close to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) M = � (bu, ξ) ∈ Bu × (0, ¯ξ + 1) : dist � SV1 �[0, ξ], ˆwu loc, V1(bu) �, Ssym � > ε/2 and πpos � SV1 �ξ, ˆwu loc, V1(bu) �� ∈ BRd �e(V1), rexit/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and let Λ1 denote a neighbourhood of V1 in the set (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) � V ∈ ν : V ≡ V1 on the closed ball BRd �e(V1), rexit �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' We may assume that this neighbourhood Λ1 is small enough so that, for every V in Λ1 and (bu, ξ) in Bu × (0, ¯ξ + 1), the following two conclusions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' if (bu, ξ) is not in M, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) either dist � SV �[0, ξ], ˆwu loc, V (bu) �, Ssym � < ε or πpos � SV �ξ, ˆwu loc, V (bu) �� ̸∈ BRd �e(V1), rexit/4 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' if (bu, ξ) is in M, then (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) dist � SV �[0, ξ], ˆwu loc, V (bu) �, Ssym � > 0 , and πpos � SV �ξ, ˆwu loc, V (bu) �� ∈ BRd �e(V1), rexit � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For V in Λ1, let (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) N = H−1 V � HV �E(V ) �� ∩ �BRd �e(V ), rexit � × Rd� \\ {E(V )} and W = ∂W s loc, V �E(V ) � = ˆws loc, V (Bs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that M, N, and W are submanifolds of R2d and since Λ1 is included in ν, it follows from inclusion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) that W is included in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, according to the condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) on V and to the inclusion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), M, N and W do actually not depend on the potential V in Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As already explained in the second remark of the beginning 47 of subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, this is mandatory to provide a setting where Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows that, according to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9), we may consider the function (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) Φ : M × Λ1 → N , (bu, ξ, V ) �→ SV �ξ, ˆwu loc, V (bu) � , which is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Notice that, even if M contains only couples (bu, ξ) for which, for V in Λ1, the position u(ξ) = πpos � SV �ξ, ˆwu loc, V (bu) � of the corresponding solution is inside BRd �e(V1), rexit � (second condition of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9)), it follows from the property (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) defining rexit that this position u(·) exits BRd �e(V1), rexit/2 � at other times, and this will provide a suitable place to perturb the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, it will be possible to modify Φ(bu, ξ, V ) by perturbing V outside of BRd �e(V1), rexit �, even if the arrival set of Φ and its image are restricted to this ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential function V in Λ1, if the image of the function M → N, V �→ Φ(m, V ) is transverse to W, then V belongs to the set ν⋔ asym stand pulses(¯ξ, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider V in Λ1 and U0 in W u V �E(V ), ¯ξ � ∩ ∂W s loc, V �E(V ) � satisfying inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the definition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) of W u V �E(V ), ¯ξ �, the point U0 is of the form (u, ˙u)(ξ) with u a standing pulse such that (u, ˙u)(0) = ˆwu loc, V (bu) and ξ in [0, ¯ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the inclusion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) satisfied by the local manifolds and the definition of ν⋔ asym stand pulses(¯ξ, ε), the implication (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) shows that (bu, ξ) belongs to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, the image Φ �(bu, ξ), V � is well defined, and it remains to notice that the transversality of Φ with W exactly corresponds to the definition Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 of the transversality of a standing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It thus follows from the definition of the set ν⋔ asym stand pulses(¯ξ, ε) that V belongs to this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining part of the proof follows exactly the same arguments as in sections 5 and 6, except for the exclusion of “case 3”, which will require a slightly different ad hoc argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Checking hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 By contrast with the previous sections, the ambient space N is now a level set of dimension 2d − 1 (instead of R2d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' however the computation is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Indeed, it follows from subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 that, on the one hand, dim(M) = dim �∂Bu E0(r) � + 1 = d − m(e0) and, on the other hand, dim(W) = d − m(e0) − 1 so that codim(W) = d + m(e0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus hypothesis 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Checking hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Take (m2, V2) in the set Φ−1(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (bu 2, ξ2) denote the components of m2, and, for every real quantity ξ, let us write U2(ξ) = �u2(ξ), v2(ξ) � = SV2 �ξ, ˆwu loc, V2(bu 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The function ξ �→ u2(ξ) is the profile of a standing pulse, connecting e(V2) to itself, for the potential V2, and, according to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9), this standing pulse is asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In 48 addition, according to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), the quantity ξ2 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let DΦ denote the full differential (with respect to m and V ) of Φ at the point (m2, V2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Hypothesis 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 follows from the next Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 (perturbation of the potential reaching a given direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every nonzero vector (φ0, ψ0) ∈ Rd × Rd belonging to TU2(ξ2)N, there exists W in Ck+1 b (Rd, R) such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) ⟨DΦ · (0, W) | (φ0, ψ0)⟩ ̸= 0 , and such that W satisfies the condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) supp(W) ∩ BRd �e(V2), rexit � = ∅ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The proof is similar to those of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let (φ2, ψ2) be a nonzero vector in Rd × Rd belonging to TU2(ξ2)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let W be a function in Ck+1 b (Rd, R), and let us assume that condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us again use the notation T(ξ, ξ′) to denote the family of evolution operators obtained by integrating the linearized differential system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='19) (for the potential function V2, and for a speed equal to 0) between the times ξ and ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every time ξ, let T ∗(ξ, ξ2) denote the adjoint operator of T(ξ, ξ2), and let �φ(ξ), ψ(ξ) � = T ∗(ξ, ξ2) · (φ2, ψ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Using the same computations as in Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, it follows from the inclusion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and the empty intersection (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) that inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12) reads (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='14) � ξ2 0 ∇W �u2(ξ) � · ψ(ξ) dξ ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, according to inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), there exists a (nonempty) open interval I included in (0, ξ2) and such that, for every ξ in I, u2(ξ) ̸∈ BRd �e(V2), rexit �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the function ξ �→ ˙u2(ξ) does not vanish on R, thus a fortiori neither on I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, three cases must be considered for the construction of the perturbation W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a time ξ† in I such that ψ(ξ†) is not collinear to ˙u2(ξ†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same construction as in the first case of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (or as in the first case of the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) can then be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in I, ψ(ξ) = α(ξ) ˙u2(ξ) with α(·) not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Again, the same construction as in the second case of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 (or as in the first case of the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) can then be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every ξ in (ξ−, ξturn), ψ(ξ) = α ˙u2(ξ) for some real (constant) quantity α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, this third case has to be precluded by a specific argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from the adjoint linearized system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='23) satisfied by φ and ψ (with c0 equal to zero) that, for every ξ in I, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='15) φ(ξ) = − ˙ψ(ξ) = −α¨u2(ξ) = −α∇V2(u2(ξ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 49 Besides, since (φ2, ψ2) was assumed to belong to TU2(ξ2)N, it follows that �φ(ξ), ψ(ξ) � belongs to TU2(ξ)H−1 V2 � HV2 �E(V2) �� for all ξ in R (the level set of the energy is invariant by the flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The orthogonal space of the tangent space to the level set N is a line spanned by the gradient of the Hamiltonian ∇HV2(U2) = (−∇V2(u2(ξ)), ˙u2(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, the condition (φ2, ψ2) ∈ TU2(ξ2)N reads (φ2, ψ2) ⊥ (−∇V2(u2(ξ)), ˙u2(ξ)) , that is α �∇V2(u2)2 + ˙u2 2 � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This implies α = 0 and thus (φ, ψ) ≡ (0, 0), a contradiction with the assumptions of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In summary, the third case cannot occur and, in both other cases, the same constructions as in the proofs of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 can be carried out, leading to a perturbation W satisfying the empty intersection (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='13) and inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='14) (and therefore also inequality (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Conclusion Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To complete the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 amounts to prove that the set ν⋔ asym stand pulses(¯ξ, ε) is dense in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 that both hypotheses 1 and 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 are fulfilled for the function Φ defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The conclusion of this theorem ensures that there exists a generic subset Λgen of Λ1 such that, for every V in Λgen, the function Φ(·, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, the set ν⋔ asym stand pulses(¯ξ, ε) is a superset of Λgen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' in particular, there exists potentials in ν⋔ asym stand pulses(¯ξ, ε) that are arbitrarily close to V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since V1 was any potential in ν, this proves the intended density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As shown at the end of subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 implies Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, which in turn implies Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Transversality of symmetric standing pulses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As it stands, the proof of the generic transversality of asymmetric standing pulses provided above does not directly apply to symmetric ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Indeed, for a symmetric standing pulse ξ �→ u(ξ), with (say) turning time 0, the condition corresponding to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='22) or (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='14) reads � ¯ξ −¯ξ ∇W �u(ξ) � · ψ(ξ) dξ ̸= 0 or equivalently � 0 −¯ξ ∇W �u(ξ) � · �ψ(ξ) + ψ(−ξ) � dξ ̸= 0 , where ¯ξ is a large enough positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This condition cannot be fulfilled if the function ξ �→ ψ(ξ) is odd and, due to the symmetry of the adjoint linear equation ¨ψ(ξ) = D2V �u(ξ) � · ψ(ξ) , this happens as soon as ψ(0) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This case, corresponding to the degeneracy of the first order derivative with respect to perturbations of the potential, can therefore not be 50 excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Possibly, the second order derivative could be investigated but the computation goes beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For that reason, the generic transversality of symmetric standing pulses is not established here and remains, to our best knowledge, an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 8 Generic non-existence of standing fronts Let us take and fix a positive quantity R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Due to the Hamiltonian invariance, precluding the existence of standing fronts is a simple task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a dense open subset of Vquad-R such that, for every potential V in this subset, there is no standing front for this potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the dense open subset Vquad-R-Morse of Vquad-R containing the functions of Vquad-R satisfying the Morse property (this notation was introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2)), and let V denote a potential in Vquad-R-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The number of critical points of such a potential is finite, and, up to applying to V an arbitrarily small localized perturbation around each of these critical points, it may be assumed that each of these critical points belongs to a level set of V containing no other critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This property is open and dense in Vquad-R-Morse, thus in Vquad-R, and, since the Hamiltonian HV defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) is constant along the profile of a standing front, it prevents the existence of a standing front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9 Proof of the main results Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 show the genericity of the properties considered in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, but only inside the space Vquad-R of the potentials that are quadratic past some radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Working in this last space is easier because it is a second countable Banach space and the flows associated to its potentials are global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In this section, the arguments will be adapted to obtain the genericity of the same properties in the space Vfull = Ck+1(Rd, R) of all potentials, endowed with the extended topology (see subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Proof of conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 Let us recall the notation FV introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), and, for every positive quantity R, let us consider the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1) FV,R = � (c, u) ∈ FV : sup ξ∈R |u(ξ)| ≤ R � of the travelling fronts of FV (invading a minimum point of V ) with a profile contained in BRd(0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As shown thereafter, the following proposition yields conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 51 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-F-R of Vfull such that, for every potential function V in this subset, V is a Morse function and every travelling front (c, u) in FV,R is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof that Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 yields conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set � R∈N∗ Vfull-⋔-F-R , is a countable intersection of generic subsets of Vfull and is therefore again a generic subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential function V in this set, V is a Morse function and every travelling front in FV belongs to FV,R as soon as R is large enough, and is therefore, according to the property of the set Vfull-⋔-F-R stated in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Statement 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The aim of subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is thus to prove Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Before doing so, here are a few preliminary comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let R be a positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 states that there exists a generic subset Vquad-R-⋔-F of Vquad-R such that, for every potential Vquad in this subset, all travelling fronts in FVquad are transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' However, due to the constraint at |u| = R, the extension to Rd of all the truncations of these potentials in BRd(0, R) is meagre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The idea is to take some margin: consider the generic subset Vquad-(R + 1)-⋔-F of Vquad-(R+1) and, using the notation introduced in definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6), consider the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) res−1 R,∞ ◦ resR,(R+1)(Vquad-(R + 1)-⋔-F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential Vfull in this set, all travelling fronts in FVfull,R are transverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' indeed, this property depends only on the values of Vfull inside the ball BRd(0, R), where Vfull must be identically equal to some potential Vquad of Vquad-(R + 1)-⋔-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It is tempting to look for an extension of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 to generic subsets, which would yield the genericity of the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Unfortunately, this corollary definitely applies to open dense subsets, and not to generic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Pursuing further in this direction, observe that, since Vquad-(R + 1)-⋔-F is a generic subset of Vquad-(R+1), there exists a countable family (ON)N∈N of dense open subsets of Vquad-(R+1) such that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) � N∈N ON ⊂ Vquad-(R + 1)-⋔-F , leading to res−1 R,∞ ◦ resR,(R+1) � � N∈N ON � ⊂ res−1 R,∞ ◦ resR,(R+1)(Vquad-R + 1-⋔-F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to general properties of functions, the following inclusion holds: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) resR,(R+1) � � N∈N ON � ⊂ � N∈N resR,(R+1)(ON) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 52 If this inclusion was an equality, then, still according to general properties of functions, the following equality would hold: res−1 R,∞ ◦ resR,(R+1) � � N∈N ON � = � N∈N res−1 R,∞ ◦ resR,(R+1)(ON) , and, since according to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 the right-hand side of this equality is a countable intersection of dense open subsets of Vfull, the intended conclusion that the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) is generic in Vfull would follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Unfortunately, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 provides no clue about the sets ON and a strict inclusion in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) cannot be precluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' However, let us make the following key observation, which enlightens the remaining of the proof: if the property “a given potential V belongs to ON” only depends on the values of V inside the ball BRd(0, R), then inclusion (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4) is actually an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The main step in the proof is thus to construct dense subsets ON of Vquad-(R+1) such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every potential Vquad in � n ON, every travelling front in FV,R is transverse, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and the property “a given potential V belongs to ON” only depends on the values of V inside the ball BRd(0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As above, let R denote a positive quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let V0 denote a potential function in Vquad-(R+1), let e−,0 and e+,0 denote a non-degenerate critical point and a non-degenerate minimum point of V0 and let c0 denote a positive speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the neighbourhoods νV0, e−,0, e+,0, c0 of V0 in Vquad-(R+1) and CV0, e−,0, e+,0, c0 of c0 in (0, +∞) provided by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 for these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Recall that those neighbourhoods are the ones from which, for every V in νV0, e−,0, e+,0, c0 and every c in CV0, e−,0, e+,0, c0, the functions ˆwu loc, c, V , the sets M and W and the functions Φu and Φs and Φ were defined in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Up to replacing the neighbourhood νV0, e−,0, e+,0, c0 by its interior, we may assume that it is open in Vquad-(R+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Similarly, we may assume that CV0, e−,0, e+,0, c0 is compact in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let N denote a non negative integer and let us consider the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) MN = Bu × Bs × (−∞, N] × CV0, e−,0, e+,0, c0 = �(bu, bs, ξ, c) ∈ M : ξ ≤ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, let us define N as (R2d)2, and let us consider the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6) OV0, e−,0, e+,0, c0, N = � V ∈ νV0, e−,0, e+,0, c0 : Φ �MN, V � is transverse to W in N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As shown in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, this set OV0, e−,0, e+,0, c0, N is made of the potential functions V in νV0, e−,0, e+,0, c0 such that every profile ξ �→ u(ξ) of a front travelling at a speed c in CV0, e−,0, e+,0, c0 and connecting e−(V ) to e+(V ) for this potential, and connecting ∂W u loc, c, V �E−(V ) � to ∂W s loc, c, V �E+(V ) � in a time not larger than N, is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set OV0, e−,0, e+,0, c0, N is a dense open subset of νV0, e−,0, e+,0, c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The density is a direct consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 which states that, generically with respect to V in νV0, e−,0, e+,0, c0, the whole image of M by the map 53 m �→ Φ(m, V ) is transverse to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To prove the openness, let us argue as in subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider a sequence (Vn)n∈N of potentials in νV0, e−,0, e+,0, c0 converging to a potential V∞ in νV0, e−,0, e+,0, c0, and such that, for every n in N, there exists mn = (bu n, bs n, ξn, cn) in MN such that the set Φ(MN, Vn) is not transverse to W at Φ(mn, Vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, according to the empty intersection (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5), ξn must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, by compactness of Bu × Bs × [0, N] × CV0, e−,0, e+,0, c0, we may assume that mn converges, as n goes to +∞, to a point m∞ of MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, by continuity, the image Φ(MN, V∞) is not transverse to W at Φ(m∞, V∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This proves that νV0, e−,0, e+,0, c0 \\ OV0, e−,0, e+,0, c0, N is closed in νV0, e−,0, e+,0, c0, and yields the intended conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Continuation of the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us make the additional assumption that the potential V0 is a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, the set of critical points of V0 is finite and depends smoothly on V in a neighbourhood νrobust(V0) of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Intersecting the sets νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 and CV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 and OV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' N above over all the possible couples (e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0) in Σcrit(V0) × Σmin(V0) provides an open neighbourhood νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 of V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' a compact neighbourhood CV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 of c0 and an open dense subset OV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' N of νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for all V ∈ OV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every front travelling at speed c ∈ CV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 and connecting the local (un)stable manifolds of two points (e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+) in Σcrit(V ) × Σmin(V ) within the “time” N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Denoting by int(A) the interior of a set A and using the notation of definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6), let us introduce the sets ˜νV0, c0 = res−1 R,∞ ◦ resR,(R+1)(νV0, c0) , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) and ˜OV0, c0, N = res−1 R,∞ ◦ resR,(R+1) �OV0, c0, N � , (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) and ˜Oext V0, c0, N = ˜OV0, c0, N ⊔ int �Vfull \\ ˜νV0, c0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) In other words, a potential ˜V of Vfull is in ˜νV0, c0 (in ˜OV0, c0, N) if it coincides, inside the ball BRd(0, R), with a potential Vquad quadratic past R + 1 and belonging to νV0, c0 (to OV0, c0, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The last set ˜Oext V0, c0, N is an extension of the open dense subset ˜OV0, c0, N of ˜νV0, c0, obtained by adding all potentials outside (the closure of) ˜νV0, c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set ˜Oext V0, c0, N is a dense open subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6, the set ˜νV0, c0 is an open subset of Vfull, and the set ˜OV0, c0, N is a dense open subset of ˜νV0, c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, according to its definition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9), the set ˜Oext V0, c0, N is a dense open subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Continuation of the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since Vquad-(R+1) is a separable space, it is second-countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus Vquad-(R+1)-Morse × (0, +∞) is also second-countable and can be covered by a countable number of products νV0, c0 × CV0, c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With symbols, there exists a countable family (V0,i, c0,i)i∈N of elements of Vquad-(R+1)-Morse × (0, +∞) so that (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10) Vquad-(R+1)-Morse × (0, +∞) = � i∈N νV0,i, c0,i × CV0,i, c0,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 54 Notice here the importance of first working with Vquad-(R+1), which is second-countable, instead of the full space Vfull, which is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the set (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) Vfull-⋔-F-R = Vfull-Morse ∩ � � � (i,N)∈N2 ˜Oext V0,i, c0,i, N � � , where Vfull-Morse is the set of potentials in Vfull which are Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every potential ˜V in the set Vfull-⋔-F-R, every travelling front (u, c) in F ˜V ,R is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ˜V be a potential function in the set Vfull-⋔-F-R and (c, u) be a travelling front in F ˜V ,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5, the map resR,(R+1) is surjec- tive, thus there exists a potential function V in Vquad-(R+1) such that V belongs to res−1 R,(R+1) ◦ resR,∞( ˜V ) (in other words V coincides with ˜V on BRd(0, R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since ˜V is a Morse function, the critical points of V in BRd(0, R) are degenerate, and up to applying to V a small perturbation in BRd(0, R + 1) \\ BRd(0, R), we may assume that its critical point in this set are also nondegenerate, so that V is actually also a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since ˜V coincides with V inside BRd(0, R) and since the travelling front u is contained in this ball, it is also a travelling front of V and it is sufficient to show that (u, c) is a transverse travelling front for V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to equality (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10), there exists an integer i such that V belongs to νV0,i, c0,i and c belongs to CV0,i, c0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, since V and ˜V coincide on BRd(0, R), ˜V belongs to ˜νV0,i, c0,i (definition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Besides, it follows from definition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) that, for every integer N, ˜V belongs to ˜Oext V0,i, c0,i, N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and since V is also in ˜νV0,i, c0,i, it follows from definition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9) that ˜V actually belongs to ˜OV0,i, c0,i, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us denote by e− and e+ the critical points of V (and ˜V ) approached by u(ξ) as ξ goes to −∞ and +∞ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the definition of the neighbourhood νV0,i, c0,i of V0,i, there exists a (unique) critical point e−,0,i and a (unique) minimum point e+,0,i of V0,i such that, if W �→ e−,i(W) and W �→ e+,i(W) denote the functions which “follow” these critical points for W in νrobust(V0,i), then e− equals e−,i(V ) and e+ equals e+,i(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us keep the notation M and Φ to denote the objects defined as in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 for the neighbourhoods νV0,i, e−,0,i, e+,0,i, c0,i of V0,i and CV0,i, e−,0,i, e+,0,i, c0,i of c0,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The travelling front (c, u) therefore corresponds to an intersection between Φ(M, V ) and W, which occurs at a certain point m of M and thus for a certain (positive) time ξ which is the time that the profile of this travelling front takes to go from the border of the local unstable manifold of e− to the border of the local stable manifold of e+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let N denote an integer not smaller than ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since ˜V belongs to ˜OV0,i, c0,i, N, there must exist (according to definition (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8)) a potential VN in νV0,i, c0,i identically equal to ˜V (and V ) on the ball BRd(0, R) and belonging to OV0,i, c0,i, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Again, (c, u) is a travelling front for VN and the previous correspondence between this front and an intersection between Φ(M, VN) and W still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since VN belongs to OV0,i, c0,i, N, the aforementioned intersection must be transverse, leading to the transversality of the front (u, c) for VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 55 Again, the three potentials ˜V and V and VN considered here have the same values along the profile of the travelling front (u, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, this front is also transverse for ˜V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' End of the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set Vfull-⋔-F-R defined in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='11) is a countable intersection of dense open subsets of Vfull, and is therefore a generic subset of Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In view of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Proof of conclusions 2 and 3 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 The proof of conclusions 2 and 3 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 is similar to the proof of conclusion 1 provided in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, only the core arguments will be reproduced here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall the notation PV introduced in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), and, for every positive quantity R, let us consider the set PV,R = � u ∈ PV : sup ξ∈R |u(ξ)| ≤ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As shown in the previous subsection for Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, the following proposition yields conclusions 2 and 3 of of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-P-R of Vfull, included in Vfull-Morse, such that, for every potential function V in Vfull-⋔-P-R, every standing pulse u in PV,R is: elementary if this standing pulse is symmetric, and transverse if this standing pulse is asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let R denote a positive quantity and let V0 denote a Morse potential function in Vquad-(R+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let e0 denote a non-degenerate critical point of V0 and let us consider an open neighbourhood νV0, e0 of V0 in Vquad-(R+1) included in both neighbourhoods provided by Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every N in N∗ and for every V in νV0, e0, let us consider the subset OV0, e0, N of νV0, e0 defined as the set of potentials V in νV0, e0 satisfying the following two conditions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every symmetric standing pulse of V , connecting ∂W u loc, V �E(V ) � to the symmetric subspace Ssym in a time not larger than N, is elementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and every asymmetric standing pulse of V , connecting ∂W u loc, V �E(V ) � to ∂W s loc, V �E(V ) � in a time not larger than N while remaining at a distance not smaller than 1/N of Ssym, is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same arguments as in the proof of Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 show that the set OV0, e0, N is a dense open subset of νV0, e0: the density follows from Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 and, regarding the openness, the key new ingredient is the condition that every asymmetric standing pulse remains at a distance at least 1/N of Ssym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Indeed, a sequence of asymmetric standing pulses (as considered in the proof) may (generally speaking) approach a symmetric standing pulse which may be non-transverse even if it is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Staying away from Ssym precludes this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 56 As on page 54, let us consider the intersections of the previous sets over all the critical points of V0: νV0 = � e0∈Σcrit(V0) νV0, e0 and OV0, N = � e0∈Σcrit(V0) OV0, e0, N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set νV0 is still open in Vquad-(R+1) and the set OV0, N is still a dense open subset of νV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in definitions (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9), these sets can be extended as follows: ˜νV0 = res−1 R,∞ ◦ resR,(R+1)(νV0) , ˜OV0, N = res−1 R,∞ ◦ resR,(R+1) �OV0, N � , and ˜Oext V0, N = ˜OV0, N ⊔ int �Vquad-(R+1) \\ ˜νV0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The end of the proof follows the same arguments as the ones of subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set Vquad-(R+1)-Morse can be covered by a countable number of subsets ˜νV0,i and the set Vfull-⋔-P-R = Vfull-Morse ∩ � � � (i,N)∈N2 ˜Oext V0,i, N � � is the generic subset the existence of which was stated in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Proof of conclusion 4 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 Let us consider the set OR of potentials V of Vfull such that all the critical points of V in BRd(0, R) are non-degenerate and have different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same arguments as in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 show that this set OR is an open dense subset of Vfull, so that the intersection ∩R∈N∗OR is generic in Vfull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Since the critical points connected by a standing front must belong to the same level set of the potential, no standing front can exist for a potential in this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Proof of conclusions 1 to 4 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Let V be a potential function belonging to the generic subset provided by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, let (c, u) be a travelling front in FV , and let e− and e+ denote the critical point and the minimum point of V connected by this travelling front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, dim � � � c′>0 {c′} × W u c′,V (E−) � � = d − m(e−) + 1 , and dim � � � c′>0 {c′} × W s c′,V (E+) � � = d + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The intersection between these two manifolds contains at least the curve {c} × U(R) corresponding to the travelling front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, the dimension of the sum of the tangent spaces to these two manifolds is not larger than the quantity �d − m(e−) + 1 � + (d + 1) − 1 = 2d + 1 − m(e−) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 57 Since according to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7 the intersection between these two manifolds is transverse in R2d+1, along the set {c} × U(R), this quantity is not smaller than 2d + 1, so that the Morse index m(e−) must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This proves conclusion 1 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Now let us assume that u is the profile of a standing pulse and let e denote the critical point of V such that this pulse connects e to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, dim (W u V (E)) = d − m(e) and dim (W s V (E)) = d − m(e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, if u is symmetric then the intersection between W u V (E) and the d−dimensional manifold Ssym is transverse in R2d, at the point U(ξturn) and this can happen only if m(e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If u is asymmetric then the intersection between W u V (E) and W s V (E) is transverse, in H−1 V �V (E) �, along the trajectory U(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The intersection of W u V (E) and W s V (E) is at least one-dimensional and the dimension of H−1 V �V (E) � is equal to 2d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Again, the transversality can happen only if m(e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This proves conclusion 2 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In all the cases considered above, the counting of the dimensions and the transversality imply that the intersections of the stable and unstable manifolds reduce to the smallest possible set, that is: the one-dimensional curve drawn by the trajectory U for travelling fronts or asymmetric pulses, and the singleton �U(ξturn) � defined by the turning point for symmetric pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' By local compactness of the unstable manifolds, this implies that the trajectories of a given class are isolated from each other (even if a family of asymmetric standing pulses may accumulate on a non-degenerate — and in this case non-transverse — symmetric pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In particular, there is only a countable number of such trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Conclusion 3 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Finally, conclusion 4 about the robustness of travelling fronts and standing pulses (the fact that they persist under small perturbations of the potential) follows from their transversality (that, is, the transversality of the intersections considered above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10 Generic asymptotic behaviour for the profiles of bistable travelling fronts and of standing pulses stable at infinity The goal of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 (and thus also conclusion 5 of Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Asymptotic behaviour of profiles Let V0 denote a potential in Vfull, let e0 denote a nondegenerate minimum point of V , and let c denote a nonnegative quantity (speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, let (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , ud) denote an orthonormal basis of Rd made of eigenvectors of D2V (e0), and let µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , µd denote the corresponding (positive) eigenvalues, with µ1 ≤ · · · ≤ µd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The statement “the smallest eigenvalue of D2V (e0) is simple”, in conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8, just mean that µ1 is smaller than µ2 (and thus also than all the other eigenvalues of D2V (e0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let 58 us make this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' With the notation of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, it follows that, for every j in {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , d}, λj,− < λ1,− < 0 < λ1,+ < λj,+ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' in other words, λ1,− and λ1,+ are, among all the eigenvalues of DFc,V (E0) (which are real), the closest ones to 0 (here E0 = (e0, 0Rd) is the equilibrium point of the flow Sc,V corresponding to e0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' If a solution ξ �→ u(ξ) of the differential system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) goes to e0 as ξ goes to −∞ (+∞), then one among the following two possible cases occurs (see Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 below for a more precise statement): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there exists a real quantity K such that u(ξ) − e0 = Keλ1,+ξu1 + oξ→−∞(eλ1,+ξ) (and u(ξ) − e0 = Keλ1,−ξu1 + oξ→+∞(eλ1,−ξ) , respectively);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' u(ξ) − e0 = oξ→−∞(eλ1,+ξ) (and u(ξ) − e0 = oξ→+∞(eλ1,−ξ), respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The words “u(ξ) approaches its limit (at ±∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point”, used in conclusion 5 of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and in conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8, mean that case 1 above occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As illustrated on Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 (see also Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), approach of equilibria “at the slowest possible rate” (case 1 above) is a generic feature among solutions of differential systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The main goal of this section is thus to provide a formal proof that this feature is indeed generic (with respect to the potential V ) for bistable travelling fronts and standing pulses stable at infinity of the parabolic system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1: Attractive node of a two-dimensional vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In the language of subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the vertical axis is the “strongly stable subspace” of the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Local strongly stable and unstable manifolds when the speed c is positive Let us keep the notation and assumptions of the previous subsection and let us assume that c is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The aim of this subsection is to provide a variant of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 59 →+ N → + + 1 +devoted to the “strongly” local stable and unstable manifolds, which are characterized by a “fast” convergence (case 2 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning the references, the same comments as in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Calling upon the notation of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1, let Esu(E0) = span �{U2,+, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , Ud,+} � and Ess(E0) = span �{U2,−, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' , Ud,−} � , and Em(E0) = span �{U1,−, U1,+} � (the superscripts “su”, “ss”, and “m” stand for “strongly unstable”, “strongly stable”, and “mild”, respectively), and βsu = λ2,+ and βss = λ2,− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, there exist norms ∥·∥su on Esu(E0) and ∥·∥ss on Ess(E0) such that inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7) (with “su” instead of “u” and “ss” instead of “s” everywhere) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity r, let us define the balls Bsu E0(r) and Bss E0(r) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8) (with the same substitutions “u”←“su” and “s”←“ss”), let Bm E0(r) denote the closed ball centred at E0 and with radius r, in the subspace Em(E0), for the usual euclidean norm on these subspace, and let BE0(r) = �Usu + Uss + Um : Usu ∈ Bsu E0(r) and Uss ∈ Bss E0(r) and Um ∈ Bm E0(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let λ3/2,− and λ3/2,+ denote real quantities satisfying λ2,− < λ3/2,− < λ1,− and λ1,+ < λ3/2,+ < λ2,+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 (local strong stable and unstable manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist a neigh- bourhood ν of V0 in Vfull, a neighbourhood C of c0 in (0, +∞) and a positive quantity r such that, for every (c, V ) in C × ν, in addition to the conclusions of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exist Ck-functions wsu loc, c, V : Bsu E0(r) → Bm E0(r) + Bss E0(r) and wss loc, c, V : Bss E0(r) → Bm E0(r) + Bsu E0(r) such that, if we consider the sets W su loc, c, V �E(V ) � = � E(V ) + Usu + wsu loc, c, V (Usu) : Usu ∈ Bsu E0(r) � and W ss loc, c, V �E(V ) � = � E(V ) + Uss + wss loc, c, V (Uss) : Uss ∈ Bss E0(r) � , then, for every U in BE0(r) the following two assertions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' U is in W su loc, c, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in (−∞, 0] and |Sc,V (ξ, U) − E(V )| = oξ→−∞(eλ3/2,+ξ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 60 and for every U in BE0(r) the following two assertions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' U ∈ W ss loc, c, V �E(V ) �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Sc,V (ξ, U) − E(V ) remains in BE0(r) for all ξ in [0, +∞) and |Sc,V (ξ, U) − E(V )| = oξ→+∞(eλ3/2,−ξ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Both differentials Dwsu loc, c0, V0(0) and Dwss loc, c0, V0(0) vanish, and both maps C × ν × Bsu E0(r) → Bm E0(r) + Bss E0(r), (c, V, Usu) �→ wsu loc, c, V (Usu) and C × ν × Bss E0(r) → Bm E0(r) + Bsu E0(r), (c, V, Uss) �→ wss loc, c, V (Uss) are of class Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Idea of the proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 The goal of this subsection is to provide a rough idea of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8, more precisely of the main conclusion of this theorem which is conclusion 2 (the proof of conclusion 1, carried out in the next subsection, is straightforward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The proof of conclusion 2 is actually almost identical to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Observe that, by contrast with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, only bistable travelling fronts and standing pulses that are stable at infinity need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In each case (bistable travelling fronts, symmetric and asymmetric standing pulses stable at infinity), the proof relies on applying Sard–Smale Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 to the same settings as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, both for potentials that are quadratic past a certain radius and for the extension to general potentials, except for the following change: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' either the unstable manifold of the left end equilibrium E−(V ) is replaced by its strongly unstable manifold, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' or the stable manifold of the right end equilibrium E+(V ) is replaced by its strongly stable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' More precisely, both replacements have to be (separately) considered both for travelling fronts and asymmetric standing pulses, while only the first replacement is relevant for symmetric standing pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us see why such change (replacement) in the setting does not affect the validity of the two assumptions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, and how its conclusions can be interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning assumption 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, this replacement leads to the following consequences: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' either the dimension of the manifold denoted by M is decreased by 1 (this is what happens for travelling fronts, be it with replacement 1 or 2, for symmetric standing pulses with replacement 1, and for asymmetric standing pulses with replacement 1), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' or the dimension of the manifold denoted by W is decreased by 1 (this is what happens for asymmetric standing pulses with replacement 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 61 In each of these cases, the dimension of the arrival manifold N is unchanged, and as a consequence, the difference dim(M) − codim(W) is exactly decreased by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' More precisely, since only bistable travelling fronts and standing pulses stable at infinity are considered, this difference is actually exactly equal to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Assumption 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 is therefore still satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Concerning assumption 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, it is also fulfilled in each of these cases, due to the key following observation: in the proof of each of the three lemmas proving that this assumption holds (Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5), the freedom provided by the variables bu and bs is not used — only the freedom provided by the time variable ξ and by the potential V are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, the fact that the unstable manifold of E−(V ) is replaced by its strongly unstable manifold does not affect the validity of the conclusion of the lemma, and neither does the fact that the stable manifold of E+(V ) is replaced by its strongly stable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, the key assumption 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In each case and for each of the two replacements 1 and 2, the conclusions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 thus still hold, and ensure that, locally generically with respect to V , the profiles of travelling fronts or of (a)symmetric standing pulses locally correspond to transverse intersections between the image of m �→ Φ(m, V ) and W in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' But the fact that dim(M) − codim(W) is now equal to −1 actually precludes the very existence of such transverse intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In other words, locally generically with respect to V , profiles of bistable travelling fronts or of (a)symmetric pulses stable at infinity approaching their limit at −∞ through its strongly stable manifold or their limit at +∞ through its strongly stable manifold do simply (locally) not exist, which is the intended conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The emptiness of such a transverse intersection due to the value −1 of the difference dim(M) − codim(W) is illustrated by Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2: Whereas the sum of the dimensions of W u(E−) and W s(E+) has the minimal value for a nonempty transverse intersection between these two manifolds to exist, for W su(E−) and W s(E+) (or for W u(E−) and W ss(E+)) this sum is smaller, so that a transverse intersection between such manifolds must be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This figure actually depicts the intersection defining a transverse asymmetric bistable standing front, but the same principle applies for bistable travelling fronts, and elementary symmetric (or transverse asymmetric) standing pulses that are stable at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The remaining arguments, ensuring the first extension to global statements for poten- tials quadratic past a certain radius (subsections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1), and then the second extension to general potentials (subsections 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 62 Wu(E-) n Ws(E+ E Wsu(E_) )nM Ws(E+To complete these arguments, a few milestones of the proof for travelling fronts are detailed in subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 Proof of conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 Let R denote a positive quantity, let us recall the notation Vquad-R introduced in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2) and Vquad-R-Morse introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2), and let us consider the set Vquad-R-Morse-ss-eig = �V ∈ Vquad-R-Morse : at every minimum point of V , the smallest eigenvalue of D2V is simple � (the subscript “ss-eig” stands for “simple smallest eigenvalue”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The set Vquad-R-Morse-ss-eig is a dense open subset of Vquad-R-Morse (and thus of Vquad-R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Openness follows from the continuity of the roots (eigenvalues of D2V at a minimum point) of a polynomial with respect to its coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To prove the density, let V be in Vquad-R-Morse, and let us assume that there exists a minimum point e of V such that the smallest eigenvalue µ1 of D2V (e) is not simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let δ denote a positive quantity, small enough so that the closed ball BRd(e, δ) contains no critical point of V but e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let ρ denote a smooth function [0, +∞) → R satisfying ρ(r) = 1 for r in [0, 1/2] and ρ(r) = 0 for r in [1, +∞) , and let ε denote a small positive quantity to be chosen below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let u1 denote an unit eigenvector of D2V (e) associated to µ1, and let us consider the perturbed potential Vpert defined as: Vpert(u) = V (u) − ε 2 �(u − e) · u1 �2ρ �|u − e| /δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, e is still a critical point of Vpert and, for every v in Rd, D2Vpert(e)(v, v) = D2V (e)(v, v) − ε(v · u1)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As a consequence, u1 is still an eigenvector of D2Vpert(e), the corresponding eigenvalue µ1 − ε is simple, and the other eigenvalues of D2Vpert(e) are the same as those of D2V (e) (the difference D2Vpert(e) − D2V (e) vanishes on the orthogonal subspace to u1 in Rd), these other eigenvalues are therefore larger than µ1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In addition, if ε is small enough, then µ1 − ε is positive (so that e is still a minimum point of Vpert) and the closed ball BRd(e, δ) contains no critical point of Vpert but e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same procedure, repeated for each minimum point of V such that the smallest eigenvalue of D2V at this minimum point is not simple, provides an arbitrarily small perturbation of V belonging to Vquad-R-Morse-ss-eig, and therefore proves the intended density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let VMorse-ss-eig denote the subset of Vfull containing Morse potentials V such that, at every minimum point point of V , the smallest eigenvalue of the Hessian D2V at this minimum point is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proceeding as in subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3, the same arguments as in the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 above show that this set VMorse-ss-eig is a generic subset of Vfull, which proves conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 Proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 for bistable travelling fronts The aim of this subsection is to complete the idea of the proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 provided in subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 with a few milestones of this proof, in the case of travelling fronts (only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' As for conclusion 1 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='7, the first goal is to prove the intended conclusion among potentials that are quadratic past a certain (positive) radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This is stated by the following proposition, which is an extension of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' It calls upon the notation FV introduced in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' There exists a generic subset of Vquad-R, included in Vquad-R-Morse-ss-eig such that, for every potential V in this subset, every travelling front (c, u) in FV is trans- verse, bistable, and its profile u approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 Reduction to a local statement Let V0 denote a potential function in Vquad-R-Morse-ss-eig, and let e−,0 and e+,0 denote non- degenerate minimum points of V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us consider the neighbourhood νrobust(V0, e−,0, e+,0) of V0 introduced in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, and let us denote by ˜νrobust(V0, e−,0, e+,0) the inter- section νrobust(V0, e−,0, e+,0) ∩ Vquad-R-Morse-ss-eig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The following proposition is a variant (extension in the case of bistable travelling fronts) of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The notation is similar, except for the “tilde” added to the symbols of the various sets, in order to differentiate them for the corresponding sets introduced in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive speed c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' there exist a neighbourhood ˜νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 of V0 in Vquad-R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' included in ˜νrobust(V0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' a neighbourhood ˜CV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 of c0 in (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' +∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' and a generic subset ˜νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' gen of ˜νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for every V in ˜νV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' gen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' every front travelling at a speed c in ˜CV0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' e+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' c0 and connecting e−(V ) to e+(V ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' for the potential V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' is transverse and its profile u approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof that Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 yields Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 already ensures the existence of a generic subset Vquad-R-⋔-F of Vquad-R such that, for every potential function V in this subset, every travelling front (c, u) in FV is transverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' According to the arguments of subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4, such a front is necessarily bistable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Thus, only the conclusion of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 relative to the asymptotic behaviour of the profile remains to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' To this end, the arguments are the same as in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' We may introduce the sets ˜νV0, c0 and CV0, c0 and νV0, c0, gen, defined exactly as the corresponding sets (without tilde) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3) (with νrobust(V0, e−,0, e+,0) replaced with ˜νrobust(V0, e−,0, e+,0)), and the same remaining arguments (replacing Vquad-R-Morse with Vquad-R-Morse-ss-eig) show the existence of a generic subset of Vquad-R, included in Vquad-R-Morse-ss-eig, such that, for every potential V in this subset, every bistable travelling front (c, u) in FV is transverse and its profile u approaches its limit at both ends of R according to the intended conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Intersecting 64 this generic subset with the one provided by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 provides a generic subset of Vquad-R for which all conclusions of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 Proof of the local statement The proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4 may be derived from the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2, up to a few changes and thanks to some key arguments, all of which are exposed in subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3 Extension to all potentials The extension to all potentials is obtained by applying the same strategy as in subsec- tion 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Let us recall the notation FV,R introduced in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The same arguments as in subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 show that the intended extension is a consequence of the following extension of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' For every positive quantity R, there exists a generic subset Vfull-⋔-F-min-rate-R of Vfull, included in VMorse-ss-eig, such that, for every potential V in this subset, every travelling front (c, u) in FV,R is transverse, bistable, and approaches its limit at +∞ (−∞) tangentially to the eigenspace corresponding to the smallest eigenvalue of D2V at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 already provides a generic subset Vfull-⋔-F-R of Vfull such that, for every potential V in this subset, every travelling front (c, u) in FV,R is transverse, and therefore bistable (subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Therefore, only the conclusion relative to the asymptotic behaviour of the profiles remains to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' The proof of this conclusion is a variation of the proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1 and follows the ideas exposed in subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3: for some potential V0 in Vquad-(R+1) and for some non-degenerate minimum points e−,0 and e+,0 of V , and for every nonnegative integer N, two variants of the set MN defined in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5) (and of the open subset OV0, e−,0, e+,0, c0, N defined in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='6)) can be introduced: one where Bu is replaced by Bsu, and one where Bs is replaced by Bss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In each of theses two cases, the condition “Φ �MN, V � is transverse to W in N” can be read as “the intersection between Φ �MN, V � and W is empty”, due to the missing dimension induced by the change in each of theses variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Then, replacing the open subset OV0, e−,0, e+,0, c0, N by the intersection of its two variants, the remaining arguments are exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' This proves Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='5 (and therefore also completes the proof of conclusion 2 of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='8 for bistable travelling fronts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Abraham and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Robbin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Transversal mappings and flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' New York- Amsterdam: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Benjamin, 1967 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10, 22, 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} 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Property for Damped Wave Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Equations 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2/3 (2003), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 571–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1023/B:JODY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='0000009749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='10737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='9d (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 36).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='2 (1976), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 431–438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1016/0022-0396(76)90130-3 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} 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(cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [26] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Joly and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Raugel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1090/S0002-9947-2010-04890-1 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [27] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Joly and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Raugel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “Generic Morse-Smale property for the parabolic equa- tion on the circle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' l’Institut Henri Poincare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Non Lin- eaire/Nonlinear Anal.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' arXiv: 1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='3186 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [28] K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1016/0378-4371(82)90178-9 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Kelley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “The stable, center-stable, center, center-unstable, unstable manifolds”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1016/0022-0396(67)90016-2 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 13, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 67 [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Knobloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “Bifurcation of degenerate homoclinic orbits in reversible and conservative systems”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1007/BF02227489 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [31] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Kupka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “Contribution à la théorie des champs génériques”.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 7, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' [52] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Zelik and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Mielke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' “Multi-pulse evolution and space-time chaos in dissipative systems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' In: Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='925 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='1090/memo/0925 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' Romain Joly Université Grenoble Alpes, CNRS UMR 5582, Institut Fourier, 38000 Grenoble, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' romain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='joly@univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='fr Emmanuel Risler Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69621 Villeurbanne, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content=' emmanuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='risler@insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} +page_content='fr 69' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndA0T4oBgHgl3EQfJv-v/content/2301.02095v1.pdf'} diff --git a/ntAzT4oBgHgl3EQfAPov/content/2301.00921v1.pdf b/ntAzT4oBgHgl3EQfAPov/content/2301.00921v1.pdf new file mode 100644 index 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a/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/2301.00706v1.pdf.txt b/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/2301.00706v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ce68454a443a645f9730a8e98e73d5025940dc1 --- /dev/null +++ b/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/2301.00706v1.pdf.txt @@ -0,0 +1,1418 @@ + +1 +Abstract— Volumetric dual photacoustic (PA) / ultrasonic +(US) +imaging +with +precise +spatial +and +temporal +coregistration can provide valuable and complementary +information for diagnosis and monitoring. Considerable +research has sought to combine 3D PA/US imaging in +configurations that +can +be +transferred +to +clinical +application but technical compromises currently result in +poor image quality either for photoacoustic or ultrasonic +modes. +Simultaneous +3D +PA/US +tomography +was +implemented here by interlacing PA and US acquisitions +during the rotate-translate scan of a 5-MHz linear array (12 +angles and 30 mm translational range to image a cylindrical +volume of 21 mm diameter and 19 mm length within 21 +seconds). Volumetric image reconstruction was performed +with synthetic aperture approaches. An original calibration +method was developed to estimate 6 geometrical parameters +and 1 temporal off-set providing sharpest and best +superimposed reconstructions. Calibration thread phantom +design and choice of metrics to build the cost function were +based on analysis of a numerical phantom and the final +selection demonstrates a high estimation accuracy of the 7 +parameters. +Experimental +estimations +validated +the +calibration repeatability. Experiments in an additional +phantom showed a superposition distance between thread +centers identified in the PA and US images to be smaller than +10% of the acoustic wavelength, and a spatial resolution on +the order of the wavelength. Dual mode 3D imaging with +high-quality co-registration and excellent, uniform spatial +resolution was further demonstrated on phantoms with +complementary contrasts, and should contribute to more +sensitive and robust imaging to detect and follow biological +changes or the accumulation of nanoagents in living +systems. +Index +Terms—Tomography, +rotate-translate +scan, +volumetric +imaging, +simultaneous +dual +imaging, +photoacoustic, ultrafast ultrasound imaging +I. INTRODUCTION +OLUMETRIC and simultaneously co-registered multimodal +imaging is increasingly developing in biomedical imaging +due to the rich, multiplexed and complementary anatomical– +functional information that can be precisely spatially and +temporally correlated [1]. The combination of positron +emission tomography (PET) and computed tomography (CT), +introduced in the 1990s, was among the first volumetric and +simultaneous bimodal imaging system to be clinically available +[2]. In this dual-modality, CT provides the anatomical context +needed to interpret the functional PET. Various combinations +of volumetric imaging have since been developed. For +example, PET has been integrated with magnetic resonance +imaging (MRI) [1], [2], photoacoustic imaging (PAI) and +optical coherence tomography (OCT) have been superimposed +[3] and PET has been combined with Doppler ultrasound +imaging [4]. +The advantages of simultaneously acquired, coregistered +multimodal images are manifold. The information gained +within one single imaging session is maximized which reduces +the scan time and increases benefits for patients, researchers +and clinical-management teams. Typically, complementary +spatio-anatomical and functional information can be obtained +through the combination of modalities for a more +comprehensive +characterization +of +the +region +under +examination. +Precisely +correlated +(both +spatially +and +temporally) +information +is +essential +for +a +precise +characterization. In addition, volumetric imaging provides a +detailed view of regions under examination from various +orientations for improved diagnosis, and facilitates the +investigation and comparison for longitudinal studies or to +monitor the progression or regression of pathology in response +to therapy. +Ultrasound imaging (USI) and PAI are complementary +imaging modalities since USI provides the anatomical +correlation for the molecular information supplied by PAI. +Indeed, USI is sensitive to differences in the mechanical +microstructures of tissues and can be used to delineate organs +and lesions. USI is used in clinical routine to obtain anatomical +information. PAI is sensitive to optical absorption and can +provide information on hemoglobin oxygenation and detect +molecular and nanoparticular contrast agents [5], [6]. Both USI +and PAI are based on the detection of ultrasonic signals and can +therefore be implemented with the same ultrasound detector for +simultaneous co-registration. Recorded ultrasonic signals are +generated in situ by the optical absorption of a laser excitation +for PAI, while, for USI, they are created with the incident +ultrasonic pulse and backscattered from structures with +different acoustic impedance. In USI the transducer generally +operates in both emission and reception modes. For 2D +imaging, simultaneously co-registration of USI and PAI has +already been demonstrated with a single ultrasound (US) +detector array [7], [8] and with a dual array configuration [9]. +However, PAI is intrinsically 3D in nature due to scattering of +the excitation light by the biological tissue, which leads to out- +of-plane artifacts in 2D imaging. Moreover, elongated +Volumetric and Simultaneous Photoacoustic and +Ultrasound Imaging with a Conventional Linear +Array in a Multiview Scanning Scheme +Clément Linger, Yoann Atlas, Remy Winter, Marine Vandebrouck, Maxime Faure, Théotim +Lucas, S. Lori Bridal, and Jérôme Gateau +V + + +2 +structures like blood vessels have a strong directionality in PAI +and may not be visible in the limited view configuration of 2D +PAI, when they emit outside of the limited angular aperture. +Spherical US detector matrices (2D arrays) with a large +angular aperture have been developed specifically for 3D PAI, +using transducers that could also emit ultrasonic waves [10], +[11]. Given the limited number of elements used to cover the +spherical surface, spherical US matrices can be considered +sparse for ultrasonic waves. A large, sparse angular aperture +can provide high quality photoacoustic (PA) images. However, +US images obtained with such PAI spherical arrays present +strong artifacts, that have been shown to be reduced using +processing to extract the Doppler signal [12], [13] or the signal +from sparse US contrast agents [12]. Indeed, pulse-echo US +images are densely filled with echogenic structures and high- +quality, 3D USI requires US matrices with a high spatial +sampling and a large number of elements. Alternatively, planar +US transducer matrices that were developed for 3D USI have +been tested for 3D PAI [14], [15]. However, the limited angular +aperture of such US transducer matrices and the poor sensitivity +of the small elements lead to limited view artifacts, limited +spatial resolution and poor sensitivity for 3D PAI. In short, +simultaneous co-registration of volumetric PAI and USI with +US matrices currently results in poor image quality for one of +the two modalities. +A possible alternative to US matrices lies in the mechanical +scanning of US linear arrays that have been designed for 2D +imaging. Several systems based on a translational scan of a +linear array have been developed recently and their ability to +provide simultaneously co-registered 3D USI / PAI has been +demonstrated [16]–[19]. In this configuration, linear arrays +operating in simultaneous 2D USI / PAI are scanned in the +elevational direction, perpendicularly to the imaging plane. The +2D images are then stacked to obtain a volume. This approach +is easy to implement and to transfer to a clinical environment. +However, the angular aperture in the elevational direction of a +linear US array is very limited, which induces strong limited +view artifacts in PAI. This issue is not addressed with the +translational scan, even when synthetic aperture focusing is +used [20], [21]. Moreover, the spatial resolution in the +translational direction is strongly degraded compared to the in- +plane resolution both in PAI and USI. Adding a rotational +motion and implementing a rotate-translate synthetic aperture +scanning of a linear US array has been shown to effectively +increase the angular aperture and to highly improve the +volumetric image quality compared to a translational scan [22], +[23]. The rotate-translate approach has been demonstrated +independently on two different imaging systems for 3D PAI +[22] and for 3D USI [23], with a significant decrease of the scan +duration between the two demonstrations. In this paper, we +demonstrate simultaneous co-registration of 3D PAI and 3D +USI with a single rotate-translate scan of a linear US array. +After a calibration step that enables us to finely determine the +position of the array in the scanning system, we validate the co- +registration performance in vitro with an imaging phantom +presenting both US and PA contrast. We also illustrate the +capabilities of the system and the complementarity of +volumetric USI and PAI by imaging phantoms that have mixed +structures with either US or a PA contrast. +II. MATERIALS AND METHODS +A. Experimental set-up +The experimental setup is presented in Fig. 1. It can be +divided in four main parts: (1) the optical excitation comprising +a nanosecond laser and an optical fiber bundle, (2) the US +acquisition system consisting of an US linear array driven by a +programmable US platform, (3) the scanning system +comprising two motorized stages and their motion controller, +and (4) the synchronization system piloted with a +programmable trigger generator. The acquisition process was +fully automated. +An optical parametric oscillator laser (SpitLight 600 OPO, +Innolas Laser GmbH, Krailling, Germany) delivering < 8 ns +pulses with a pulse repetition frequency (PRF) of 20 Hz was +used to generate the optical excitation at 700 nm. A bifurcated +fiber bundle (CeramOptec GmbH, Bonn, Germany) guided the +light toward the imaged volume to obtain a bilateral +illumination. The mean laser energy at each fiber output was +estimated to be around 6 mJ. The fixed PRF of the Laser sets +the time base for the acquisition sequence. The pulse energy +was recorded using a pyrometer incorporated in the laser. +The US transducer array was a 128-element linear array (L7– +4, ATL) driven by a programmable, 64-channel US machine +(Vantage, Verasonics, WA, USA). For all the transmit events +and all the receive events, only the 64, central elements of the +array were used. Each Laser pulse triggered a receive-only +event to record the PA data. Between two laser pulses, transmit +events produced “plane wave” ultrasound emissions (a beam +that is unfocused in the lateral direction of the array) [24]. 6.25 +ms after each laser pulse, five tilted, plane waves were emitted +with steered angles of -4°, -2°, 0° (the 64 elements in the +transducer array were fired at the same time), 2° and 4° at a +PRF of 160 Hz. Transmitted US pulses were 1 cycle long at 5.2 +MHz. Pulse-echo US signals and PA signals were recorded at +a sampling frequency of 20 MS/s and 62.5 MS/s, respectively. +A constant gain was adjusted for the PA and US signals to +ensure a good amplitude digitalization of received signals +without saturation (no time gain compensation was applied). + + +3 +The US array was mounted on a rotate-translate scanning +system described in detail in ref [23]. In brief, a rotation stage +was mounted on a translation stage (Physik instrumente, +Karlsruhe, Germany). The array axis (axis along the row of its +elements) was aligned with the rotation axis. The rotation stage +was moved to 12 different angles with a nominal angular +sampling period of Δα = 4°. For each angle, the imaged volume +was scanned with a translation range L = 30 mm and a linear +sampling period Δℓ = 1 mm. Relative to parameters used in ref +[23], the scan parameters, in particular the translation velocity, +had to be adjusted to adapt for the PRF of the Laser. The motion +was continuous and, for each laser emission and each US plane +wave emission, the motor positions were recorded and stored +in the motor controller. The scan was automated and initiated +with an external trigger. The motor positions are presented in +Fig. 1(b). +To synchronize the instruments and to coordinate the +acquisition sequence, a trigger generator (BNC Model 577, +Berkeley Nucleonics, San Rafael, CA, USA) was used to send +external +triggers +simultaneously +to +the +Laser, +the +programmable US system and the motion controller. +The imaged sample, the array elements and the outputs of the +fiber bundle were immersed in a water bath filled with tap +water. The speed of sound in the bath was derived from the +water temperature [25], measured with a thermometer +(HI98509, Hanna instruments, Lingolsheim, France). +For a volumetric acquisition, a total of 411 laser pulses were +fired for a total duration of 21 s. Video 1 presents a video of the +acquisition and illustrates the rotate-translate scan of the array. +At the end of the acquisition, the radiofrequency data of the PA +and US events, the motor positions and the pyrometer values +were transferred to a computer for signal processing and image +reconstruction. +B. Image reconstruction +1) Image Reconstruction Algorithm +The 3D image grid was defined in a fixed Cartesian +coordinate system (O, ex, ey, ez). The vectors ey and ez +correspond to the rotation axis and the radial direction, +respectively, when the rotation angle equals zero. The vector ex +completes the orthonormal basis. The origin O is chosen so that +xO and zO are the coordinates of the rotation axis, when the +translation stage is at its center position (ℓ=0), and yO is the +coordinate of the center of the array. In this grid, the voxel +dimensions were chosen equal to px × py × pz = 71 µm × +143 µm × 71 µm, with an anisotropy reflecting the best- +expected resolutions. The image volume is defined by a +diamond-shaped, cross-sectional area (DSCA, Fig. 2) in the xz- +plane (length of the diagonal L= 30 mm, and centered at (x=0, +z =25 mm) ) and the active length of the array (19 mm) along +the y-axis. +PA signals were divided by the corresponding pyrometer +value to compensate for the pulse-to-pulse energy fluctuations +of the Laser. PA signals were bandpass filtered between 2 MHz +and 10 MHz (Butterworth, order 3). +Image reconstruction was performed with simple delay-and- +sum beamforming algorithms. The one-way (PA) and two-way +(US) travel times between the US transducer element positions +(xn, yn, zn) and each imaged voxel (xp, yp, zp) were computed, +assuming a constant speed of sound, c, in the medium. Then, +the value of each voxel was computed by summing the signal +values assessed at the voxel-associated travel time, over all the +elements of the array and all the different tomographic +positions. The US image reconstruction algorithm was detailed +in ref [23]. The same apodizations were used for PA and US +image reconstruction. +Three-dimensional, +envelope-detected +images +were +obtained. The 3-D images were displayed using maximum +amplitude projection (MAP) images along the axes of the +coordinate system. Rotating MAP images around the z axis +were obtained with the 3D project option of ImageJ [26] . The +colorscale used for the images is displayed in Fig. 1(c). + +2) Time delays for the reconstruction +The recording of the ultrasound signals was set to start at the +same time as the Laser emission for the PA acquisition or at the +same time as the ultrasound emission for the US acquisition. +However, due to the lens effects [27] and additional time delays +induced by the acquisition hardware, we found that two time +parameters t0 PA and t0 US needed to be determined to convert the +voxel-associated travel times into the time indexes of the +recorded signals. t0 PA and t0 US correspond to effective offsets +of the recorded signals. +t0 PA was determined experimentally. Two 20-µm diameter +black nylon threads (NYL02DS, Vetsuture, France) were +positioned perpendicularly to the imaging plane of the array +and in the vicinity of the elevation focus (around 25 mm from +the face of the array). They were illuminated with the Laser +light and PA signals were recorded. Given their small diameter + +Fig. 1. (a) Annotated picture of the experimental setup. (b) Motor +positions for the PA acquisitions over one scan. For a better readability, +positions corresponding to the US and PA acquisitions are indicated +only for the inset in the upper right corner of the graph. Positions shown +in grey (drop out) were not used for the reconstruction, but are acquired +due to the continuous motion of the motors (c) Colormap used for all +images: PA images are represented in shades of blue and US images +in shades of orange, the sum of the two leading to white. PA signals +are presented on a linear scale while US signals are presented within +the range from -40 to 0 dB. + +Programmable +Us system +Linear +USarray +Rotation +S1308 +Motion +Optical fibers +cantroller +Phantom +Translation +Laser +stage +0000 +-16 +20 +e [mm] +US +. +PA +0.5 +US +Dropout +PA ++ +Target +-20 +20 +40 +4 +with regards to the wavelength at the center frequency of the +array (Λc≈300µm), the threads can be assumed to be point +absorbers in 2D. 2D images (py × pz = 71 µm × 71 µm) were +beamformed for different values of t0 PA, with the speed of +sound determined with the temperature of the water bath. The +one-dimension Brenner’s gradient of the images was computed +along the lateral dimension of the array: +𝐹𝐵𝑟𝑒𝑛𝑛𝑒𝑟 1𝐷 = ∑ +(𝑓(𝑦 + 2, 𝑧) − 𝑓(𝑦, 𝑧))2 +y,z + +(1) +where f(y,z) represents the gray level intensity in the 2D +image. The Brenner's gradient provides a quantitative measure +of image sharpness and was shown to be an efficient metric for +the speed of sound calibration in PA [28]. It was found to be +maximized for t0 PA = − 1.3 µs. In the recorded signals, |t0 PA| +corresponded also to the time of arrival of PA signals generated +by a metalized mylar film (space blanket) pressed against the +face of the array. +3) Spatial transformation from motor to array positions +For each tomographic position, the experimental acquisition +gives the motor positions in their own coordinate systems +(integrated position sensors), while the reconstruction +algorithm requires the position and orientation of the array in +the fixed coordinate system (O, ex, ey, ez). Therefore, the spatial +transformation matrix from the mobile Cartesian coordinate +system (Oa, u, v, w) attached to the transducer array to (O, ex, +ey, ez) needs to be assessed. The vector u corresponds to the +elevation direction, the vector v to the long axis along the row +of the elements and the vector w to the axial direction of the +linear array. The origin Oa is the center of the active transducer +aperture. Oa is located on the interface between the transducer +array and the water. +A tomographic position is described by the linear and +angular motor positions (ℓ, α), respectively. The translation +length equals zero (ℓ=0) when the translation stage is at the +center position, and the rotation angle equal zero (α=0) when +the radial axis is vertical. The position of the center of the +element number n of the array (𝑛 ∈ ⟦1, 𝑁⟧ with N=64) in (O, +ex, ey, ez) can be decomposed as: +( +𝑥𝑛 +𝑦𝑛 +𝑧𝑛 +) = ( +𝑥𝑂𝑎(ℓ, α) +𝑦𝑂𝑎(ℓ,α) +𝑧𝑂𝑎(ℓ, α) +) + (𝑛 − 1 − +𝑁−1 +2 ) ∙ 𝑝 ∙ ( +𝑣𝑥(α) +𝑣𝑦(α) +𝑣𝑧(α) +) +(2) +With p the interelement spacing of the array. Here, p = +298 µm. Moreover, the transformation matrices are: +( +𝑢𝑥(α) +𝑣𝑥(α) +𝑤𝑥(α) +𝑢𝑦(α) +𝑣𝑦(α) +𝑤𝑦(α) +𝑢𝑧(α) +𝑣𝑧(α) +𝑤𝑧(α) +) = ℛ𝛼 ∙ ℛ𝑃𝑖𝑡𝑐ℎ ∙ ℛ𝑌𝑎𝑤 ∙ ℛ𝑅𝑜𝑙𝑙 (3) + +( +𝑥𝑂𝑎(ℓ, α) +𝑦𝑂𝑎(ℓ, α) +𝑧𝑂𝑎(ℓ, α) +) = ℛ𝛼 ∙ ( +Δ𝑥 +0 +Δ𝑧 +) + ℓ ∙ ( +cos(θ) ∙ cos(φ) +sin(θ) +cos(θ) ∙ sin(φ) +) +(4) + +With ℛ𝛼 the rotation matrix around the y-axis due to the +motion of the rotation motor. ℛ𝑅𝑜𝑙𝑙, ℛ𝑃𝑖𝑡𝑐ℎ and ℛ𝑌𝑎𝑤 are +rotation matrices around the 1st-axis, the 2nd-axis and the 3rd- +axis, respectively. Δx and Δz are the coordinates of Oa when +α=0 and ℓ=0. The product of these rotation matrices and the +offsets models the misalignment of the array axis v compared +to the rotation axis of the stage ey. The angles θ and φ account +for misalignment of the translation axis with ex. Therefore, a +total of 7 geometrical parameters independent of (ℓ, α) need to +be determined: Roll, Pitch, Yaw, Δx, Δz, θ, φ. +C. Calibration method +The inadequate estimation of the reconstruction parameters +results in degraded image quality in terms of sharpness and a +misalignment of the PA and US images. Therefore, we +developed a calibration method for the determination of the +required parameters. The calibration method needs to be +applied once and the parameters can be used to reconstruct +subsequent acquisitions as long as the setup is not mechanically +misaligned. +The calibration method combines a calibration phantom and +an optimization algorithm to estimate the 7 parameters: t0 US, +Roll, Pitch, Yaw, Δx, Δz, θ. The parameter φ was fixed equal +to zero because the perpendicularity between the translation +axis and ez was precisely ensured by the mechanical design, and +because of equivalent solutions for our optimization algorithm +and for different sets of (Pitch, φ) values. +1) Calibration Phantom +The required specifications for the calibration phantom were +that it be simple to build, easy to use and does not require an +absolute and tedious positioning procedure. We developed a +wire phantom, inspired by phantoms used for the calibration of + +Fig. 2. (a) Imaged region (also called region of interest (ROI)) of the +calibration phantom in a 3D coordinate system with dimensions in mm. +(b) Picture of the calibration phantom. Four 20-µm nylon threads are +mounted on a yellow 3D-printed frame. To ease the readability, the +threads have been highlighted with the same color as in (a). A +projection of the ROI is shown in red. (c) Volumetric images +reconstructed with a set of optimized parameters: first row: +photoacoustic image; second row: ultrasound image; Third row: +Combined PA/US image. The colorscale is presented in Fig. 1 (c). Each +image is a maximum amplitude projection (MAP) image. The visible +part of the diamond-shaped, cross-sectional area (DSCA) is shown +with yellow dashed lines in the MAP images along y. + + +10 +10 +X(mT) +MAP along x +MAP alongy +MAP along z +PA +US +5mm +PA+US +5 +freehand 3D ultrasound systems such as Z-fiducial phantoms +[29]. Wires or threads have several advantages. First, black +threads have a dual contrast PA and US compared to water, and +are therefore expected to be superimposed on the dual-modality +images. Second, straight threads provide elongated and uniform +structures that can be easily intersected and identified in a +volumetric image even with a sparse sampling in one direction +(as opposed to small spheres for instance). Third, the +orientation can be varied. Finally, with a few non-crossing and +well-separated threads, the segmentation of the 3D image +allows local assessments of the image quality, in particular, the +image sharpness in slices intersecting the threads. +Our calibration phantom is presented in Fig. 2. It is +comprised of four threads with two orientations: two threads +parallel to one another and positioned in a horizontal plane +(Thread 1 and 3), and two threads rotated by ± 22° placed in +parallel planes below and above, respectively (Thread 2 and 4). +The angle was chosen to provide a good sensitivity to the +different parameters to be estimated. The spacing between the +threads was chosen so that the phantom fits inside the imaged +volume when the parallel threads are roughly aligned along the +y-axis and at z ≈ 25 mm. The orientation of the phantom was +chosen so that the threads appear as points in xz-planes whose +reconstruction sharpness is highly sensitive to the tomographic +positions. +The experimental calibration phantom was implemented +with 20-µm diameter black nylon threads (NYL02DS, +Vetsuture, France) mounted on a 3D-printed frame (Fig. 2 (b)). +A numerical phantom was also designed in order to validate the +calibration method with a known set of parameters. For this +numerical phantom, each thread was discretized into point +sources (PA) or point scatterers (US) spaced by Λc/5 where Λc +is the ultrasound wavelength at the center frequency of the +transducer (Λc≈300µm). The numerical simulation is based on +the sum over all the points of time signals delayed by the travel +times of the ultrasound waves. A one cycle sinusoidal signal at +5 MHz with a gaussian envelop was used for the US simulation, +and the derivative of a gaussian pulse (standard deviation 17 +ns) for the PA simulation. The finite size (width and height) and +cylindrical focusing of the array elements were modeled with a +cylindrical transducer discretized with point transducers spaced +by Λc/5. Additionally, the 7 geometrical parameters and the two +time-offsets were considered in the simulation. +2) Calibration algorithm +The calibration algorithm is based on an optimization +algorithm that minimizes a cost function. +a) +Cost function +For one set of the 7 parameters and one volumetric acquisition +comprising PA and US (5 steered angles) data, the cost function +was assessed using the images of 5 slices located at y = i×3 mm +with 𝑖 ∈ ⟦−2; 2⟧, i.e. centered and distributed to avoid edge +effects. For each slice, we reconstructed two 30-mm width +square images (one PA and one US) centered around (x = 0, +z = 25 mm) and with pixel sizes py × pz = 71 µm × 71 µm using +the 3D reconstruction algorithm. Each envelope-detected +image was thresholded at one fourth of its maximum pixel +value to produce a binary image. The fourth largest connected +components of the binary image were identified as regions and +each region is expected to correspond to one thread. The cost +was set to zero if less than four connected components were +counted. For each region, the centroid position (𝑥𝑗, 𝑧𝑗) with +weights based on the grayscale image intensity value was +computed. The distance between the centroids determined in +the US image and in the PA image in slice i was: +𝑑US−PA(𝑖, 𝑗) = √(𝑥𝑗 +𝑈𝑆 − 𝑥𝑗 +𝑃𝐴) +2 + (𝑧𝑗 +𝑈𝑆 − 𝑧𝑗 +𝑃𝐴) +2 +(4) +In each image, the four centroids were sorted by increasing +angular position and each was attributed to one of the threads. +For Threads 2 and 4, several quantities were computed. First, +a linear regression was performed with the five centroids (one +per slice) of each thread and the coefficient of determination +was computed. The mean over the two threads of the coefficient +of determination gave R2US and R2PA for the US images and the +PA images, respectively. R2US and R2PA equals one when the +threads are reconstructed as straight structures in the volume +and when the image quality allows adequate segmentation of +the threads. R2US and R2PA were found to be sensitive to errors +for the parameters Pitch, Δx and Δz. Second, the mean distance +of dPA-US, named DPA-US, was computed over the two threads +and the five slices. DPA-US evaluated the superposition of the PA +and US images. DPA-US is expected to be equal to zero for the +correct set of parameters. DPA-US was found to be sensitive to +errors in the parameter t0 US. DPA-US is expressed in mm and the +cost was set equal to zero for DPA-US > 1 mm. Finally, the local +normalized variance of the US images was computed in a +square region of 2 mm-width (twice the translation step) around +the centroid. The normalized variance quantifies variations in +the pixel values about the mean. It is equal to the variance of +the pixel values over their mean. This measurement of the +image sharpness was reported for an autofocus method [28]. +The mean of the local normalized variance over the two threads +and over the five slices was named NV US. NV US is expected to +be maximal for the correct set of parameters. NV US was found +to be sensitive to errors in the parameters: Pitch, Yaw, Δx, Δz, +θ. +Finally, a normalized two-dimension squared gradient of the +entire US images (four threads) was computed as a sharpness +metric: +𝑆𝑁 2𝐷 = +1 +∑ +𝑓(𝑥,𝑧) +𝑥,𝑧 +(∑ +(𝑓(𝑥 + 1, 𝑧) − 𝑓(𝑥, 𝑧))2 +𝑥,𝑧 + + + ∑ +(𝑓(𝑥, 𝑧 + 1) − 𝑓(𝑥, 𝑧))2 +𝑥,𝑧 + ) +(5) +where f(x,z) represents the gray level intensity in the 2D +ultrasound image. The mean of the normalized squared gradient +over the five slices was named SN US. SN US was found to be +sensitive to errors in the parameters: Roll, Pitch, Yaw, Δx, Δz. +The metrics R2US, R2PA, DPA-US, NV US and SN US were selected +based on numerical simulations and observations of the +variations in the metric values induced by each parameter +individually. Among the diversity of tested metrics, we kept the +most sensitive and we made sure that all the parameters were +covered by at least one metric. We found that the normalized +variance and normalized squared gradient were more efficient +for US images than for PA images. This could be explained by +the fact that the US image reconstruction combines data for five +steered angles, which may induce stronger variations when the +parameters are away from their expected values. We combined +the metrics with a product and the cost was then defined by: + + +6 +𝐶𝑜𝑠𝑡 = { +0 for DPA−US ≥ 1 mm +−𝑅𝑈𝑆 +2 ∙ 𝑅𝑃𝐴 +2 ∙ (1 − DPA−US)2 ∙ 𝑁𝑉 𝑈𝑆 ∙ 𝑆𝑁 𝑈𝑆 + +(6) +b) Optimization algorithm +The calibration algorithm relies on the cost function. An +initial combination of 7 parameters was given as an input. Two +steps were then applied. First, several combinations of +parameters were proposed, in which each parameter was drawn +at random following a normal distribution around the initial +guess. Costs of these combinations were computed until +reaching a total of 100 combinations with non-zero cost. The +combination with the smallest cost was used for the second +step: the application of a Particle Swarm optimization +algorithm. +With the numerical calibration phantom, we found a +variability of the output combination after the calibration +algorithm and of the associated cost. This indicates local +minima of the cost function. To mitigate this variability, we +chose to run the optimization algorithm 20 times on the same +dataset. The 20 combinations were then sorted by increasing +cost and the final combination was obtain by calculating the +median of each parameter on the combinations with the 5 +lowest costs. + +3) Metrics for the variability +a) +Acceptability range +The variability of the determination of the parameters was +compared to an acceptability range. Three acceptability ranges +were defined depending on the unit of the parameter. The +acceptability range for length parameters (Δx, Δz) was set equal +to the wavelength Λc. For t0 US, we used the wave period at the +central frequency of the transducer: 0.2 µs. For the angles, we +considered an axial deviation of Λc seen from the lateral +aperture of the array as a significant error. As the pitch of the +array is equal to Λc, the acceptability range for angles was set +to sin−1 1 +64 = 15.6 mrad. +b) +Variability quantifications +To assess the variability induced by the optimization +algorithm, the numerical calibration phantom was used. The +simulated combination reflected the experimental data (mean +over 10 experiments). Absolute difference between obtained +and expected parameters were computed and divided by the +acceptability threshold to be expressed as a percentage. This +metric evaluated the accuracy of the calibration method and +therefore is named Ac. +To assess inter-acquisition variability on experimental data, +parameters were obtained for 10 acquisitions. These parameters +were compared to the mean parameter over the 10 acquisitions. +For each parameter, the mean absolute difference of the +obtained outputs compared to the mean value was computed +(also named mean absolute deviation) and divided by the +acceptability threshold to be expressed as a percentage. This +parameter assesses the repeatability of the entire calibration +process and is called Rp. + +4) Metrics for the spatial resolution and superposition +We quantified the superposition of the US and PA images +using a second phantom comprised of threads (see Ph1, in +section II.D) with a dual contrast. The quantification method +assesses the distance between the images of each thread in USI +and in PAI. As for the image processing used in the calculation +of DUS-PA on the calibration phantom, we determined the +positions of the centroids (both in the US images and the PA +images) for each thread and 5 slices located at y = i×3 mm with +𝑖 ∈ ⟦−2; 2⟧. Each slice was reconstructed with pixel sizes py × +pz = 71 µm × 71 µm. The mean distance of dUS-PA (see equation +(4)) and its standard deviation over the slices and over the +threads of the same material (nylon or polyester) measured the +superposition quality. +For each nylon thread, the spatial resolution was estimated +by fitting the images in each slice i with a 2D-Gaussian model +of equation: +𝑓(𝑥, 𝑧) = 𝐴 exp (− ( +(𝑥−𝑥0)2 +2𝜎𝑥2 + +(𝑧−𝑧0)2 +2𝜎𝑧2 )) +(7) +where 𝐴 is the amplitude, 𝑥0, 𝑧0 are the centroid positions and +𝜎𝑥, 𝜎𝑧 are the standard deviations along 𝑥 and along 𝑧 . The full +width half maximum was calculated along 𝑥 and 𝑧 +as 𝐹𝑊𝐻𝑀𝑖 = 2√2ln 2 × 𝜎𝑖 . As nylon threads can be +considered small compared to Λc, the FWHM is an estimate of +the width of the Line Spread Function (LSF). +D. Imaging phantoms +The first phantom (Ph1) was a wire phantom designed to +evaluate the co-registration capabilities using a phantom with a +geometry different than the calibration phantom and in addition +with a different material. Three 20-µm diameter black nylon +threads (NYL02DS, Vetsuture, France) and three black +polyester threads (Coat Epic 150) were mounted on a 3D- +printed frame similar to Fig. 2(b). They were arranged +symmetrically with respect to the center of the frame and with +various angles. +The second phantom (Ph2) was prepared with agar powder +2% w/v (A1296, Sigma Aldrich, St. Louis, MO, USA) and +cellulose powder 1% w/v (Sigmacell cellulose Type 20, Sigma +Aldrich) in water. Cellulose particles (20 µm) act as ultrasound +scatterers to mimic the scattering properties of biological +tissues for US imaging. The gel was molded in a cylindrical +mold (diameter of 20 mm) with three cylindrical solid +inclusions of 5 mm in diameter and of the same length as the +mold. The agar-cellulose solution was heated to 85°C and +poured into the mold. When the mold was half full, 100-µm- +diameter black polyethylene microspheres (BKPMS 90–106 +um, Cospheric, Santa, Barbara, CA, USA) were spread on the +superior interface and were trapped at the interface during the +solidification of the gel. The mold was then filled with the hot +agar solution. When the gel solidified, the cylindrical +inclusions were removed and filled with water. The +microspheres remained embedded in the gel. Ph2 was placed so +that the cylindrical holes and the plane of spheres were parallel +and perpendicular to the rotation axis, respectively. +The third phantom (Ph3) was prepared with agar powder (2% +w/v) and cellulose powder (1% w/v) in water for the first half +and with agar powder (2% w/v) for the second half. Two +crossed 20-µm diameter black nylon threads were embedded in + + +7 +the gel. +III. RESULTS +A. Accuracy and repeatability of the calibration +The accuracy and the repeatability of the calibration outputs +are presented in Table. I, in percentage of the acceptability +range. First, it can be noticed that all the 7 parameters are fully +within the acceptability ranges, for both Ac and Rp. +For the accuracy, the calibration outputs were compared to +the ground truth thanks to a numerical simulation. The mean +Ac over the 7 parameters was found equal to 26% and Ac had +a maximum of 55% for the Roll parameter. We can then +consider that the developed calibration method enables to +accurately determine the set of parameters. +To evaluate the repeatability, ten acquisitions performed on +various days (distributed in three imaging sessions over one +week) were used. For each acquisition, the optical fibers and +the phantom were repositioned to avoid any bias. Rp estimates +the mean absolute deviation of each parameter. The mean Rp +over the 7 parameters is around 16% and a maximum of 39% +was reached for Δz. Therefore, the calibration is repeatable and +stable over time. +Despite slight variations in the evaluation of each +parameters, the accuracy and the repeatability of the calibration +method are highly satisfying. We can therefore consider the +developed calibration method to be reliable and robust. +B. Superimposition quality and spatial resolution +Fig. 3 presents PA/US images of the slice y =0 (center of the +linear array) of Ph1. Ph1 aims to test if the experimental +calibration remains valid for a phantom different than the +calibration phantom in the spatial arrangement of the threads, +but also in the thread material. The phantom Ph1 is comprised +of three nylon threads and three polyester threads arranged so +that each thread has a different orientation. Only the threads +numbered 2 were set parallel to the y-axis. For an easier +comparison between the two materials, Ph1 was built so that +each of the three nylon threads had a symmetrical polyester +thread with respect to the center of the frame. In Fig. 3, +symmetrical threads have the same number and the suffixes ‘n’ +and ‘p’ refer to nylon and polyester threads, respectively. For +instance, the thread 1n is symmetrical to 1p, with respect to the +center of the image. The nylon threads were grouped on the top +part of the phantom while the polyester threads are grouped on +the bottom part. Video 2 displays rotating MAP images around +z of Ph1 and therefore shows the spatial arrangement of the +threads. As nylon threads are thinner than polyester ones, both +US and PA signals were weaker for nylon. The amplitude in the +reconstructed image was 3 times smaller in PA. To facilitate +the visualization in Fig. 3, the upper and the lower part of the +volumetric images were normalized by their local maximum +and not the global one. The separation between the two parts is +illustrated by the horizontal dashed white line in Fig. 3. We can +visually see that polyester threads appear larger than the nylon +ones both on the PA and US images. This is expected given +their larger diameter. However, the superposition of the images +can be observed for the two materials and for all the threads +regardless of their orientation or position in space (Fig. 3 and +Video2). The blue lateral halo around the white spots in the +combined image (Fig. 3 left) indicates that the lateral resolution +is wider for PA images. +For a more quantitative description, Table. II presents the +superposition distance and the FWHM calculations. The +superposition distance between the center of a thread in PA and +in US is small compared to the US wavelength Λc (less than +10% of Λc) and smaller than the spot obtained for each object +in the image. The superposition distance is not significantly +different for the two materials. This result indicates that the +calibration enables the superposition of nylon thread with +different orientations than in the calibration phantom, and it +validates that the calibration ensures the superpositions for +objects with a dual contrast but in another material. The +standard deviation computed over the three threads and five +imaging planes is small as well compared to Λc. This result +shows the low dispersion of the superimposition distance both +with the thread orientation and with the spatial position in the +imaged volume. +The FWHMs calculated for the nylon threads provides an +estimate of the LSF because the thread diameter is much +smaller than Λc. We first notice that for both US and PA and +for both x and z directions, the FWHM is on the order of +magnitude of Λc. Along the x-direction, the resolution is limited +by the diffraction. The high resolution in the x-direction results +from the large angular aperture provided by the rotation scan +and the synthetic aperture approach. For comparison, the +FWHMx was on the order of 1 mm [23] for a translation-only +scan with the same US array. The low standard deviation of +FWHMx indicates that the resolution is independent of the +position of the object in the volume and its orientation. The +spatial homogeneity associated with the rotate-translation +scheme and previously observed independently in USI [23] and +PAI [22] is then confirmed for the simultaneously co-registered +imaging. +In each direction, the US resolution is slightly better than the +PA resolution (70-100% of Λc for US vs 130% of Λc for PA). + +Fig. 3. Image reconstruction of the central plane of Ph1 (y=0). The +horizontal white dashed line represents the separation between nylon +and polyester zones. Threads with the same number are symmetrical +and n stands for nylon and p for polyester. PA image (left) is presented +on a linear scale while the US image (center) is presented in dB with a +threshold at -30dB. +TABLE I +ACCURACY AND REPEATABILITY STUDY ON THE CALIBRATION + +Yaw +Pitch +Roll +θ +Δx +Δz +𝑡0 𝑈𝑆 +Ac (%) a +2.8 +18 +55 +20 +25 +32 +24 +Rp (%) a +28 +8.3 +8.6 +8.5 +17 +39 +1.1 +a Ac and Rp are expressed in percentage of the acceptability ranges. +TABLE II +RESOLUTION AND SUPERPOSITION ASSESSMENT + +FWHM nylon (µm) +Superposition (µm) + +US +PA +nylon +polyester +along x (mean ± std) +212 ± 18 +387 ± 30 +15 ± 9 +23 ± 17 +along z (mean ± std) +296 ± 24 +379 ± 21 + + +nylon +PAnylon +USr +nylon +PA+US +2n +3n +3p +2p +Ip +polyester +mm +polyester +polyester +8 +Along x, two main factors can explain the FWHMx differences +between US and PA. First, PA images rely on US signals +produced by the illuminated object and not on backscattered US +signals generated by the US array. For small objects, the US +frequency spectrum recorded by the array is then usually +broader in PAI than in USI, and especially contains low +frequencies which may decrease the diffraction-limited +resolution. Second, for one laser excitation, five tilted plane +waves are emitted, which increases the number of independent +views and the spatial sampling in USI compared to PAI. This +sampling factor has been shown to influence the resolution +along x [23]. Along z, which corresponds to the axial direction, +the LSF is mainly influenced by the pulse duration. The pulse- +echo mechanism can explain the better resolution of the US +images compared to PA. +C. Complementary distributions of US and PA contrasts +To further demonstrate the advantages of the dual modality +imaging, phantoms with complementary contrasts were +designed and produced. The images of Ph2 and Ph3 are +displayed in Fig. 4 and 5, respectively. +For the ultrasound contrast, Ph2 is a homogeneously +scattering medium (agar with cellulose) with three cylindrical +and anechoic holes filled with water. For the PA contrast, +numerous black-dyed micro-spheres are arranged in a central +plane (Fig. 4). The optical absorption of the agar gel and the +water are negligible. In the first row of the Fig. 4, a MAP image +along y is shown for PAI, USI, and the superposition of both. +Rotating MAP images round the z-axis are presented in Video +3. The 20-mm diameter cylindrical shape of the phantom could +be retrieved both in PAI and USI. One can note the +homogeneous image quality in the xz-plane for both modalities. +In the second row of Fig. 4, images of a slice perpendicular to +the z-axis are presented. The slice was chosen to cut two of the +three holes. The homogeneity of the USI image along the y-axis +can be observed. As expected, the three holes appear with a +negative contrast for both PAI and USI independently of their +spatial position. The holes allow to further validate the +superimposition of the two modalities. They are concentric in +USI and in PAI. We can notice that the outlines of the holes are +blurrier and the diameter of holes seems smaller on the US +images. This effect can be attributed to the stronger side lobes +in the US images induced the log compression (display in dB) +of the color scale compared to the linear scale used in PAI. Such +side lobes are also visible in Fig. 2(c). The microspheres were +hardly visible in the agar matrix in USI, or only when numerous +spheres are gathered together, while they appear with a strong +contrast in PAI. With the superposition, both the agar and the +microspheres are visible. This is of interest to assess the +distribution of PA contrast agents in an organ which is +homogenously echoic for instance. The US images gives the +contour of the phantom, which is similar to the anatomical +context, while the PA image gives the distribution of the +marked spheres which are analogous to a contrast agent. In a +clinical application, the PA contrast agents could be +therapeutic nanoagents accumulating locally. Holes are +mimicking bodies with a negative contrast such as cysts. +For the second phantom Ph 3, media with two different +ultrasound contrasts were used: agar with cellulose is more +echogenic that agar alone. Two black nylon threads bring the +optical absorption contrast. Images of Ph3 are presented in Fig. +5. The US image in the first row (MAP along z) and the rotating +MAP images in Video 4 clearly show the contrast between the +two blocks of gel. The nylon threads are visible in the MAP US +images, but mainly due to their elongated shape as they are +barely differentiable from the surrounding speckle in the slice +perpendicular to the y-axis and intersecting the two threads in +the agar with cellulose part (second row of Fig. 5). Agar and +agar with cellulose do not have any contrast in PA. However, +the threads show a homogenous and high contrast in PAI over +the two agar blocks. This phantom could typically mimic a +blood vessel (black thread), highly visible in PAI, perfusing +two different organs with different echogenicity. +IV. DISCUSSION +We +demonstrated +high-quality, +volumetric +and +simultaneously co-registered PAI and USI. The simultaneous +dual imaging was made possible by the use of a linear US +transducer array, and the high quality (resolution, contrast, +visibility) of the volumetric images resulted from the large +synthetic angular aperture enabled by the rotate-translate scan + + +Fig. 5. Image reconstruction of Ph3. Schematic drawing of Ph3 are +displayed in the left column. The first row displays MAP images along +z while the second row shows a slice perpendicular to the y-axis. US +images were thresholded at -40 dB. Along y-axis, left part is agar with +cellulose and right part is agar alone. + + + +Fig. 4. Image reconstruction of Ph2. Schematic drawing of Ph2 are +displayed in the left column. The first row displays MAP images along +y while the second row shows a slice perpendicular to the z-axis. PA +images are presented in linear color scale. However, because a few +bright spots (probably clusters of microspheres) were dominating the +color scale and hiding the rest, voxel values were saturated at 40% of +their maximum before normalization. US images were thresholded at - +40 dB. + +5mm5mm +9 +geometry. Images showed a homogenous quality over a large +imaged volume (cylinder of diameter of 21 mm and length of +19 mm). The effective synthetic aperture for both imaging +modalities and the superposition of the PA and US images for +features having a dual contrast required an accurate +determination of the positions of the US array. To this end, we +developed and validated a calibration method, which was +determined to be both accurate and repeatable. This initial +calibration process allowed for the subsequent acquisitions of +images without the need of fiducial markers on every imaged +volume. We demonstrated the superposition of PA and US +images with phantoms having a dual contrast and the +complementarity of the mapped information with phantoms +having complementary spatial distributions of US and PA +contrast agents. +The calibration was based on the combination of a dedicated +calibration phantom, a cost function and an optimization +process. 7 parameters were determined for our scan geometry +to obtain superimposed and sharp PA and US images. The +mechanical mount and the position sensors of the stages ensure +that the determined parameters remain valid for subsequent +scans as long as no deformation or accidental motion of the +array occurs. For this study, no significant variation could be +noticed over a period of one week. The calibration phantom +was easy to build and was comprised of four well-separated +threads for a facilitated identification and measurements of +local properties. Our cost function used only five imaging +planes to avoid the reconstruction of the entire volume and the +associated long computation time. Our selection of metrics with +the sharpest variations with regards to the parameters implied +that the cost was dominated by the properties of the US image, +but with an influence of the properties of PA images. For the +optimization algorithm, we found that the selection of the initial +guess was a crucial step due to the presence of local minima in +the cost function. Additionally, the particle swarm optimization +was found to be more effective to determine a solution close to +the ground truth than a downhill simplex method. The +refinement of the optimization algorithm was beyond the scope +of this study as we focused on the effectivity of the method, but +it will be considered in a future study. +This calibration procedure was found effective for our +system although it requires the initial imaging of a dedicated +phantom. In the past decade, several multiperspective or +multiview imaging systems with US transducer arrays have +been developed both in PA imaging and in US imaging, +independently, leading to various calibration procedures. For +US imaging, coherent compounding of images acquired with +two US transducer arrays have been developed in 2D [30] and +in 3D [31]. Dedicated calibration phantoms were also used that +consisted in isotropic scatterers (5 wires in 2D and 3 spheres in +3D). The phantoms were combined with a cost function linked +to the coherence of the echoes from the isotropic scatterers on +the different elements of the array to determine geometrical +parameters (4 parameters in 2D and 6 in 3D). A simplex search +method was used. The calibration showed an improvement in +the contrast and in the resolution of the images but the use of +isotropic scatterers resulted in long computation time of +volumetric images during the calibration process. Incoherent +compounding of US images was also investigated for 2D [32] +and 3D [33] imaging. The spatial transformation matrix +between the positions of the arrays were assessed directly using +the images acquired on the sample of interest. This approach +could enable free hand scanning and imaging of moving organs +but could not achieve a synthetic aperture approach which +therefore limits the final image quality. In PA imaging, +multiperspective or multiview imaging is almost inherent to the +modality as PA tomography with spherical and hemispherical +scans of an US detector has been investigated in the early PA +scanners [34]. However, planar detection geometries able to +acquire 3D PA images have been angulated to enhance the +image view with a synthetic aperture approach [35], [36]. In ref +[35], two planar arrays were assembled in a rigid configuration. +An initial calibration was performed by imaging a dedicated +phantom comprised of three threads with different orientations +with each array independently, segmenting the images of the +threads and determining the rigid transformation between +arrays. In ref [36], fiducial markers were incorporated in the +imaged region. Multiview imaging was also performed by +stitching volumetric images [37], which is similar to incoherent +compounding and relies on the features obtained in partially +overlapping images. In brief, no standard calibration exists +either in USI or PAI, but only dedicated calibration phantoms +were shown to allow synthetic aperture without fiducial +markers. Additionally, while for a single imaging modality, +some parameters could compensate each other, which put less +constraints on the calibration method, in volumetric and +simultaneous PAI-USI, the superposition and sharpness of the +dual images could only be obtained with accurately determined +parameters, in particular because of the different travel times in +PAI and in USI between the US transducers and the voxels. To +the best of your knowledge, we presented here the first dual +PA-US 3D calibration method. +The co-registered volumetric maps of PAI and USI allow to +obtain complementary information (Fig. 4 and 5). In the frame +of longitudinal studies with different imaging sessions, the US +image is expected to give the anatomical reference to better +understand and to co-register images acquired at different time +points, while PA images could reveal molecular or functional +phenomena with a slow kinetic, such as the accumulation of a +nanoparticular contrast agent with a long circulation time. +For other simultaneously co-registered multimodal imaging +such as PET/CT, the anatomical imaging modality contains +information that can be used to improve the image quality of +the functional imaging modality. For PAI/USI, recent work on +the reconstruction of 2D PA images showed an improvement +of the PA image quality of blood vessels using structural +information from ultrasound images [38]. Additionally, light +fluence distribution could be modeled using US images and +used to improve PA image quality [39]. +The study presented here was performed at a single optical +wavelength. Multispectral approaches [40] will be investigated +with the developed scanner to enable discrimination between +different chromophores and contrast agents and to evaluate +physiological parameters such as the oxygen saturation. On the +US side, the system presented here operated at an US center +frequency of 5 MHz. However, the rotate-translate scan can be +scaled to other ultrasound frequencies to gain sensibility and +information on other spatial scale [22], [41]. The spatial + + +10 +resolution increases with the US frequency of the array for both +PAI and USI, but higher frequencies will also improve the +sensitivity to small absorbing regions in PAI. Linear US arrays +are commercially available in a wide range of center +frequencies. +Building on the in vitro proof of concept presented here, we +are currently adapting the scanner for in vivo imaging with the +addition of an acoustic coupling system to remove the large +water tank. 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Ntziachristos, “Ultra- +wideband three-dimensional optoacoustic tomography,” Opt. +Lett., vol. 38, no. 22, pp. 4671–4674, Nov. 2013, doi: +10.1364/OL.38.004671. + + + + diff --git a/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/load_file.txt b/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..777fc209571beaf242a5c7416affe28740e2b575 --- /dev/null +++ b/qtAyT4oBgHgl3EQfzvl6/content/tmp_files/load_file.txt @@ -0,0 +1,1041 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf,len=1040 +page_content='1 Abstract— Volumetric dual photacoustic (PA) / ultrasonic (US) imaging with precise spatial and temporal coregistration can provide valuable and complementary information for diagnosis and monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Considerable research has sought to combine 3D PA/US imaging in configurations that can be transferred to clinical application but technical compromises currently result in poor image quality either for photoacoustic or ultrasonic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Simultaneous 3D PA/US tomography was implemented here by interlacing PA and US acquisitions during the rotate-translate scan of a 5-MHz linear array (12 angles and 30 mm translational range to image a cylindrical volume of 21 mm diameter and 19 mm length within 21 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Volumetric image reconstruction was performed with synthetic aperture approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' An original calibration method was developed to estimate 6 geometrical parameters and 1 temporal off-set providing sharpest and best superimposed reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Calibration thread phantom design and choice of metrics to build the cost function were based on analysis of a numerical phantom and the final selection demonstrates a high estimation accuracy of the 7 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Experimental estimations validated the calibration repeatability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Experiments in an additional phantom showed a superposition distance between thread centers identified in the PA and US images to be smaller than 10% of the acoustic wavelength, and a spatial resolution on the order of the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Dual mode 3D imaging with high-quality co-registration and excellent, uniform spatial resolution was further demonstrated on phantoms with complementary contrasts, and should contribute to more sensitive and robust imaging to detect and follow biological changes or the accumulation of nanoagents in living systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Index Terms—Tomography, rotate-translate scan, volumetric imaging, simultaneous dual imaging, photoacoustic, ultrafast ultrasound imaging I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' INTRODUCTION OLUMETRIC and simultaneously co-registered multimodal imaging is increasingly developing in biomedical imaging due to the rich, multiplexed and complementary anatomical– functional information that can be precisely spatially and temporally correlated [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The combination of positron emission tomography (PET) and computed tomography (CT), introduced in the 1990s, was among the first volumetric and simultaneous bimodal imaging system to be clinically available [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In this dual-modality, CT provides the anatomical context needed to interpret the functional PET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Various combinations of volumetric imaging have since been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For example, PET has been integrated with magnetic resonance imaging (MRI) [1], [2], photoacoustic imaging (PAI) and optical coherence tomography (OCT) have been superimposed [3] and PET has been combined with Doppler ultrasound imaging [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The advantages of simultaneously acquired, coregistered multimodal images are manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The information gained within one single imaging session is maximized which reduces the scan time and increases benefits for patients, researchers and clinical-management teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Typically, complementary spatio-anatomical and functional information can be obtained through the combination of modalities for a more comprehensive characterization of the region under examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Precisely correlated (both spatially and temporally) information is essential for a precise characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In addition, volumetric imaging provides a detailed view of regions under examination from various orientations for improved diagnosis, and facilitates the investigation and comparison for longitudinal studies or to monitor the progression or regression of pathology in response to therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Ultrasound imaging (USI) and PAI are complementary imaging modalities since USI provides the anatomical correlation for the molecular information supplied by PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Indeed, USI is sensitive to differences in the mechanical microstructures of tissues and can be used to delineate organs and lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' USI is used in clinical routine to obtain anatomical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PAI is sensitive to optical absorption and can provide information on hemoglobin oxygenation and detect molecular and nanoparticular contrast agents [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Both USI and PAI are based on the detection of ultrasonic signals and can therefore be implemented with the same ultrasound detector for simultaneous co-registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Recorded ultrasonic signals are generated in situ by the optical absorption of a laser excitation for PAI, while, for USI, they are created with the incident ultrasonic pulse and backscattered from structures with different acoustic impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In USI the transducer generally operates in both emission and reception modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For 2D imaging, simultaneously co-registration of USI and PAI has already been demonstrated with a single ultrasound (US) detector array [7], [8] and with a dual array configuration [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, PAI is intrinsically 3D in nature due to scattering of the excitation light by the biological tissue, which leads to out- of-plane artifacts in 2D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Moreover, elongated Volumetric and Simultaneous Photoacoustic and Ultrasound Imaging with a Conventional Linear Array in a Multiview Scanning Scheme Clément Linger, Yoann Atlas, Remy Winter, Marine Vandebrouck, Maxime Faure, Théotim Lucas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Lori Bridal, and Jérôme Gateau V 2 structures like blood vessels have a strong directionality in PAI and may not be visible in the limited view configuration of 2D PAI, when they emit outside of the limited angular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Spherical US detector matrices (2D arrays) with a large angular aperture have been developed specifically for 3D PAI, using transducers that could also emit ultrasonic waves [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Given the limited number of elements used to cover the spherical surface, spherical US matrices can be considered sparse for ultrasonic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A large, sparse angular aperture can provide high quality photoacoustic (PA) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, US images obtained with such PAI spherical arrays present strong artifacts, that have been shown to be reduced using processing to extract the Doppler signal [12], [13] or the signal from sparse US contrast agents [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Indeed, pulse-echo US images are densely filled with echogenic structures and high- quality, 3D USI requires US matrices with a high spatial sampling and a large number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Alternatively, planar US transducer matrices that were developed for 3D USI have been tested for 3D PAI [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, the limited angular aperture of such US transducer matrices and the poor sensitivity of the small elements lead to limited view artifacts, limited spatial resolution and poor sensitivity for 3D PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In short, simultaneous co-registration of volumetric PAI and USI with US matrices currently results in poor image quality for one of the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A possible alternative to US matrices lies in the mechanical scanning of US linear arrays that have been designed for 2D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Several systems based on a translational scan of a linear array have been developed recently and their ability to provide simultaneously co-registered 3D USI / PAI has been demonstrated [16]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In this configuration, linear arrays operating in simultaneous 2D USI / PAI are scanned in the elevational direction, perpendicularly to the imaging plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The 2D images are then stacked to obtain a volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This approach is easy to implement and to transfer to a clinical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, the angular aperture in the elevational direction of a linear US array is very limited, which induces strong limited view artifacts in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This issue is not addressed with the translational scan, even when synthetic aperture focusing is used [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Moreover, the spatial resolution in the translational direction is strongly degraded compared to the in- plane resolution both in PAI and USI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Adding a rotational motion and implementing a rotate-translate synthetic aperture scanning of a linear US array has been shown to effectively increase the angular aperture and to highly improve the volumetric image quality compared to a translational scan [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The rotate-translate approach has been demonstrated independently on two different imaging systems for 3D PAI [22] and for 3D USI [23], with a significant decrease of the scan duration between the two demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In this paper, we demonstrate simultaneous co-registration of 3D PAI and 3D USI with a single rotate-translate scan of a linear US array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' After a calibration step that enables us to finely determine the position of the array in the scanning system, we validate the co- registration performance in vitro with an imaging phantom presenting both US and PA contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We also illustrate the capabilities of the system and the complementarity of volumetric USI and PAI by imaging phantoms that have mixed structures with either US or a PA contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' MATERIALS AND METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Experimental set-up The experimental setup is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' It can be divided in four main parts: (1) the optical excitation comprising a nanosecond laser and an optical fiber bundle, (2) the US acquisition system consisting of an US linear array driven by a programmable US platform, (3) the scanning system comprising two motorized stages and their motion controller, and (4) the synchronization system piloted with a programmable trigger generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The acquisition process was fully automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' An optical parametric oscillator laser (SpitLight 600 OPO, Innolas Laser GmbH, Krailling, Germany) delivering < 8 ns pulses with a pulse repetition frequency (PRF) of 20 Hz was used to generate the optical excitation at 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A bifurcated fiber bundle (CeramOptec GmbH, Bonn, Germany) guided the light toward the imaged volume to obtain a bilateral illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean laser energy at each fiber output was estimated to be around 6 mJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The fixed PRF of the Laser sets the time base for the acquisition sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The pulse energy was recorded using a pyrometer incorporated in the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The US transducer array was a 128-element linear array (L7– 4, ATL) driven by a programmable, 64-channel US machine (Vantage, Verasonics, WA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For all the transmit events and all the receive events, only the 64, central elements of the array were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Each Laser pulse triggered a receive-only event to record the PA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Between two laser pulses, transmit events produced “plane wave” ultrasound emissions (a beam that is unfocused in the lateral direction of the array) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='25 ms after each laser pulse, five tilted, plane waves were emitted with steered angles of -4°, -2°, 0° (the 64 elements in the transducer array were fired at the same time), 2° and 4° at a PRF of 160 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Transmitted US pulses were 1 cycle long at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Pulse-echo US signals and PA signals were recorded at a sampling frequency of 20 MS/s and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='5 MS/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A constant gain was adjusted for the PA and US signals to ensure a good amplitude digitalization of received signals without saturation (no time gain compensation was applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3 The US array was mounted on a rotate-translate scanning system described in detail in ref [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In brief, a rotation stage was mounted on a translation stage (Physik instrumente, Karlsruhe, Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The array axis (axis along the row of its elements) was aligned with the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The rotation stage was moved to 12 different angles with a nominal angular sampling period of Δα = 4°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each angle, the imaged volume was scanned with a translation range L = 30 mm and a linear sampling period Δℓ = 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Relative to parameters used in ref [23], the scan parameters, in particular the translation velocity, had to be adjusted to adapt for the PRF of the Laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The motion was continuous and, for each laser emission and each US plane wave emission, the motor positions were recorded and stored in the motor controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The scan was automated and initiated with an external trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The motor positions are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To synchronize the instruments and to coordinate the acquisition sequence, a trigger generator (BNC Model 577, Berkeley Nucleonics, San Rafael, CA, USA) was used to send external triggers simultaneously to the Laser, the programmable US system and the motion controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The imaged sample, the array elements and the outputs of the fiber bundle were immersed in a water bath filled with tap water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The speed of sound in the bath was derived from the water temperature [25], measured with a thermometer (HI98509, Hanna instruments, Lingolsheim, France).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For a volumetric acquisition, a total of 411 laser pulses were fired for a total duration of 21 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Video 1 presents a video of the acquisition and illustrates the rotate-translate scan of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' At the end of the acquisition, the radiofrequency data of the PA and US events, the motor positions and the pyrometer values were transferred to a computer for signal processing and image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Image reconstruction 1) Image Reconstruction Algorithm The 3D image grid was defined in a fixed Cartesian coordinate system (O, ex, ey, ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The vectors ey and ez correspond to the rotation axis and the radial direction, respectively, when the rotation angle equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The vector ex completes the orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The origin O is chosen so that xO and zO are the coordinates of the rotation axis, when the translation stage is at its center position (ℓ=0), and yO is the coordinate of the center of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In this grid, the voxel dimensions were chosen equal to px × py × pz = 71 µm × 143 µm × 71 µm, with an anisotropy reflecting the best- expected resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The image volume is defined by a diamond-shaped, cross-sectional area (DSCA, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2) in the xz- plane (length of the diagonal L= 30 mm, and centered at (x=0, z =25 mm) ) and the active length of the array (19 mm) along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA signals were divided by the corresponding pyrometer value to compensate for the pulse-to-pulse energy fluctuations of the Laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA signals were bandpass filtered between 2 MHz and 10 MHz (Butterworth, order 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Image reconstruction was performed with simple delay-and- sum beamforming algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The one-way (PA) and two-way (US) travel times between the US transducer element positions (xn, yn, zn) and each imaged voxel (xp, yp, zp) were computed, assuming a constant speed of sound, c, in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Then, the value of each voxel was computed by summing the signal values assessed at the voxel-associated travel time, over all the elements of the array and all the different tomographic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The US image reconstruction algorithm was detailed in ref [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The same apodizations were used for PA and US image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Three-dimensional, envelope-detected images were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The 3-D images were displayed using maximum amplitude projection (MAP) images along the axes of the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Rotating MAP images around the z axis were obtained with the 3D project option of ImageJ [26] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The colorscale used for the images is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2) Time delays for the reconstruction The recording of the ultrasound signals was set to start at the same time as the Laser emission for the PA acquisition or at the same time as the ultrasound emission for the US acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, due to the lens effects [27] and additional time delays induced by the acquisition hardware, we found that two time parameters t0 PA and t0 US needed to be determined to convert the voxel-associated travel times into the time indexes of the recorded signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' t0 PA and t0 US correspond to effective offsets of the recorded signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' t0 PA was determined experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Two 20-µm diameter black nylon threads (NYL02DS, Vetsuture, France) were positioned perpendicularly to the imaging plane of the array and in the vicinity of the elevation focus (around 25 mm from the face of the array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' They were illuminated with the Laser light and PA signals were recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Given their small diameter Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' (a) Annotated picture of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' (b) Motor positions for the PA acquisitions over one scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For a better readability, positions corresponding to the US and PA acquisitions are indicated only for the inset in the upper right corner of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Positions shown in grey (drop out) were not used for the reconstruction, but are acquired due to the continuous motion of the motors (c) Colormap used for all images: PA images are represented in shades of blue and US images in shades of orange, the sum of the two leading to white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA signals are presented on a linear scale while US signals are presented within the range from -40 to 0 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Programmable Us system Linear USarray Rotation S1308 Motion Optical fibers cantroller Phantom Translation Laser stage 0000 16 20 e [mm] US .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='5 US Dropout PA + Target 20 20 40 4 with regards to the wavelength at the center frequency of the array (Λc≈300µm), the threads can be assumed to be point absorbers in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2D images (py × pz = 71 µm × 71 µm) were beamformed for different values of t0 PA, with the speed of sound determined with the temperature of the water bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The one-dimension Brenner’s gradient of the images was computed along the lateral dimension of the array: 𝐹𝐵𝑟𝑒𝑛𝑛𝑒𝑟 1𝐷 = ∑ (𝑓(𝑦 + 2, 𝑧) − 𝑓(𝑦, 𝑧))2 y,z (1) where f(y,z) represents the gray level intensity in the 2D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=" The Brenner's gradient provides a quantitative measure of image sharpness and was shown to be an efficient metric for the speed of sound calibration in PA [28]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' It was found to be maximized for t0 PA = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='3 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In the recorded signals, |t0 PA| corresponded also to the time of arrival of PA signals generated by a metalized mylar film (space blanket) pressed against the face of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3) Spatial transformation from motor to array positions For each tomographic position, the experimental acquisition gives the motor positions in their own coordinate systems (integrated position sensors), while the reconstruction algorithm requires the position and orientation of the array in the fixed coordinate system (O, ex, ey, ez).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Therefore, the spatial transformation matrix from the mobile Cartesian coordinate system (Oa, u, v, w) attached to the transducer array to (O, ex, ey, ez) needs to be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The vector u corresponds to the elevation direction, the vector v to the long axis along the row of the elements and the vector w to the axial direction of the linear array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The origin Oa is the center of the active transducer aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Oa is located on the interface between the transducer array and the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A tomographic position is described by the linear and angular motor positions (ℓ, α), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The translation length equals zero (ℓ=0) when the translation stage is at the center position, and the rotation angle equal zero (α=0) when the radial axis is vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The position of the center of the element number n of the array (𝑛 ∈ ⟦1, 𝑁⟧ with N=64) in (O, ex, ey, ez) can be decomposed as: ( 𝑥𝑛 𝑦𝑛 𝑧𝑛 ) = ( 𝑥𝑂𝑎(ℓ, α) 𝑦𝑂𝑎(ℓ,α) 𝑧𝑂𝑎(ℓ, α) ) + (𝑛 − 1 − 𝑁−1 2 ) ∙ 𝑝 ∙ ( 𝑣𝑥(α) 𝑣𝑦(α) 𝑣𝑧(α) ) (2) With p the interelement spacing of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Here, p = 298 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Moreover, the transformation matrices are: ( 𝑢𝑥(α) 𝑣𝑥(α) 𝑤𝑥(α) 𝑢𝑦(α) 𝑣𝑦(α) 𝑤𝑦(α) 𝑢𝑧(α) 𝑣𝑧(α) 𝑤𝑧(α) ) = ℛ𝛼 ∙ ℛ𝑃𝑖𝑡𝑐ℎ ∙ ℛ𝑌𝑎𝑤 ∙ ℛ𝑅𝑜𝑙𝑙 (3) ( 𝑥𝑂𝑎(ℓ, α) 𝑦𝑂𝑎(ℓ, α) 𝑧𝑂𝑎(ℓ, α) ) = ℛ𝛼 ∙ ( Δ𝑥 0 Δ𝑧 ) + ℓ ∙ ( cos(θ) ∙ cos(φ) sin(θ) cos(θ) ∙ sin(φ) ) (4) With ℛ𝛼 the rotation matrix around the y-axis due to the motion of the rotation motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' ℛ𝑅𝑜𝑙𝑙, ℛ𝑃𝑖𝑡𝑐ℎ and ℛ𝑌𝑎𝑤 are rotation matrices around the 1st-axis, the 2nd-axis and the 3rd- axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Δx and Δz are the coordinates of Oa when α=0 and ℓ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The product of these rotation matrices and the offsets models the misalignment of the array axis v compared to the rotation axis of the stage ey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The angles θ and φ account for misalignment of the translation axis with ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Therefore, a total of 7 geometrical parameters independent of (ℓ, α) need to be determined: Roll, Pitch, Yaw, Δx, Δz, θ, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Calibration method The inadequate estimation of the reconstruction parameters results in degraded image quality in terms of sharpness and a misalignment of the PA and US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Therefore, we developed a calibration method for the determination of the required parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The calibration method needs to be applied once and the parameters can be used to reconstruct subsequent acquisitions as long as the setup is not mechanically misaligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The calibration method combines a calibration phantom and an optimization algorithm to estimate the 7 parameters: t0 US, Roll, Pitch, Yaw, Δx, Δz, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The parameter φ was fixed equal to zero because the perpendicularity between the translation axis and ez was precisely ensured by the mechanical design, and because of equivalent solutions for our optimization algorithm and for different sets of (Pitch, φ) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1) Calibration Phantom The required specifications for the calibration phantom were that it be simple to build, easy to use and does not require an absolute and tedious positioning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We developed a wire phantom, inspired by phantoms used for the calibration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' (a) Imaged region (also called region of interest (ROI)) of the calibration phantom in a 3D coordinate system with dimensions in mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' (b) Picture of the calibration phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Four 20-µm nylon threads are mounted on a yellow 3D-printed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To ease the readability, the threads have been highlighted with the same color as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A projection of the ROI is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' (c) Volumetric images reconstructed with a set of optimized parameters: first row: photoacoustic image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' second row: ultrasound image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Third row: Combined PA/US image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The colorscale is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Each image is a maximum amplitude projection (MAP) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The visible part of the diamond-shaped, cross-sectional area (DSCA) is shown with yellow dashed lines in the MAP images along y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 10 10 X(mT) MAP along x MAP alongy MAP along z PA US 5mm PA+US 5 freehand 3D ultrasound systems such as Z-fiducial phantoms [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Wires or threads have several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' First, black threads have a dual contrast PA and US compared to water, and are therefore expected to be superimposed on the dual-modality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Second, straight threads provide elongated and uniform structures that can be easily intersected and identified in a volumetric image even with a sparse sampling in one direction (as opposed to small spheres for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Third, the orientation can be varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Finally, with a few non-crossing and well-separated threads, the segmentation of the 3D image allows local assessments of the image quality, in particular, the image sharpness in slices intersecting the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Our calibration phantom is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' It is comprised of four threads with two orientations: two threads parallel to one another and positioned in a horizontal plane (Thread 1 and 3), and two threads rotated by ± 22° placed in parallel planes below and above, respectively (Thread 2 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The angle was chosen to provide a good sensitivity to the different parameters to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The spacing between the threads was chosen so that the phantom fits inside the imaged volume when the parallel threads are roughly aligned along the y-axis and at z ≈ 25 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The orientation of the phantom was chosen so that the threads appear as points in xz-planes whose reconstruction sharpness is highly sensitive to the tomographic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The experimental calibration phantom was implemented with 20-µm diameter black nylon threads (NYL02DS, Vetsuture, France) mounted on a 3D-printed frame (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A numerical phantom was also designed in order to validate the calibration method with a known set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For this numerical phantom, each thread was discretized into point sources (PA) or point scatterers (US) spaced by Λc/5 where Λc is the ultrasound wavelength at the center frequency of the transducer (Λc≈300µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The numerical simulation is based on the sum over all the points of time signals delayed by the travel times of the ultrasound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A one cycle sinusoidal signal at 5 MHz with a gaussian envelop was used for the US simulation, and the derivative of a gaussian pulse (standard deviation 17 ns) for the PA simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The finite size (width and height) and cylindrical focusing of the array elements were modeled with a cylindrical transducer discretized with point transducers spaced by Λc/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Additionally, the 7 geometrical parameters and the two time-offsets were considered in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2) Calibration algorithm The calibration algorithm is based on an optimization algorithm that minimizes a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' a) Cost function For one set of the 7 parameters and one volumetric acquisition comprising PA and US (5 steered angles) data, the cost function was assessed using the images of 5 slices located at y = i×3 mm with 𝑖 ∈ ⟦−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2⟧, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' centered and distributed to avoid edge effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each slice, we reconstructed two 30-mm width square images (one PA and one US) centered around (x = 0, z = 25 mm) and with pixel sizes py × pz = 71 µm × 71 µm using the 3D reconstruction algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Each envelope-detected image was thresholded at one fourth of its maximum pixel value to produce a binary image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The fourth largest connected components of the binary image were identified as regions and each region is expected to correspond to one thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The cost was set to zero if less than four connected components were counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each region, the centroid position (𝑥𝑗, 𝑧𝑗) with weights based on the grayscale image intensity value was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The distance between the centroids determined in the US image and in the PA image in slice i was: 𝑑US−PA(𝑖, 𝑗) = √(𝑥𝑗 𝑈𝑆 − 𝑥𝑗 𝑃𝐴) 2 + (𝑧𝑗 𝑈𝑆 − 𝑧𝑗 𝑃𝐴) 2 (4) In each image, the four centroids were sorted by increasing angular position and each was attributed to one of the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For Threads 2 and 4, several quantities were computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' First, a linear regression was performed with the five centroids (one per slice) of each thread and the coefficient of determination was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean over the two threads of the coefficient of determination gave R2US and R2PA for the US images and the PA images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' R2US and R2PA equals one when the threads are reconstructed as straight structures in the volume and when the image quality allows adequate segmentation of the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' R2US and R2PA were found to be sensitive to errors for the parameters Pitch, Δx and Δz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Second, the mean distance of dPA-US, named DPA-US, was computed over the two threads and the five slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' DPA-US evaluated the superposition of the PA and US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' DPA-US is expected to be equal to zero for the correct set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' DPA-US was found to be sensitive to errors in the parameter t0 US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' DPA-US is expressed in mm and the cost was set equal to zero for DPA-US > 1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Finally, the local normalized variance of the US images was computed in a square region of 2 mm-width (twice the translation step) around the centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The normalized variance quantifies variations in the pixel values about the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' It is equal to the variance of the pixel values over their mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This measurement of the image sharpness was reported for an autofocus method [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean of the local normalized variance over the two threads and over the five slices was named NV US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' NV US is expected to be maximal for the correct set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' NV US was found to be sensitive to errors in the parameters: Pitch, Yaw, Δx, Δz, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Finally, a normalized two-dimension squared gradient of the entire US images (four threads) was computed as a sharpness metric: 𝑆𝑁 2𝐷 = 1 ∑ 𝑓(𝑥,𝑧) 𝑥,𝑧 (∑ (𝑓(𝑥 + 1, 𝑧) − 𝑓(𝑥, 𝑧))2 𝑥,𝑧 + ∑ (𝑓(𝑥, 𝑧 + 1) − 𝑓(𝑥, 𝑧))2 𝑥,𝑧 ) (5) where f(x,z) represents the gray level intensity in the 2D ultrasound image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean of the normalized squared gradient over the five slices was named SN US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' SN US was found to be sensitive to errors in the parameters: Roll, Pitch, Yaw, Δx, Δz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The metrics R2US, R2PA, DPA-US, NV US and SN US were selected based on numerical simulations and observations of the variations in the metric values induced by each parameter individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Among the diversity of tested metrics, we kept the most sensitive and we made sure that all the parameters were covered by at least one metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We found that the normalized variance and normalized squared gradient were more efficient for US images than for PA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This could be explained by the fact that the US image reconstruction combines data for five steered angles, which may induce stronger variations when the parameters are away from their expected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We combined the metrics with a product and the cost was then defined by: 6 𝐶𝑜𝑠𝑡 = { 0 for DPA−US ≥ 1 mm −𝑅𝑈𝑆 2 ∙ 𝑅𝑃𝐴 2 ∙ (1 − DPA−US)2 ∙ 𝑁𝑉 𝑈𝑆 ∙ 𝑆𝑁 𝑈𝑆 (6) b) Optimization algorithm The calibration algorithm relies on the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' An initial combination of 7 parameters was given as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Two steps were then applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' First, several combinations of parameters were proposed, in which each parameter was drawn at random following a normal distribution around the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Costs of these combinations were computed until reaching a total of 100 combinations with non-zero cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The combination with the smallest cost was used for the second step: the application of a Particle Swarm optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' With the numerical calibration phantom, we found a variability of the output combination after the calibration algorithm and of the associated cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This indicates local minima of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To mitigate this variability, we chose to run the optimization algorithm 20 times on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The 20 combinations were then sorted by increasing cost and the final combination was obtain by calculating the median of each parameter on the combinations with the 5 lowest costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3) Metrics for the variability a) Acceptability range The variability of the determination of the parameters was compared to an acceptability range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Three acceptability ranges were defined depending on the unit of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The acceptability range for length parameters (Δx, Δz) was set equal to the wavelength Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For t0 US, we used the wave period at the central frequency of the transducer: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the angles, we considered an axial deviation of Λc seen from the lateral aperture of the array as a significant error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' As the pitch of the array is equal to Λc, the acceptability range for angles was set to sin−1 1 64 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='6 mrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' b) Variability quantifications To assess the variability induced by the optimization algorithm, the numerical calibration phantom was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The simulated combination reflected the experimental data (mean over 10 experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Absolute difference between obtained and expected parameters were computed and divided by the acceptability threshold to be expressed as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This metric evaluated the accuracy of the calibration method and therefore is named Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To assess inter-acquisition variability on experimental data, parameters were obtained for 10 acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' These parameters were compared to the mean parameter over the 10 acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each parameter, the mean absolute difference of the obtained outputs compared to the mean value was computed (also named mean absolute deviation) and divided by the acceptability threshold to be expressed as a percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This parameter assesses the repeatability of the entire calibration process and is called Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4) Metrics for the spatial resolution and superposition We quantified the superposition of the US and PA images using a second phantom comprised of threads (see Ph1, in section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='D) with a dual contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The quantification method assesses the distance between the images of each thread in USI and in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' As for the image processing used in the calculation of DUS-PA on the calibration phantom, we determined the positions of the centroids (both in the US images and the PA images) for each thread and 5 slices located at y = i×3 mm with 𝑖 ∈ ⟦−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2⟧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Each slice was reconstructed with pixel sizes py × pz = 71 µm × 71 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean distance of dUS-PA (see equation (4)) and its standard deviation over the slices and over the threads of the same material (nylon or polyester) measured the superposition quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each nylon thread, the spatial resolution was estimated by fitting the images in each slice i with a 2D-Gaussian model of equation: 𝑓(𝑥, 𝑧) = 𝐴 exp (− ( (𝑥−𝑥0)2 2𝜎𝑥2 + (𝑧−𝑧0)2 2𝜎𝑧2 )) (7) where 𝐴 is the amplitude, 𝑥0, 𝑧0 are the centroid positions and 𝜎𝑥, 𝜎𝑧 are the standard deviations along 𝑥 and along 𝑧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The full width half maximum was calculated along 𝑥 and 𝑧 as 𝐹𝑊𝐻𝑀𝑖 = 2√2ln 2 × 𝜎𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' As nylon threads can be considered small compared to Λc, the FWHM is an estimate of the width of the Line Spread Function (LSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Imaging phantoms The first phantom (Ph1) was a wire phantom designed to evaluate the co-registration capabilities using a phantom with a geometry different than the calibration phantom and in addition with a different material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Three 20-µm diameter black nylon threads (NYL02DS, Vetsuture, France) and three black polyester threads (Coat Epic 150) were mounted on a 3D- printed frame similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' They were arranged symmetrically with respect to the center of the frame and with various angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The second phantom (Ph2) was prepared with agar powder 2% w/v (A1296, Sigma Aldrich, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Louis, MO, USA) and cellulose powder 1% w/v (Sigmacell cellulose Type 20, Sigma Aldrich) in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Cellulose particles (20 µm) act as ultrasound scatterers to mimic the scattering properties of biological tissues for US imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The gel was molded in a cylindrical mold (diameter of 20 mm) with three cylindrical solid inclusions of 5 mm in diameter and of the same length as the mold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The agar-cellulose solution was heated to 85°C and poured into the mold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' When the mold was half full, 100-µm- diameter black polyethylene microspheres (BKPMS 90–106 um, Cospheric, Santa, Barbara, CA, USA) were spread on the superior interface and were trapped at the interface during the solidification of the gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mold was then filled with the hot agar solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' When the gel solidified, the cylindrical inclusions were removed and filled with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The microspheres remained embedded in the gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Ph2 was placed so that the cylindrical holes and the plane of spheres were parallel and perpendicular to the rotation axis, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The third phantom (Ph3) was prepared with agar powder (2% w/v) and cellulose powder (1% w/v) in water for the first half and with agar powder (2% w/v) for the second half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Two crossed 20-µm diameter black nylon threads were embedded in 7 the gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Accuracy and repeatability of the calibration The accuracy and the repeatability of the calibration outputs are presented in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' I, in percentage of the acceptability range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' First, it can be noticed that all the 7 parameters are fully within the acceptability ranges, for both Ac and Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the accuracy, the calibration outputs were compared to the ground truth thanks to a numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean Ac over the 7 parameters was found equal to 26% and Ac had a maximum of 55% for the Roll parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We can then consider that the developed calibration method enables to accurately determine the set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To evaluate the repeatability, ten acquisitions performed on various days (distributed in three imaging sessions over one week) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For each acquisition, the optical fibers and the phantom were repositioned to avoid any bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Rp estimates the mean absolute deviation of each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mean Rp over the 7 parameters is around 16% and a maximum of 39% was reached for Δz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Therefore, the calibration is repeatable and stable over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Despite slight variations in the evaluation of each parameters, the accuracy and the repeatability of the calibration method are highly satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We can therefore consider the developed calibration method to be reliable and robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Superimposition quality and spatial resolution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3 presents PA/US images of the slice y =0 (center of the linear array) of Ph1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Ph1 aims to test if the experimental calibration remains valid for a phantom different than the calibration phantom in the spatial arrangement of the threads, but also in the thread material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The phantom Ph1 is comprised of three nylon threads and three polyester threads arranged so that each thread has a different orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Only the threads numbered 2 were set parallel to the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For an easier comparison between the two materials, Ph1 was built so that each of the three nylon threads had a symmetrical polyester thread with respect to the center of the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3, symmetrical threads have the same number and the suffixes ‘n’ and ‘p’ refer to nylon and polyester threads, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For instance, the thread 1n is symmetrical to 1p, with respect to the center of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The nylon threads were grouped on the top part of the phantom while the polyester threads are grouped on the bottom part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Video 2 displays rotating MAP images around z of Ph1 and therefore shows the spatial arrangement of the threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' As nylon threads are thinner than polyester ones, both US and PA signals were weaker for nylon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The amplitude in the reconstructed image was 3 times smaller in PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To facilitate the visualization in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3, the upper and the lower part of the volumetric images were normalized by their local maximum and not the global one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The separation between the two parts is illustrated by the horizontal dashed white line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We can visually see that polyester threads appear larger than the nylon ones both on the PA and US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This is expected given their larger diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, the superposition of the images can be observed for the two materials and for all the threads regardless of their orientation or position in space (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3 and Video2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The blue lateral halo around the white spots in the combined image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3 left) indicates that the lateral resolution is wider for PA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For a more quantitative description, Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' II presents the superposition distance and the FWHM calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The superposition distance between the center of a thread in PA and in US is small compared to the US wavelength Λc (less than 10% of Λc) and smaller than the spot obtained for each object in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The superposition distance is not significantly different for the two materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This result indicates that the calibration enables the superposition of nylon thread with different orientations than in the calibration phantom, and it validates that the calibration ensures the superpositions for objects with a dual contrast but in another material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The standard deviation computed over the three threads and five imaging planes is small as well compared to Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This result shows the low dispersion of the superimposition distance both with the thread orientation and with the spatial position in the imaged volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The FWHMs calculated for the nylon threads provides an estimate of the LSF because the thread diameter is much smaller than Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We first notice that for both US and PA and for both x and z directions, the FWHM is on the order of magnitude of Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Along the x-direction, the resolution is limited by the diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The high resolution in the x-direction results from the large angular aperture provided by the rotation scan and the synthetic aperture approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For comparison, the FWHMx was on the order of 1 mm [23] for a translation-only scan with the same US array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The low standard deviation of FWHMx indicates that the resolution is independent of the position of the object in the volume and its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The spatial homogeneity associated with the rotate-translation scheme and previously observed independently in USI [23] and PAI [22] is then confirmed for the simultaneously co-registered imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In each direction, the US resolution is slightly better than the PA resolution (70-100% of Λc for US vs 130% of Λc for PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Image reconstruction of the central plane of Ph1 (y=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The horizontal white dashed line represents the separation between nylon and polyester zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Threads with the same number are symmetrical and n stands for nylon and p for polyester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA image (left) is presented on a linear scale while the US image (center) is presented in dB with a threshold at -30dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' TABLE I ACCURACY AND REPEATABILITY STUDY ON THE CALIBRATION Yaw Pitch Roll θ Δx Δz 𝑡0 𝑈𝑆 Ac (%) a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='8 18 55 20 25 32 24 Rp (%) a 28 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='5 17 39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content='1 a Ac and Rp are expressed in percentage of the acceptability ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' TABLE II RESOLUTION AND SUPERPOSITION ASSESSMENT FWHM nylon (µm) Superposition (µm) US PA nylon polyester along x (mean ± std) 212 ± 18 387 ± 30 15 ± 9 23 ± 17 along z (mean ± std) 296 ± 24 379 ± 21 nylon PAnylon USr nylon PA+US 2n 3n 3p 2p Ip polyester mm polyester polyester 8 Along x, two main factors can explain the FWHMx differences between US and PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' First, PA images rely on US signals produced by the illuminated object and not on backscattered US signals generated by the US array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For small objects, the US frequency spectrum recorded by the array is then usually broader in PAI than in USI, and especially contains low frequencies which may decrease the diffraction-limited resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Second, for one laser excitation, five tilted plane waves are emitted, which increases the number of independent views and the spatial sampling in USI compared to PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This sampling factor has been shown to influence the resolution along x [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Along z, which corresponds to the axial direction, the LSF is mainly influenced by the pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The pulse- echo mechanism can explain the better resolution of the US images compared to PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Complementary distributions of US and PA contrasts To further demonstrate the advantages of the dual modality imaging, phantoms with complementary contrasts were designed and produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The images of Ph2 and Ph3 are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the ultrasound contrast, Ph2 is a homogeneously scattering medium (agar with cellulose) with three cylindrical and anechoic holes filled with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the PA contrast, numerous black-dyed micro-spheres are arranged in a central plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The optical absorption of the agar gel and the water are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In the first row of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4, a MAP image along y is shown for PAI, USI, and the superposition of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Rotating MAP images round the z-axis are presented in Video 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The 20-mm diameter cylindrical shape of the phantom could be retrieved both in PAI and USI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' One can note the homogeneous image quality in the xz-plane for both modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4, images of a slice perpendicular to the z-axis are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The slice was chosen to cut two of the three holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The homogeneity of the USI image along the y-axis can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' As expected, the three holes appear with a negative contrast for both PAI and USI independently of their spatial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The holes allow to further validate the superimposition of the two modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' They are concentric in USI and in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We can notice that the outlines of the holes are blurrier and the diameter of holes seems smaller on the US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This effect can be attributed to the stronger side lobes in the US images induced the log compression (display in dB) of the color scale compared to the linear scale used in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Such side lobes are also visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The microspheres were hardly visible in the agar matrix in USI, or only when numerous spheres are gathered together, while they appear with a strong contrast in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' With the superposition, both the agar and the microspheres are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This is of interest to assess the distribution of PA contrast agents in an organ which is homogenously echoic for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The US images gives the contour of the phantom, which is similar to the anatomical context, while the PA image gives the distribution of the marked spheres which are analogous to a contrast agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In a clinical application, the PA contrast agents could be therapeutic nanoagents accumulating locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Holes are mimicking bodies with a negative contrast such as cysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the second phantom Ph 3, media with two different ultrasound contrasts were used: agar with cellulose is more echogenic that agar alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Two black nylon threads bring the optical absorption contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Images of Ph3 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The US image in the first row (MAP along z) and the rotating MAP images in Video 4 clearly show the contrast between the two blocks of gel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The nylon threads are visible in the MAP US images, but mainly due to their elongated shape as they are barely differentiable from the surrounding speckle in the slice perpendicular to the y-axis and intersecting the two threads in the agar with cellulose part (second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Agar and agar with cellulose do not have any contrast in PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, the threads show a homogenous and high contrast in PAI over the two agar blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This phantom could typically mimic a blood vessel (black thread), highly visible in PAI, perfusing two different organs with different echogenicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' DISCUSSION We demonstrated high-quality, volumetric and simultaneously co-registered PAI and USI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The simultaneous dual imaging was made possible by the use of a linear US transducer array, and the high quality (resolution, contrast, visibility) of the volumetric images resulted from the large synthetic angular aperture enabled by the rotate-translate scan Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Image reconstruction of Ph3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Schematic drawing of Ph3 are displayed in the left column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The first row displays MAP images along z while the second row shows a slice perpendicular to the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' US images were thresholded at -40 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Along y-axis, left part is agar with cellulose and right part is agar alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Image reconstruction of Ph2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Schematic drawing of Ph2 are displayed in the left column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The first row displays MAP images along y while the second row shows a slice perpendicular to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' PA images are presented in linear color scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, because a few bright spots (probably clusters of microspheres) were dominating the color scale and hiding the rest, voxel values were saturated at 40% of their maximum before normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' US images were thresholded at - 40 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 5mm5mm 9 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Images showed a homogenous quality over a large imaged volume (cylinder of diameter of 21 mm and length of 19 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The effective synthetic aperture for both imaging modalities and the superposition of the PA and US images for features having a dual contrast required an accurate determination of the positions of the US array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To this end, we developed and validated a calibration method, which was determined to be both accurate and repeatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This initial calibration process allowed for the subsequent acquisitions of images without the need of fiducial markers on every imaged volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' We demonstrated the superposition of PA and US images with phantoms having a dual contrast and the complementarity of the mapped information with phantoms having complementary spatial distributions of US and PA contrast agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The calibration was based on the combination of a dedicated calibration phantom, a cost function and an optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 7 parameters were determined for our scan geometry to obtain superimposed and sharp PA and US images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The mechanical mount and the position sensors of the stages ensure that the determined parameters remain valid for subsequent scans as long as no deformation or accidental motion of the array occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For this study, no significant variation could be noticed over a period of one week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The calibration phantom was easy to build and was comprised of four well-separated threads for a facilitated identification and measurements of local properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Our cost function used only five imaging planes to avoid the reconstruction of the entire volume and the associated long computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Our selection of metrics with the sharpest variations with regards to the parameters implied that the cost was dominated by the properties of the US image, but with an influence of the properties of PA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the optimization algorithm, we found that the selection of the initial guess was a crucial step due to the presence of local minima in the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Additionally, the particle swarm optimization was found to be more effective to determine a solution close to the ground truth than a downhill simplex method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The refinement of the optimization algorithm was beyond the scope of this study as we focused on the effectivity of the method, but it will be considered in a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This calibration procedure was found effective for our system although it requires the initial imaging of a dedicated phantom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In the past decade, several multiperspective or multiview imaging systems with US transducer arrays have been developed both in PA imaging and in US imaging, independently, leading to various calibration procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For US imaging, coherent compounding of images acquired with two US transducer arrays have been developed in 2D [30] and in 3D [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Dedicated calibration phantoms were also used that consisted in isotropic scatterers (5 wires in 2D and 3 spheres in 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The phantoms were combined with a cost function linked to the coherence of the echoes from the isotropic scatterers on the different elements of the array to determine geometrical parameters (4 parameters in 2D and 6 in 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A simplex search method was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The calibration showed an improvement in the contrast and in the resolution of the images but the use of isotropic scatterers resulted in long computation time of volumetric images during the calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Incoherent compounding of US images was also investigated for 2D [32] and 3D [33] imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The spatial transformation matrix between the positions of the arrays were assessed directly using the images acquired on the sample of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' This approach could enable free hand scanning and imaging of moving organs but could not achieve a synthetic aperture approach which therefore limits the final image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In PA imaging, multiperspective or multiview imaging is almost inherent to the modality as PA tomography with spherical and hemispherical scans of an US detector has been investigated in the early PA scanners [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, planar detection geometries able to acquire 3D PA images have been angulated to enhance the image view with a synthetic aperture approach [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In ref [35], two planar arrays were assembled in a rigid configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' An initial calibration was performed by imaging a dedicated phantom comprised of three threads with different orientations with each array independently, segmenting the images of the threads and determining the rigid transformation between arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In ref [36], fiducial markers were incorporated in the imaged region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Multiview imaging was also performed by stitching volumetric images [37], which is similar to incoherent compounding and relies on the features obtained in partially overlapping images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In brief, no standard calibration exists either in USI or PAI, but only dedicated calibration phantoms were shown to allow synthetic aperture without fiducial markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Additionally, while for a single imaging modality, some parameters could compensate each other, which put less constraints on the calibration method, in volumetric and simultaneous PAI-USI, the superposition and sharpness of the dual images could only be obtained with accurately determined parameters, in particular because of the different travel times in PAI and in USI between the US transducers and the voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' To the best of your knowledge, we presented here the first dual PA-US 3D calibration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The co-registered volumetric maps of PAI and USI allow to obtain complementary information (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' In the frame of longitudinal studies with different imaging sessions, the US image is expected to give the anatomical reference to better understand and to co-register images acquired at different time points, while PA images could reveal molecular or functional phenomena with a slow kinetic, such as the accumulation of a nanoparticular contrast agent with a long circulation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For other simultaneously co-registered multimodal imaging such as PET/CT, the anatomical imaging modality contains information that can be used to improve the image quality of the functional imaging modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For PAI/USI, recent work on the reconstruction of 2D PA images showed an improvement of the PA image quality of blood vessels using structural information from ultrasound images [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Additionally, light fluence distribution could be modeled using US images and used to improve PA image quality [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The study presented here was performed at a single optical wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Multispectral approaches [40] will be investigated with the developed scanner to enable discrimination between different chromophores and contrast agents and to evaluate physiological parameters such as the oxygen saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' On the US side, the system presented here operated at an US center frequency of 5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' However, the rotate-translate scan can be scaled to other ultrasound frequencies to gain sensibility and information on other spatial scale [22], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' The spatial 10 resolution increases with the US frequency of the array for both PAI and USI, but higher frequencies will also improve the sensitivity to small absorbing regions in PAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Linear US arrays are commercially available in a wide range of center frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' Building on the in vitro proof of concept presented here, we are currently adapting the scanner for in vivo imaging with the addition of an acoustic coupling system to remove the large water tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' A small water tank with an acoustically transparent membrane sealing the bottom [16] will be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtAyT4oBgHgl3EQfzvl6/content/2301.00706v1.pdf'} +page_content=' For the translation in clinics, the bulky motorized stages will be replaced by a scanning system 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0000000000000000000000000000000000000000..c592317059de88255f4c84388136d7d15103d6c1 --- /dev/null +++ b/qtE4T4oBgHgl3EQfvw2m/content/2301.05245v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:60dd68ee13161897174e7c3c2443cade8c1a877fb21a8721b63243ea779943ee +size 1837822 diff --git a/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/2301.03711v1.pdf.txt b/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/2301.03711v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3875a504bd5dca2d4883c010ff0c1a05a0b5b16 --- /dev/null +++ b/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/2301.03711v1.pdf.txt @@ -0,0 +1,2079 @@ +Springer Nature 2021 LATEX template +3D Shape Perception Integrates Intuitive +Physics and Analysis-by-Synthesis +Ilker Yildirim1,2,3*†, Max H. Siegel4,5*†, Amir A. +Soltani4,5,6†, Shraman Ray Chaudhari7 and Joshua B. +Tenenbaum4,5* +1*Department of Psychology, Yale University, New Haven, CT. +2Department of Statistics & Data Science, Yale University, New +Haven, CT. +cWu-Tsai Institute, Yale University, New Haven, CT. +4Department of Brain & Cognitive Sciences, MIT, Cambridge, +MA. +5The Center for Brains, Minds, and Machines, MIT, Cambridge, +MA. +6Department of Psychology, Boston College, Chestnut Hill, +Massachusetts, USA. +7Department of Electrical Engineering & Computer Science, +MIT, Cambridge, MA. +*Corresponding author(s). E-mail(s): ilker.yildirim@yale.edu; +maxs@mit.edu; jbt@mit.edu; +†These authors contributed equally to this work. +Abstract +Many surface cues support three-dimensional shape perception, but peo- +ple can sometimes still see shape when these features are missing – in +extreme cases, even when an object is completely occluded, as when +covered with a draped cloth. We propose a framework for 3D shape per- +ception that explains perception in both typical and atypical cases as +analysis-by-synthesis, or inference in a generative model of image for- +mation: the model integrates intuitive physics to explain how shape can +be inferred from deformations it causes to other objects, as in cloth- +draping. Behavioral and computational studies comparing this account +with several alternatives show that it best matches human observers in +1 +arXiv:2301.03711v1 [q-bio.NC] 9 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Shape perception integrates intuitive physics and analysis-by-synthesis +both accuracy and response times, and is the only model that corre- +lates significantly with human performance on difficult discriminations. +Our results suggest that bottom-up deep neural network models are +not fully adequate accounts of human shape perception, and point to +how machine vision systems might achieve more human-like robustness. +Introduction +For more than a century, vision scientists have studied the many cues that +humans or machines use to recover shape. Edges or bounding contours, gradi- +ents of shading or texture, stereo disparity, and motion parallax are just a few +of the cues that can be computed from the visible surface of an object and that +can reliably indicate an object’s three-dimensional (3D) shape across different +views (Bulthoff, 1991). When available, surface cues effectively support shape +perception in humans and machines. However, a set of recent studies (Lit- +tle & Firestone, 2021; Phillips & Fleming, 2020; Wong, Bi, Soltani, Yildirim, +& Scholl, 2022; Yildirim, Siegel, & Tenenbaum, 2016) present a challenge to +the classical cue-based theory of shape perception: Even when a surface is +obscured, humans can sometimes perceive shape, without directly seeing the +object at all. Consider the synthesized images of cloth-covered objects in Fig. +1A and B, in which each object is completely occluded by a thin, cotton-like +fabric draping over it. Although the draped shapes look very different from +the comparison objects (shown in randomly chosen orientations), observers +can nonetheless pick out which unoccluded airplane or chair matches the 3D +shape of the appropriate occluded object. 1 +Here we ask: How can people perceive object shape (and pose, size, cat- +egory, etc.) in these images, when all the classic visual cues are mostly or +entirely absent? Even those image cues that are present may be highly mis- +leading, as they belong not to the underlying object’s shape but the surface +of the occluding cloth. Somehow we are able to interpret the shape of the +cloth as an interaction between the underlying rigid object’s geometry and the +way the cloth drapes and deforms upon contact. While sculptors have long +exploited this capacity of the visual system to depict human faces and figures, +only recently have detailed behavioral studies provided convincing evidence +that humans somehow “undo” (Phillips & Fleming, 2020; Wong et al., 2022) +the effect of the cloth to access the hidden object. Thus far, a computational +account of vision that can explain shape perception, even in the absence of +surface cues, remains absent. +There are at least two ways one might explain how people perceive the +shape of draped objects, corresponding to two contemporary frameworks which +each seek to advance beyond classical cue-based approaches to 3D shape. One +1The correct shape matches in Fig. 1A and B are: (A) left airplane on top goes with left on +bottom, right on top goes with right on bottom; (B) left chair on top goes with right on bottom, +right on top goes with left on bottom. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +3 +Fig. 1 Seeing 3D shape through a cloth. (A) Bottom row: two different airplanes. Top row: +the same airplanes, draped with cloth and presented in random order and in a random pose. +(B) Same as (A) but for two different chairs. Despite the variation in viewpoint and complete +occlusion of the cloth-draped objects, human observers can still match the cloth-draped and +unoccluded object pairs. (See note in main text for correct answer.) (C-E) Sculptors have +long displayed their skill in works that are crafted from a single rigid material but convey +both an illusory effect of cloth draping and rich 3D shape for the object under the illusory +cloth, as in Giovanni Strazza’s (c. 1850s) “Veiled Virgin” (C, marble), Gabriel Klasmer’s +(2000) “Car in the Sun” (D, fiberglass), or Wendell Castle’s (1985) “Ghost Clock” (E, +mahogany, partially bleached). +possibility is that instead of a relatively constrained set of interpretable, mean- +ingful cues, often derived from an analysis of the geometry of objects and of +image formation processes, the human visual system might engage a much +larger set of cues which are obtained through some black-box learning mecha- +nism (and are therefore difficult to write down or interpret). Recent computer +vision models based on deep convolutional neural networks (DCNNs) have +demonstrated learned feature hierarchies which facilitate impressive object +recognition capabilities (Krizhevsky, Sutskever, & Hinton, 2012; LeCun, Ben- +gio, & Hinton, 2015) and which are relatively robust to variation in appearance +and pose even though the model training objective does not explicitly include +these goals. Moreover, these same features have been shown to enable many +other seemingly disparate visual tasks, including shape perception, with only +minor adaptation (e.g. fine-tuning, or adding one or a small number of addi- +tional output layers) (Hong, Yamins, Majaj, & DiCarlo, 2016). Perhaps these +features are sufficiently robust to generalize across even more extreme image +transformations, such as cloth occlusion. +A second possibility is that we see 3D shape via “analysis by synthesis”, or +inference in a physics-based generative model of how scenes form and give rise +to images (Mumford, 1994; Yuille & Kersten, 2006). On this view, shape per- +ception is not driven solely or primarily by a fixed, universal set of image cues, +computed bottom-up from any image and sufficient for any downstream task. +Rather, we infer 3D shape through a top-down interpretation process based + +A +E +B +CSpringer Nature 2021 LATEX template +4 +Shape perception integrates intuitive physics and analysis-by-synthesis +on an internal model of how images are formed and the role that shape plays +in that model. The generative model approach sees cloth draping as just one +exemplar of a potentially unbounded space of atypical presentations of objects, +in which some aspects of the physics of scenes and images grossly alter an +object’s appearance from its typical form while remaining easily interpretable +by humans: consider as other examples viewing an object such as the chair or +airplane in Fig. 1 outside in a rainstorm, or under ten feet of cloudy water, +or through colored plastic wrap, or in the light of a full moon at night. The +open-ended compositionality of the visual world may imply that it is difficult +or impossible to specify or learn a single set of bottom-up image cues or fea- +tures which reliably and robustly encode an object’s 3D shape even in such +atypical, rare conditions. Instead, a system should model individual, scene- +level causes – the physical objects and processes that generate images – and +how they combine and yield visual input. Then, by reversing their effects, it +may recover the original physical scene. Thus a visual system could still iden- +tify a draped object and even perceive its fine-grained 3D shape if it were able +to model and somehow invert cloth physics. +Our goal in this paper is to use the perception of objects under cloth +as a case study to evaluate concrete versions of each of these accounts of +shape perception. The theoretical merits of the pure bottom-up and top- +down approaches have been extensively debated in the literature, but it has +been difficult to find strong evidence distinguishing the bottom-up cue-based +and top-down model-based approaches; until recently, neither discriminative +classifiers nor Bayesian generative models performed well in realistic visual +tasks, so comparisons with biological vision were limited to controlled sce- +narios with simplified, non-naturalistic stimuli (Liu, Knill, & Kersten, 1995). +Advances in algorithms and computing hardware, however, have led to DCNN +and analysis-by-synthesis models that achieve good performance with complex +natural images and can now be rigorously evaluated as models of how we per- +ceive 3D shape in challenging cases with rich naturalistic stimuli. They also +let us explore various hybrid accounts that to date have received very little +direct evaluation in human psychophysics: in particular, we compare human +judgments with top-down analysis-by-synthesis models attempting to match +images at either the level of raw pixels or intermediate-level representations +based on DCNN features. +To our knowledge, only one empirical study has studied human percep- +tion in light of these improved methods. Erdogan and Jacobs (2017) defined a +compositional generative model of 3D shape and compared human judgments +of shape similarity2 with those derived from bottom-up classifiers and from +top-down inference in a 3D generative model, concluding that the latter might +underlie shape perception because it correlated better with human responses. +But this study, while pioneering, provides only limited evidence for top-down +2Erdogan and Jacobs (2017) elicited graded similarity judgments between different objects (so +that the study and target items could be similar but not identical). In contrast, we use forced- +choice judgments of same or different object identity; the correct shape is always a response +option. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +5 +analysis-by-synthesis in perception. The best bottom-up model was also quan- +titatively predictive of human judgments and performed almost as well as the +top-down model. With more training and improved DCNNs, the quantitative +gap between these models might be expected to narrow even further. In addi- +tion, neither model improved dramatically over a simple baseline matching test +to target images at the pixel level. Here, we demonstrate a stronger, qualitative +distinction between model classes enabled by our completely occluded cloth- +draped stimuli: standard DCNNs and pixel-based observers, unlike people and +our generative models, perform no better than chance on harder instances and +even with extensive specialized training show little improvement. +Our design choices offer several other advantages. Because we chose uncom- +mon stimuli with variable difficulty, we find meaningful variance in human +performance and response time, which allows for finer-grained model evalua- +tions and comparisons of humans and models on trial-by-trial speed as well as +accuracy. The generative model that we consider performs iterative inference +with significant stimulus-driven variation in the number of computational steps +and therefore can be directly compared with subjects’ reaction times, poten- +tially revealing signatures and roles for feedback or recurrence in biological +shape perception. +Results +The Object-Under-Cloth Task +While in some cases humans can recognize draped objects from a single image +(e.g., Fig. 1C, D), we chose as our experimental setting a visual match-to- +sample task that allows us to directly address the above three possibilities. We +choose this setting as we are primarily interested in how generative models +can support online, detailed 3D shape perception, rather than object catego- +rization or any kind of memory-based process. The essence of our proposal +is that observers may perceive the 3D shape of cloth-covered objects in arbi- +trary orientations by approximately simulating in their minds the process of +how cloth drapes over the object in three dimensions, and imagining what +the resulting 2D image would look like. So we constructed an experimental +task that should be directly solvable via this mechanism: We show observers +an unoccluded matching object along with a target draped shape (i.e., the +initial matching object rendered in computer graphics under a simulation of +cloth draping) and an unoccluded distractor object, in a two-alternative forced +choice (2AFC) task. We call this the “occluded” condition to contrast with a +control “unoccluded” condition (see below). +We chose 10 different everyday object categories (airplane, bicycle, bus, +car, chair, guitar, motorcycle, pistol, rifle, table) and sampled object meshes +for each category from a large repository of 3D shapes (Chang et al., 2015). +We used 24 unique exemplars from each category, yielding 120 visual-matching +trials; each trial used two shapes, and each unique shape appeared in one +trial. Each trial presented an unoccluded target shape, a distractor shape, and + +Springer Nature 2021 LATEX template +6 +Shape perception integrates intuitive physics and analysis-by-synthesis +Fig. 2 Matching a target shape to one of two unoccluded test objects. (A) Left: A trial in +the occluded task condition. The top image shows a “target item”, the bottom-left image +shows a “matching test item” and the bottom-right image shows a “distractor test item”. +Right: Trials from the occluded task. Each triplet displays, from left-to-right, target item; +matching test item; and distractor test item for one trial. We show “easier” trials, with +different-category distractor and matching test items, and “harder” trials, where both test +items are of the same category. (B) Left: A trial from the unoccluded task condition, spatial +configuration as in (A). Right: Each triplet shows a trial from the unoccluded task, showing +instances of easier and harder trials. +the target shape after cloth draping. We varied the similarity between the +distractor and matching items (Fig. 2A, right) to generate visual-matching +triplets ranging in difficulty. In half of the trials, the target and distractor +objects were drawn from the same category; these we term “harder” trials +because same-category shapes are generally more difficult to distinguish than +different-category objects, which we call “easier”. +To create the cloth-occluded stimuli, we simulated cloth draping via a +particle-based physics engine (Macklin, M¨uller, Chentanez, & Kim, 2014); we +chose simulation parameters (e.g., number of iterations) and the mechani- +cal and material properties of the simulated cloth (e.g., stiffness and mass) +to enable efficient, stable simulation of natural-looking cotton-like cloth (see +Materials & Methods). +In the unoccluded condition, we use the same objects but show the tar- +get shape without cloth; see Fig. 2B. In this version of the task, viewpoint +variability and the shape similarity between the matching and distractor test +items are the only confounding variables. + +A +Different category distractor +Same category distractor +Occluded +(Easier trials) +(Harder trials) +Which is the same object as above? +B +Unoccluded +Which is the same object as above?Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +7 +Physics-Based Analysis-By-Synthesis (PbAS) +We formalize the problem of matching a cloth covered object with its unoc- +cluded counterpart as approximate Bayesian inference in a causal generative +model. Our physics-based analysis-by-synthesis (PbAS) method combines +physics and graphics knowledge with statistical inference and optimization. +The model consists of three components: a generative model for scenes +and images, feature extraction for approximately Bayesian inference (using +a pseudo-likelihood approach), and a simulator-in-the-loop inference engine +based on Bayesian optimization. As an account of how people can perceive +the shapes of objects under cloth (or other challenging viewing conditions), +we posit that each of these three components has some analog in the mind +and brain, and that they operate and interact in something like the ways we +specify here – not precisely as we have implemented them, but close enough +that the speed and accuracy characteristics of the PbAS model can be quan- +titatively compared with human behavior, along with different model variants +and alternative accounts. +The generative model in PbAS captures the physical scene variables, includ- +ing object shape and pose, cloth properties, and the mechanics of how they +interact, which together produce the geometry of the occluding cloth surface. +It further includes a model of graphics – how surface geometry, material, and +light interact to generate an image (some factors, like optics, are handled +implicitly; see e.g. Koch, Baig, and Zaidi (2018) for an explicit treatment, +including modeling, of human representation of visual scene geometry). Given +a hypothesized 3D object shape in a hypothesized pose, the model produces +a synthesized or hypothetical image which may be compared with the image +actually observed. In the analysis by synthesis framework, perception requires +inverting this process to recover the object shape and pose likely to have given +rise to the observed image (Fig. 3A, B). +Like most generative models, PbAS is too complex to invert exactly. A +ubiquitous approximation algorithm, Markov Chain Monte Carlo (MCMC), +iteratively constructs samples from a target distribution like the posterior, but +in our case requires far too many iterations to work because each step includes +costly physics simulation. We sought instead to maximize the posterior using +Bayesian optimization (Snoek, Larochelle, & Adams, 2012), which relative to +MCMC provides a guided inference scheme where the next scene hypothesis +to evaluate is informed by all (instead of only the current) evaluations of the +posterior function (Cranmer, Brehmer, & Louppe, 2020). BayesOpt simulta- +neously estimates and optimizes the posterior, providing an algorithm which +efficiently samples increasingly more probable hypotheses for object shape and +rotation given an input occluded image (Fig. 3B). (Psychologically, BayesOpt +can be seen as implementing a kind of goal-conditioned mental imagery; see +Hamrick and Griffiths (2013) for an application in the context of mental rota- +tion.) The probability of a scene hypothesis is computed by comparing its +corresponding rendered hypothesis image with the input, using a feedforward + +Springer Nature 2021 LATEX template +8 +Shape perception integrates intuitive physics and analysis-by-synthesis +feature hierarchy fenc implemented as the first fully-connected layer of a pre- +trained DCNN (Krizhevsky et al., 2012). While the goal of inference in our +model is posterior probability maximization, the optimization trajectory is +also of interest for comparison with human behavior. +PbAS can arrive at a reasonable percept rapidly (Fig. 3C, D) compared +to sampling-based methods like standard MCMC. It therefore provides a +more plausible quantitative standard for understanding average human accu- +racy, how accuracy improves with longer viewing time, and stimulus-driven +variability in response time. +Synthesis: Generative Model +The generative model consists of (i) latent variables describing the scene: a 3D +object shape S and its rotation R; (ii) a forward physics simulator along with +cloth parameters: cloth size, position, stiffness, mass, and friction, denoted fΨ; +and (iii) a rendering function and lighting parameters, together denoted fΓ. +We set the physics simulation parameters fΨ and renderer parameters fΓ to +the same values as used for stimuli generation (see Section “The object-under- +cloth task” and Materials & Methods). While the model is designed to perceive +cloth-covered objects, it applies to unoccluded objects, as in the unoccluded +task condition, as a special case by setting fΨ to the identity function. +Given an occluded input observation (indicated as “Input” in red frame, +Fig. 3B) and an unoccluded “context object” (in blue frame, Fig. 3B), we +wish to estimate the object shape S and rotation R that best explains the +input image. More formally, we wish to invert the generative model to find +scene hypotheses that explain perceptual input using Bayesian inference, which +amounts to finding the posterior +Pr(S, R | Iobs) ∝ Pr(Iobs | S, R, fΨ, fΓ)Pru(S)Pr(R)δfΨδfΓ += Pr(Iobs | Ihyp)Pru(S)Pr(R), +(1) +where Pr(Iobs | S, R, fΨ, fΓ) is a likelihood term induced by the physics engine +fΨ and rendering function fΓ, and the delta functions select fixed physics fΨ +and rendering fΓ parameters. For brevity, in the equality in Eq. 1 and below +we write Ihyp = fΨ(fΓ(S, R)) for the hypothesis image given latent scene +parameters and suppress the delta notation. We next explain each term. +The context object allows observers to form a distribution Pru(S) over +the possible shape of the draped object; because the context object is pre- +sented as a 2D rendering, its shape is uncertain. Even though human observers +do not need auxiliary shape information to process cloth-occluded images, +this accompanying context object provides a computationally tractable shape +hypothesis space for generative modeling (see the Discussion for future direc- +tions relaxing this constraint). We represent this shape uncertainty Pru(S) +using a categorical distribution over the K nearest neighbors of the actual con- +text object (excluding the context object itself) in a large repository of shapes +(the ShapeNet dataset (Chang et al., 2015); see Fig. 3B). In our simulations + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +9 +we take K = 4 and each neighbor is assigned a probability as a function of +its distance rank; see Materials & Methods. We place a uniform prior over +rotations Pr(R) covering the half-sphere centered at canonical pose. +Executing the physics simulator fΨ with a scene hypothesis (a sampled +shape S and its rotation R) results in a draped cloth geometry G (Fig. 3B). +Passing the resulting scene to the rendering function fΓ in turn yields an image +Ihyp = fΓ(G) of the cloth-draped object (Fig. 3B), i.e. a hypothesis image +which may be compared with an input observed image Iobs to evaluate its +likelihood under the scene hypothesis. +The detailed geometry resulting from cloth simulation (e.g. the partic- +ular pattern of wrinkles) can vary significantly with even small changes in +the values of random variables (Macklin et al., 2014); therefore, calculating +an accurate likelihood (through marginalization) for any scene hypothesis is +computationally intractable (Cranmer et al., 2020). As a result, we define a +pseudo-likelihood function Pr(Iobs | Ihyp) based on the distance D(Iobs, Ihyp) +between the input and hypothesis images in a suitable feature space arising +from an encoder fenc(·); here, we set D = ℓ1 and adopt the features com- +puted by the first fully-connected layer of AlexNet (Krizhevsky et al., 2012) +as the encoder fenc. The PbAS (pseudo-) likelihood for an input image, given +a hypothesis image rendered from a scene proposal, is then +Pr(Iobs | Ihyp) ∝ exp (− ∥(fenc(Iobs) − fenc(Ihyp)∥1) +With these choices for the prior and likelihood, the posterior Pr(S, R | Iobs) +of Eq. 1 depends only on two terms: the discrepancy Pr(Iobs | Ihyp) between +the observed image and the rendered latent parameters, and the uncertainty +Pru(S) over the shape of the context image. +By measuring the discrepancy between rendered scene hypotheses and +observed images in terms of DCNN encoder-based features, the PbAS model as +described is an instance of a hybrid top-down/bottom-up (or model-based/cue- +based) approach to 3D shape recovery (see also Wang, Mei, Yuille, and +Kortylewski (2021)). We also consider a purely top-down analysis-by-synthesis +approach which is identical except that image discrepancies are computed in +terms of raw pixel deviations. The likelihood is then simply +Pr(Iobsms | Ihyp) ∝ exp +� +− ∥Iobs − Ihyp∥1 +� +We refer to this alternative as the “Pixel Likelihood PbAS” model, or “Pixel- +PbAS” for short. +Analysis: Inference Using Bayesian Optimization +The posterior Pr(S, R | Iobs) of Eq. 1 contains all information that our +model extracts from an observed image Iobs, but computing this distribu- +tion is intractable. Standard simulation-based inference methods based on +MCMC ensure eventual convergence to the full posterior but in practice spend + +Springer Nature 2021 LATEX template +10 +Shape perception integrates intuitive physics and analysis-by-synthesis +Fig. 3 Overview of Physics-based Analysis-by-Synthesis (PbAS). (A) Application of the +PbAS model to solve the object-under-cloth task. Given an image triplet, two PbAS models +are run in parallel; each execution takes as input a test item and the target item. On each +iteration, the two executions of the model are compared to determine how well the target +item is explained by each test candidate. (B) Interpreting an input cloth-covered image (red +frame), with a context unoccluded object (u; blue frame) supplying a prior over object shape +Pru(S). Bayesian optimization (BayesOpt) efficiently guides inference, improving shape (S) +and rotation (R) hypotheses across iterations. S and R proposals initialize the cloth drap- +ing simulation, then are evaluated by computing the distance D between the current scene +hypothesis (rendered to a hypothesis image) and the input in a suitable feature encoding +space fenc. (C) Visualization of three inference (panel B) trajectories over time. Rows are +independent runs of PbAS each with input as in A, B (red frame) and show the cumulative +best scene hypothesis at each iteration. Blocks show hypotheses visualized with (upper; ”Pre- +diction”) and without (lower; ”Without cloth”) cloth occlusion. Model estimation accuracy +improves with increasing iteration number, but some uncertainty remains as the model (like +people) cannot in general perfectly identify the shape or pose of a draped object. (D) Evolu- +tion of model accuracy averaged across multiple runs in the occluded task condition. Model +predictions by iteration for 15 same-category “harder” and 15 different-category “easier” +trials. +too many iterations in low probability regions (Cranmer et al., 2020). We +focus instead on the maximum a posteriori (MAP) setting: finding the best +single scene interpretation rather than the full posterior over all possible +latent variable settings. Following previous work in simulation-based inference +(J¨arvenp¨a¨a et al., 2019; Kandasamy, Schneider, & P´oczos, 2015; Tamura & +Hukushima, 2018), we employ Bayesian optimization (or BayesOpt (Snoek et + +A +c +Iterations +30 +60 +90 +120 +Iterations +Run 1 +PbAS +Prediction +Run 2 +Which unoccluded object better +explains the cloth-covered image? +Run 3 +PbAS +Run 1 +B +Without cloth +Run 2 +Unoccluded +context +K=4 nearest shapes +object, u +Pru(S) +Run 3 +D[fene(Input), +BayesOpt +fenc(Prediction) +Pr(R) +D +1.0 +Input +[S, R} to +0.8 +try next +0.6 +0.4 +Prediction +A +Different category +at iter. j +0.2 +Same category +0.0 +1 +40 +80 +120 +160 +G (draped +Model iterations +geometry)Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +11 +al., 2012)); unlike gradient-based algorithms, BayesOpt allows us to optimize +functions which include procedures, such as our scene renderer, which do not +expose or do not support gradients. See Materials & Methods for an overview +and details of BayesOpt applied in PbAS. +Solving the Object-Under-Cloth Task Using the Model +Human participants see two unoccluded context objects (i.e., test items) and +one target object on each trial. Recall that by its design, the PbAS model +interprets a target (i.e., cloth-draped or unoccluded depending on experiment +condition) object in the context of an unoccluded object. Thus, to model a +given trial, we form two pairs, each consisting of a context object (either the +matching item or the distractor item) and the target object, and apply PbAS +to each pair (Fig. 3A). Each PbAS run aims to explain the same input image, +but with different shape hypotheses derived from either the matching object +or the distractor object. At every iteration, we save the current best parameter +estimates (i.e., shape and rotation) and the log posterior score for that scene +hypothesis. Using the odds ratio decision rule, we obtain the model’s best +estimate of the underlying shape for each inference step. +We ran the PbAS model 32 times on each trial, for 200 iterations each, +and treated each of these runs as a simulated participant (although with finer +temporal resolution). At each of the 200 iterations, we averaged the binary +decisions across runs to obtain mean accuracy predictions – i.e., simulating the +accuracy of participants’ average shape choices. In our analysis we compare the +dynamics of model choice with human decisions sampled at three different time +intervals, corresponding to three different presentation durations that varied +across experimental conditions (see the Section “Iterative refinement in PbAS +explains Human Accuracy and Response Time” for comparisons of models and +human behavior). Fig. 3D shows how the average model performance changes +as a function of iteration for a subset of our stimuli. +Bottom-Up Models Based On DCNNs +To help evaluate the PbAS model and its correspondence with human per- +ception, we considered several well-studied bottom-up models as comparisons +for human and model performance. Recent computer vision models based +on DCNNs learn powerful visual feature hierarchies achieving state-of-the-art +object recognition performance. These feature hierarchies are relatively robust +to variation in pose and lighting, can predict certain aspects of variance in +neural and behavioral data, and are considered the “current best models of +the primate visual stream” (Schrimpf et al., 2018). Moreover they are useful +for visual tasks beyond object recognition; these features have been used for a +number of other vision problems e.g. object localization and pose estimation +(Yamins & DiCarlo, 2016), among others, with minor or no modification. In +testing these pretrained models, our goal is not to establish whether DCNNs, +considered as a model class, can perform the object-under-cloth task. DCNNs +are universal function approximators; with enough data, enough compute, and + +Springer Nature 2021 LATEX template +12 +Shape perception integrates intuitive physics and analysis-by-synthesis +the right architecture and optimization procedure, they are likely able to learn +to perform our visual-matching task. Instead, our goal is to assess whether +the features learned from categorizing objects in natural scenes can suffice to +perceive cloth-occluded shapes as well. +Because our synthesized stimuli and task design differ from those used +for the pretrained DCNNs, we also test the same networks after fine-tuning +them using images similar to our experimental stimuli. We tested the following +architectures, each pretrained using ImageNet (Deng et al., 2009): AlexNet +(Krizhevsky et al., 2012), ResNet-50 (He, Zhang, Ren, & Sun, 2016), and +VGG16 (Simonyan & Zisserman, 2014). Each DCNN was fine-tuned separately +for the cloth-occluded and unoccluded conditions. The task was the same visual +matching problem presented to humans: given an image containing two unoc- +cluded test shapes and one target object (a “triplet”; objects sampled from a +total of 50 shapes), determine which test shape corresponds to the target. We +repeated this process 32 times; thus we fine-tuned 32 copies (to match the num- +ber of PbAS runs per trial) of each architecture for each occlusion condition. +We report the average accuracy of these 32 fine-tuned networks. See Materials +& Methods for dataset generation, fine-tuning, and evaluation procedures. +For both the pretrained and fine-tuned conditions, we found that no archi- +tecture was more accurate than AlexNet (see Fig. S1). Therefore, we use +both the pretrained AlexNet and our fine-tuned variant in our comparisons of +bottom-up models with behavior. +Iterative Refinement in PbAS Explains Human Accuracy +and Response Times +To evaluate PbAS as a candidate model for human perception, we compared +its predictions on the object-under-cloth task with two key behavioral mea- +sures: average accuracy and response time. We recruited human subjects and +assigned them to either the occluded or unoccluded condition (see Fig. 2A, B +left panels). Participants were also divided into three presentation time con- +ditions: the two fixed (1 or 2 second) time conditions and the unlimited time +condition, which presents stimuli until subjects respond. In total, the exper- +iment consisted of 2 occlusion × 3 presentation time = 6 conditions in a +between-subjects design. +As is typical in modeling studies, we compared the average accuracy of +PbAS and alternative models with that of humans. Because accuracy measures +alone might simply favor models that are more performant, we also examined +how PbAS “response times” – the number of inference iterations used per trial +– might explain human response times on the same trials. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +13 +Explaining Human Accuracy Across Presentation Time +Conditions +We first established that behavioral performance is significantly affected by +task setting. While participants performed well above chance across all occlu- +sion and presentation time conditions, their performance varied with respect +to these design parameters. Most obviously, human performance was better in +the unoccluded setting (p < .05). With longer presentation time, average per- +formance significantly improved (p < .05, Fig. S2A; see Fig. S2B for results +broken down by occlusion condition) and response times increased (p < .05, +Fig. S2C; see Fig. S2D for results broken down by occlusion condition). We +also note that there was no learning effect throughout the experiment, with +participants’ average performance remaining fairly constant across trials (Fig. +S5). +The design of our behavioral experiment offers a multifaceted view of +human performance in terms of presentation time, trial difficulty (defined as +whether test items are of same or different category; Fig. 2), and occlusion +condition. In Fig. 4A, we present average human accuracy levels for each pre- +sentation time, pooled with respect to the two difficulty types (“Different +Category” vs. “Same Category”) and two occlusion conditions (occluded vs. +unoccluded). Observers performed significantly above chance even in the most +challenging setting with cloth occlusion, same-category distractors, and the +briefest presentation time (1 second). Note also that performance improved +with longer presentation time in the same-category distractor trials where, +unlike the easier different-category distractor case, performance does not reach +ceiling even with unlimited presentation time. We now ask whether PbAS and +bottom-up models can explain these nuanced results. + +Springer Nature 2021 LATEX template +14 +Shape perception integrates intuitive physics and analysis-by-synthesis +Fig. 4 PbAS explains how human accuracy increases with longer stimulus presentation +time. (A) Behavioral accuracy for each presentation time, occlusion condition, and difficulty. +(A trial is said to be hard if the distractor test item is of the same category as the target +item, and easy if the distractor test item is of a different category than the target item.) (B) +Divergence between model and human performance at each model iteration. Colored lines +show ℓ2 distance between PbAS model and human accuracy levels in indicated presentation +time condition. Human accuracy at each increasing presentation time is best matched by +model at correspondingly greater iteration (colored triangles). (C) Accuracy of the PbAS +model at the three iteration numbers chosen to be close to the best matching iterations +marked by colored triangles in (B). (We show results for 50 rather than 48 iterations; see text +for details.) Evolution of PbAS accuracy levels over these snapshots closely matches human +accuracy levels at the corresponding presentation times; compare (A). (D) Performance of +the bottom-up network (BU) and the fine-tuned (FT) variants; FT model reports ensemble +average. Unlike humans and the PbAS model, in harder cloth-occluded trials with same- +category distractors, the BU and FT models remain close to chance (dashed lines). (E) +Performance of the “Pixel-PbAS” model, which performs a more top-down form of analysis- +by-synthesis by attempting to explain input images at the pixel level, by excluding the +bottom-up image encoding module. Relative to the PbAS model, this model requires more +iterations to reach human-level performance; more critically, it qualitatively misses a key +aspect of behavior by performing equally well across occlusion conditions, specifically in +the harder same category trials. Error bars in panels A, C and E show standard deviation; +significance determined using independent-sample t-tests. (F) ℓ2 distance between human +accuracy (A) and models: PbAS, bottom-up network pretrained (BU) and after fine-tuning +(FT), and PbAS without image encoding (using pixels for likelihood computation; “Pixel- +PbAS”). For the PbAS and Pixel-PbAS models, for each presentation time, we present the +distances based on their corresponding best-matching iteration number. Error bars show +95% bootstrapped confidence intervals; “***”: p < .001; “*”: p < .05; “n.s.”: not significant +(p > .05). + +A +B +Human Behavior +distance +1 sec +2 secs +Unlimited +0.6 +Same +Different +Different +Same +Different +Same +Category +Category +Category +Category +Category +Category +0.5 +1 sec +*** +1.0 +2 secs + behavior C +0.4 +Unlimited +0.8 +Accuracy +0.3 +0.6 +0.2 + to +0.4 +PbAS +0.1 +0.2 +0.0 +1 +20406080100120140160180 +0.0 +# of iterations in PbAs +un +un. +D +Physics-based Analysis-by-Synthesis (PbAS) +Bottom-up +Fine-tuned +(BU) +(FT) +Iteration 50 +Iteration 80 +Iteration 110 +Different +Same +Different +Same +Different +Same +Category +Category +Category +Category +Category +Different +Same +Different +Same +Category +Category +Category +Category +Category +*** +1.0 +1.0 +0.8 +0.8 +Accuracy +Accuracy +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +ocC +ocC +Un +ocC +Un +E +F +Pixel Likelihood PbAS +Distance between accuracy +Iteration 80 +Iteration 110 +Iteration 140 +levels of models and humans +Different +Same +Different +Same +Different +Same +1 sec +2 secs +Unlimited +Category +Category +Category +Category +Category +Category +0.6 +n.s. +1.0 +n.s. +0.5 +0.8 +Distance +0.4 +Accuracy +0.6 +0.3 +0.2 +0.2 +0.1 +0 +0.0 +Un +ocC +Un +ocC +Un +occ +ocC +OCCSpringer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +15 +We compared average human accuracy levels for each presentation time +condition (collapsing over occlusion and difficulty) with PbAS accuracy at each +model iteration. The comparison used the ℓ2 distance. We found that the longer +the presentation time, the more model iterations are needed to best match +behavior: the fit for 1 second data requires fewer (48) iterations than are needed +for the 2 second condition (80), and even more iterations (110) are needed +to match the unlimited time data (Fig. 4B). The performance of the PbAS +model at the best-fitting iteration numbers for each presentation time closely +matches their corresponding behavioral accuracies (compare Fig. 4A and C, +which shows model performance at iterations 50, 80, and 110 for simplicity; +model accuracy levels at 48 and 50 iterations are essentially identical). In +particular, the correspondence between PbAS and behavior (measured as the +ℓ2 distance between behavioral and model accuracy levels) is stronger than it +is for any other model (Fig. 4F, p < .05, except PbAS vs. FT in the 1 second +condition; see below). +Unlike the PbAS model, the bottom-up features derived from pre-trained +DCNNs failed to explain human accuracy levels, nor did they after fine- +tuning these networks separately for each occlusion condition (Fig. 4D, F). As +expected, the performance of the pretrained bottom-up network declined sub- +stantially under occlusion, but it did so even for the easier different category +distractor trials (Fig. 4D, “Bottom-up (BU)”). For the harder cloth-draped, +same-category trials, the performance of the bottom-up model reduced to +chance (Fig. 4D, “Bottom-up (BU)”). Fine-tuning this network improved its +overall performance, but most of this improvement manifested in the different- +category trials and indeed its performance remained near chance in the harder +cloth-occluded trials with same category distractors (Fig. 4D, “Fine-tuned +(FT)”). These results are reflected in the correspondence between human and +network accuracy. In all but one condition, the discrepancy between bottom-up +and fine-tuned models, and human behavior, is higher than it is for PbAS (Fig. +4F). (In the 1-second condition, the fine-tuned model is statistically insepara- +ble from PbAS, but it decouples from behavior in finer-grained trial-by-trial +analysis, as we explain in the next section (see also Figs. 5, S3, S4)). +Overall, unlike PbAS, the discrepancy between human and network accu- +racy levels increased with presentation time, suggesting the need for additional +computations beyond the bottom-up processing implemented in these DCNN +models (Fig. 4F). +These results provide support for the role of top-down computations (the +generative model) in the hybrid architecture embodied in PbAS: The DCNN +feature hierarchies that alone cannot explain behavior are useful when they +guide inference (by defining the likelihood) in the generative model. Is this +bottom-up component necessary to explain behavior? We evaluated a model +that removed the image encoding module. This ablation – referred to as the +Pixel Likelihood PbAS (or “Pixel-PbAS” for short) – computes likelihood in +the pixel space, keeping everything else unchanged from PbAS. We found that +this ablation fails to reproduce an important aspect of behavior: Unlike the + +Springer Nature 2021 LATEX template +16 +Shape perception integrates intuitive physics and analysis-by-synthesis +PbAS model and human judgments, the Pixel-PbAS model performs equally +well in the harder (i.e., same category) occluded and unoccluded trials (Fig. +4E). Moreover, it takes longer to reach human level accuracy relative to PbAS, +requiring about 30 more iterations for each presentation time condition (Fig. +4E). Finally, this model does not match behavior as well as PbAS; using its +best-fitting iteration numbers, the distance to behavior is greater than that +of PbAS (Fig. 4F; p < .05 for each pairwise comparison, using direct boot- +strap hypothesis testing). However, we note that unlike the bottom-up models, +the distance from the Pixel-PbAS model to behavior is constant or decreases +slightly across presentation time conditions, indicating that the iterative refine- +ment of scene hypotheses is still crucial to explain how behavioral performance +improves with longer exposure times. These results establish that both top- +down and bottom-up components of the PbAS architecture are needed to +account for behavior. Relative to the bottom-up models, PbAS’s superior +account of behavior is not merely a result of its better task performance, but is +instead due to its making similar perceptual judgments, and errors, as humans. +The next two sections provide further evidence for these conclusions using +fine-grained error and response time analyses. +Explaining Trial-Level Human Accuracy +Fig. 5 +Fine-grained analysis of human accuracy at the level of individual trials in the unlim- +ited time condition. PbAS explains behavior better than alternative models. (A) Trial-level +average accuracy correlations between models and humans in the unlimited time condition. +PbAS, bottom-up network pretrained (BU) and after fine-tuning (FT), and the PbAS model +without image encoding (“Pixel-PbAS”). (The fine-tuned model reports ensemble average +of multiple fine-tuned networks.) Error bars show bootstrapped 95% confidence intervals. +Statistical comparisons are made using direct bootstrap hypothesis testing (“***”: p < .001; +“n.s.”: p > .05). (B) The hardest, same-category trials reveal that only the PbAS model +consistently correlates with behavior in both the unoccluded and occluded conditions. The +x-axis values are normalized to range between 0 and 1. Correlation coefficients are indicated +on each scatter plot; bootstrapped 95% confidence intervals in brackets. +Next, we evaluated the ability of the models to explain average human accu- +racy at the level of individual trials. In the unlimited time condition, we found + +A +Unoccluded, +Occluded, +B +Occluded, same category trials +1.0 +all trials +all trials +r = .44 [.33, .54] +r=.11 [.03, -19] +r = .02 [-.07, -10] +r=.22 [.12, .31] + accuracy +*** +*** +behavior +1.0 +1.0 +n.s. +*** +0.8 +0.8 +0.8 +0.8 +0.8 +*** +*** +0.6 +0.6 - +0.6 +0.6 +0.4 +0.4 - +0.4 +0.4 +0.6 +、 +0.2 - +0.2 - +0.2 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.4 +Unoccluded, same category trials +accuracy +:.44[.36,.51] +r=.33 [.27,.39] +r= .46 [.37, .55] +r = .24 [.16, .32] +0.2 +1.0 +1.0 +0.8 +0.8 +0.8 +0.8 - +0.0 +Human +0.6 +0.6 +0.6 - +S +0.4 - +0.4 +0.4 - +PbA +pbp +PbA +0.2 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +el +0.5 +1.0 +BO +PbAS +Bottom-up +Fine-tuned +Pixel-PbASSpringer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +17 +that the trial-by-trial accuracy of the PbAS model at the best-fitting iteration +(iteration 110, marked by the dark blue triangle in Fig. 4B) correlated well +with behavior, and did so consistently in both occlusion conditions (0.55 and +0.62 in unoccluded and cloth-occluded conditions; Fig. 5A). In the unoccluded +condition, PbAS better correlated with behavior relative to the pretrained +bottom-up network features (p < .001, using bootstrap direct hypothesis test- +ing), but fine-tuning was effective in closing the gap; PbAS and the fine-tuned +model showed no difference (p = .31). However, in the occluded condition, the +PbAS model better explained behavior relative to both the pretrained and +fine-tuned alternatives (p < .001; Fig. 5A). PbAS also correlated with behav- +ior better than the Pixel-PbAS model in both the unoccluded and occluded +conditions (p < .001; Fig. 5A; see Fig. S3 for qualitatively similar results in +the other two presentation time conditions.) Despite the superior quantitative +account of PbAS, we note that none of the models considered could explain +all of the reproducible variance in the behavioral data. Split-half correlations +across participants (see Materials & Methods) in the unlimited presentation +time condition were around r = .80 for both occlusion conditions, significantly +higher (p < .05) than the correlation achieved by PbAS. +What underlies the PbAS model’s ability to consistently account for +behavioral accuracy at the trial-level across both occlusion conditions? We +hypothesize that both top-down generative knowledge and our bottom-up fea- +ture embedding are crucial. To address, we first notice that in the easier, +different category trials, humans performance is at ceiling, especially in the +unlimited time condition (see the Different Category bars in Fig. 4A). There is +therefore little variance to explain in these easier trials. Thus, we focus on the +difficult same-category trials where there is appreciable variance in behavioral +accuracy across trials. We find that in these difficult trials, when compared to +the bottom-up models, only PbAS can account for behavior in both occlusion +conditions (Unoccluded: r = .44[.36, .51]; Occluded: r = .44[.33, .54], where +[l, u] indicates lower/upper 95% confidence intervals). In the regular, unoc- +cluded condition, the fine-tuned model (and to some extent the pretrained +model) can explain some of these fine-grained behavioral patterns, however, +these models, especially the fine-tuned model, decouple from behavior under +cloth occlusion (Fig. 5B). The Pixel-PbAS model also falls short of the perfor- +mance of the full PbAS model in both occlusion conditions (p ¡ .001; Fig. 5B; +see Fig. S4 for qualitatively similar results in the other two presentation time +conditions), further demonstrating the necessity of both top-down generative +knowledge and the bottom-up image embedding for successful prediction of +behavior. +Explaining Trial-Level Response Times As Iterative Inference +Our analyses have so far focused on accuracy. Here, we analyze human response +times to ask whether the time course of inference in PbAS can explain the evo- +lution of observers’ perceptual decision-making at the level of individual trials +– how long they decide to view a stimulus before making their choice. Thus, + +Springer Nature 2021 LATEX template +18 +Shape perception integrates intuitive physics and analysis-by-synthesis +in the unlimited time condition, we compare the number of iterations required +for the model to arrive at a decision on a given trial (in a given experimental +condition) with the average human response time for that trial. To do so, we +devised a simple decision rule in the model that applies to individual trials. At +each model iteration, this decision rule compares the average model accuracy +to a criterion set to the average participant accuracy within the trial’s con- +dition. We record the earliest iteration that PbAS performance exceeds that +criterion (or the maximum iteration number, 200, otherwise) and take it as a +predictor for that trial’s average response time. This is akin to a drift-diffusion +model3 (Usher & McClelland, 2001) where evidence accumulation naturally +arises from the iterative refinement of scene hypotheses in the PbAS model. +The results are response time predictions for each trial of each condition in +the experiment. +Fig. 6 Trial-level response time comparisons. Trial-by-trial average human response times +(log milliseconds) are explained by the PbAS model (log number of iterations) based on +a simple decision threshold (see text for details). The PbAS model captures significantly +more variance than the ablated Pixel-PbAS model in each occlusion condition (p < .001, +using direct bootstrap hypothesis testing). For each comparison, the mean correlation and +bootstrapped 95% confidence intervals (in brackets) are shown. +Despite the simplicity of this decision rule, we found a remarkable corre- +spondence between the number of iterations needed to solve a trial in PbAS +and the time humans took to respond on that trial (Fig. 6); the relationship +holds for both occlusion conditions. No parameters (beyond taking human per- +formance as criterion for each condition) were fit to explain response times. +Because the Pixel-PbAS model also performs iterative inference, we can test +its ability to explain response time data as we did with PbAS. We found that +PbAS gave a better account of response time data than the ablated model in +each occlusion condition (Fig. 6). +Discussion +We presented evidence for the use of generative model computations in visual +perception, in the form of physics-based mental simulations. Our behavioral +3Unlike standard drift diffusion models, the drift rate and other parameters arise from model +inference; no parameters are fit save the criterion. + +Unoccluded +Occluded +,.73] +r = .67 [.52, .77] +=.71 67, +r = .55 [52, .58] +r=.48 [.35, -56] +ms) +8.25 +8.25 +8.5 +8.5 +601 +8.00 +8.00- +8.0 +7.75 +7.75 - +8.0- +RT +7.50 +7.50 +luman +7.5 +7.5 +7.25 +7.0 +7.0 +7.00 +7.00 +工 +2 +4 +0 +2 +0 +2 +0 +2 +4 +0 +4 +PbAS RT +Pixel-PbAS RT +PbAS RT +Pixel-PbAS RT +(log # iterations) +(log # iterations) +(log # iterations) +(log # iterations)Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +19 +results as well as recent related literature (Little & Firestone, 2021; Phillips & +Fleming, 2020; Yildirim et al., 2016) raise a fundamental question: How is it +possible to perceive the shape of an object when none of the classic visual cues +to shape are visible? We proposed that the mind and brain exploit internal +representations of the physical processes which form scenes and images. Our +Physics-based Analysis by Synthesis (PbAS) model incorporates knowledge +of scene structure and dynamics to explain, through online optimization and +physics simulation, why a cloth-covered object appears the way it does – as the +result of dropping a cloth on an inferred shape in an inferred pose. We tested +PbAS in a shape matching task which required subjects to match a cloth- +draped object with its unoccluded (and randomly rotated) counterpart, in the +presence of a distractor. The PbAS model predicts not only overall human +accuracy in this visual matching task, but also how performance improves with +longer stimulus presentation times. Crucially, the number of inference steps +needed to reach a behaviorally-determined performance threshold predicts, on +a trial-by-trial basis, average participant response times. +Our work adds to the growing literature showing that perception in the +brain can be understood as efficient approximate inference in generative mod- +els, or analysis-by-synthesis (Echeveste, Aitchison, Hennequin, & Lengyel, +2020; Erdogan & Jacobs, 2017; Yildirim, Belledonne, Freiwald, & Tenenbaum, +2020; Yuille & Kersten, 2006). Past studies have examined some predictions of +this theory, but have not provided quantitative evidence that such rich gener- +ative models – incorporating shape, object interaction dynamics, and sensory +features – are used online during perception. PbAS also differs from previ- +ously considered generative models in its focus on scene elements and causal +processes, which when composed allow it to interpret images which are out- +side typical perceptual experience. In this way, our work identifies the flexible +use of ad hoc dynamic scene properties in perception, such as cloth mechan- +ics, that only indirectly influence image formation and are not usually seen +as cues to 3D shape. Perceiving shape through cloth occlusion highlights how +such “nuisance” variables can play a central role in 3D object perception. Our +work argues that the compositional use of generative models provides the best +way of understanding how these factors influence perception. +Bottom-up models based on DCNNs performed poorly both in the object- +under-cloth task and in mimicking human behavior. A DCNN that has been +fine-tuned on thousands of images of cloth-occluded objects produces behavior +with roughly similar average accuracy as humans in our briefest presentation +conditions (1 sec), but unlike the PbAS model fails to explain how performance +improves with time and does not correlate at all with trial-by-trial accuracy +in the most challenging conditions (occluded with cloth and same-category +distractors, for all presentation conditions tested). DCNNs, as a model class, +should in principle be able to learn any mapping from inputs to outputs, +but our fine-tuning results show that in practice, the data requirements can +be substantial (and likely exceed human experience) and the best results far +from human-like. Given the broader context of the many atypical, challenging + +Springer Nature 2021 LATEX template +20 +Shape perception integrates intuitive physics and analysis-by-synthesis +viewing conditions that the visual system may encounter, these findings under- +score the importance of generalization and robustness, ongoing challenges for +DCNNs, and illustrate how top-down knowledge can enable perception in dif- +ficult novel contexts. Bottom-up models do, however, play an important role +in our framework; relative to the ablated Pixel-PbAS model, the hybrid archi- +tecture implemented in PbAS demonstrates that powerful feature hierarchies +can usefully facilitate or guide inference in generative models. This perspec- +tive is compatible with much research on “core” object recognition showing +the explanatory power of bottom-up models (DiCarlo, Zoccolan, & Rust, 2012; +Yamins et al., 2014). Future work should also evaluate continuing developments +in DCNNs, trained using alternative loss functions, architectures or datasets +(Geirhos et al., 2021; Konkle & Alvarez, 2022), which may show improved +generalization to difficult perceptual tasks. +The PbAS model suggests that perception of cloth-covered objects in the +brain relies on a combination of feedforward, feedback, and recurrent compu- +tation. We believe that this is valuable as, relative to the case of feedforward +processing, there is little evidence to constrain or generate hypotheses regard- +ing the role of feedback and recurrent computation in visual scene analysis +(Gilbert, 2013). PbAS suggests a new computational goal for feedback and +recurrence in the brain, which is in some ways related to pattern theory as +expressed in Mumford (1994): Such processing might implement the progres- +sive unfolding of one or a number of physical simulations. It is likely that these +forms of neural computation implement multiple computational goals needed +for such diverse functions as attention, learning, and perception (Gilbert, +2013). The hypothesis suggested by PbAS – internal simulations of physical +processes – is not exclusive of the others and future work should explore their +combination. +The present implementation of the PbAS model accounts for behavior +in the specific matching task we studied here (Fig. 1A, B; Fig. 2). Future +work should exploit its modular architecture to address other experimental +paradigms and perceptual problems. For example, from an image of a single +draped object (without a comparison unoccluded object), humans can often +infer its category, approximate pose, and partial shape (Fig. 1C-E). While +evaluating PbAS in this more difficult scenario is beyond the scope of the +present paper, the framework readily extends to this setting; see Fig 7 for a +demonstration from a proof-of-concept implementation. +Our results suggest that shape perception under cloth draping involves +mental operations beyond the rapid, bottom-up processing believed character- +istic of traditional object recognition (Grill-Spector & Kanwisher, 2005). To +what extent are the computations hypothesized by PbAS – 3D shape infer- +ence, mental rotation, or mental simulations of physical and image formation +processes – also engaged in rapid, automatic visual processing? And how do +they relate to other cognitive mechanisms supporting dynamic processing such +as visual routines (Ullman, 1987) and mental imagery (Shepard & Metzler, +1971)? + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +21 +Fig. 7 Seeing the shape of a single cloth-draped object, without the aid of unoccluded +candidates (cf. Fig. 2A, B). By expanding its shape hypothesis space to contain a large +set of category-specific objects (as opposed to the four nearest neighbors of the available +context object as in our main model) and removing the unoccluded inference module, PbAS +can obtain plausible estimates of 3D pose and geometry. Rows show (from left) input image +containing target cloth-covered object and four inferred shape/pose hypotheses under this +modified PbAS model, ordered from high to low posterior probability. +Recent psychophysical work suggests that these computations might be +implemented in the visual system as part of spontaneous processing of sen- +sory data. Wong et al. (2022) studied cloth-covered object perception across +a battery of visual tasks, finding evidence that scenes are rapidly and auto- +matically parsed as the appropriate physical causes. In addition to behavioral +probes, cognitive neuroscience can address where in the brain the computa- +tions specified by the PbAS model might be implemented (see Shams and +Beierholm (2022) for a recent review of Bayesian causal models and inference +in the brain). For example, fMRI studies (Fischer, Mikhael, Tenenbaum, & +Kanwisher, 2016) have identified brain regions supporting intuitive physical +judgments in a dorsal frontoparietal network; it is of significant interest to + +Springer Nature 2021 LATEX template +22 +Shape perception integrates intuitive physics and analysis-by-synthesis +answer whether the same or similar brain regions are also recruited during the +perception of cloth-draped objects. +The PbAS framework discussed and supported here may also play a broader +role in visual processing beyond our cloth-draped object setting, unifying +competencies beyond traditional shape and object perception. A common +computational engine may therefore support perception of the dynamical prop- +erties of objects, such as the relative masses of colliding rigid bodies or single +objects reacting to the application of external forces (Sanborn, Mansinghka, & +Griffiths, 2013; Schwettmann, Tenenbaum, & Kanwisher, 2019; Wu, Yildirim, +Lim, Freeman, & Tenenbaum, 2015); the stiffness of deformable objects under- +going natural transformations (Bi, Shah, Wong, Scholl, & Yildirim, 2021; +Paulun & Fleming, 2020; Paulun, Schmidt, van Assen, & Fleming, 2017); vis- +cosity and flow of liquids (Bates, Yildirim, Tenenbaum, & Battaglia, 2019; +Kubricht et al., 2017; Van Assen, Barla, & Fleming, 2018); and in general the +perception of the physical (i.e., non-intentional) causal history of an object +(Chen & Scholl, 2016; Fleming & Schmidt, 2019; Schmidt, Phillips, & Flem- +ing, 2019). In each of these cases, it is at least plausible that the brain uses +generative models to simulate the physical processes that could have produced +the observed scene, and compare the results of these simulations to the sensory +input. A better understanding of how the brain supports these abilities could +also lead to more robust, and more human-like, machine vision systems. +Materials and Methods +Generative model +Cloth Simulations +We used the FLeX engine, a particle-based physics engine, for cloth physics +simulation (Macklin et al., 2014). Simulation parameters as well as the +mechanical-material properties of the cloth were chosen so as to achieve fast, +stable simulation of natural-looking, cotton-like cloth. Simulation parameters +were as follows. Iterations: 4; subiterations: 19; particle radius: 0.0078; collision +distance: 0.0078; shape collision margin: 0.00078; particle collision margin: 0.0; +relaxation mode: default; relaxation factor: 1.3; drag: 0.09; damping: 0.0; dis- +sipation: 0.0; restitution: 0.0. The mechanical-material properties of the cloth +were as follows. Strength stiffness: 0.8; bend stiffness: 0.64; shear stiffness: 0.4; +particle mass: 1.0; static friction: 0.18; dynamic friction: 1.1. +To increase simulation efficiency, we simplified the geometry of the +ShapeNet meshes using Blender (Blender Online Community, 2015)). First, +we corrected the surface normals on each mesh by ensuring that they were +consistent and pointed outwards. Second, we used Blender’s “Solidify” mesh +modifier with the thickness parameter set to −0.0001. Finally, we merged faces +that were adjacent and approximately coplanar (with surface normals differing +by less than 0.02 rad ≈ 1.15°). + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +23 +We initialized simulations by placing a square cloth (represented compu- +tationally with 210 × 210 particles) just above the geometric center of the +rotated object to be draped. We then ran the simulation for 150 steps, suffi- +cient to fully drape all objects we tested. Each cloth simulation took between +3 and 40 seconds on a NVIDIA 2080TI GPU, on the order of 1000 times +faster than alternative implementations using CPU-based cloth simulation and +unsimplified meshes. +Image Rendering +The scene was lit to minimize shadows. We placed 14 point lights with energy +0.5 on a sphere with radius 1.22 (object radius normalized to 1), with lights +distributed approximately equidistant using the Fibonacci sphere algorithm. +We rendered these scenes to 224×224 images using Blender’s internal renderer. +To equate the texture appearance of the draped and unoccluded images, we +replaced the optical materials associated with the original ShapeNet meshes +with a diffuse material (diffuse color 0.75 in each RGB channel, diffuse intensity +0.75, and specular intensity 0.07). We used a very similar material to render +draped cloths (diffuse color 0.8 in each RGB channel, diffuse intensity 0.8, and +specular intensity 0.05). We reasoned that equating the texture appearance +in this way would aid the bottom-up neural network models in emphasizing +shape over texture (Geirhos et al., 2018). +The experimental stimuli underlying the object-under-cloth task are pub- +licly +available: +https://github.com/CNCLgithub/intuitive-physics-3d-shape +-perception-stimuli. +Approximating Shape Distance +Given two shapes from ShapeNet Si and Sj, we define a shape distance met- +ric by (1) rendering each object in a standard canonical pose, (2) passing each +image through a pretrained AlexNet (Krizhevsky et al., 2012) and extracting +feature activations at the first fully-connected layer (i.e., applying fenc as for +pseudo-likelihood evaluation during inference), and (3) evaluating the ℓ1 dis- +tance between the feature activations for each shape. The resulting measure is +similar to that used when calculating the pseudo-likelihood. +Shape Prior +Given an unoccluded context object s0, we modeled the observer’s shape +uncertainty Pru(S) as a categorical distribution over the K = 4 shapes near- +est to s0. Let dsk be the weight of the kth closest shape sk to s0; then +Pru(S = sk) ∝ exp(−dsk) with 1 ≤ k ≤ K. The Shapenet database forms +a sparse approximation to the space of all object shapes, and we found that +the distance between an object and its closest neighbors could vary wildly; +one reason is that some object classes have many more exemplars than others. +Therefore, a prior defined solely using shape distance showed high variance +across trials and was unsuitable for our purposes (e.g., it induces arbitrary + +Springer Nature 2021 LATEX template +24 +Shape perception integrates intuitive physics and analysis-by-synthesis +bias towards either the distracting or matching object from trial to trial). The +unnormalized weights for each nearest shape were instead assigned based on +the rank order of their distance to the context object, starting at ds1 = 750 +and increasing at increments of 75 so that dsk = 750 + (k − 1)75. The scale +of these weights was chosen so that the relative contributions of the prior and +likelihood were comparable. +Inference Using Bayesian Optimization +In comparison with traditional inference schemes based on random-walk +MCMC (Cranmer et al., 2020), Bayesian optimization provides a more guided +or “active” approach to inference, where the next scene hypothesis to evaluate +the posterior on is informed by all of the previous evaluations of the posterior. +In adopting Bayesian optimization, we forego full posterior estimation (which +MCMC can provide in principle) in favor of a good point maximum a posteriori +(MAP) estimate. This choice is further motivated by the computational cost +of cloth simulation, which is responsible for nearly all of the work our model +must do. BayesOpt requires many fewer iterations, and therefore cloth sim- +ulations, than random-walk MCMC. It trades expensive overhead (compared +to other methods) in choosing search candidates for greater search efficiency +(Snoek et al., 2012). +Following Kandasamy et al. (2015), we sought to learn a function from +latent scene variables (i.e., shape and rotation) to their (unnormalized) log +posterior scores. By specifying a tractable Gaussian Process (GP) prior over +functions and conditioning on all available data, BayesOpt yields an online +strategy for adaptively choosing parameter settings to evaluate and prescribes +how the results update the GP posterior. The uncertainty in the GP approxi- +mation of the log posterior score decreases as the number of inference iterations +increases (i.e., as more evaluations of the posterior are observed). This prob- +abilistic approximation is computationally cheap to evaluate and has support +over the entire range of scene hypotheses (i.e., can be evaluated for any scene +hypothesis including those that are previously not evaluated). +BayesOpt requires specification of the GP kernel, which encodes prior +assumptions about (e.g) the smoothness of functions (Rasmussen & Williams, +2006), and an acquisition function which selects the next hypothesis given +the results of all previous evaluations. In our work, we used a Mat´ern kernel +with ν = 1.5 (the Mat´ern 3/2 kernel) and Automatic Relevance Deter- +mination (Rasmussen & Williams, 2006) to learn a probabilistic mapping +from latent scene hypothesis onto posterior scores; and we use the Expected +Improvement (EI) as our acquisition function, which favors scene hypothe- +ses that are expected to most improve the posterior score. Each iteration of +BayesOpt consists of (i) updating the estimated regression function (from +scene hypotheses to posterior scores) and (ii) optimizing the acquisition func- +tion to determine which scene hypothesis to evaluate in the next iteration. We +next describe each of these two components. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +25 +Scene hypotheses are coded specially for BayesOpt. We represent rotations +using normalized Euler angles R = {Rx, Ry, Rz}, with each orthogonal axis +taking values in [0, 1] and together spanning the half-sphere of rotations. The +shape variable is discrete, which we transform to a continuous encoding using +vector quantization. We map the interval [0, 0.25) to shape hypothesis (i.e., +nearest neighbor) 1; [0.25, 0.50) maps to the next nearest neighbor, shape 2, +and so on up to shape 4. The GP therefore learns a regression function from +a 4-dimensional input (three numbers for rotation, one for shape) to a scalar, +the log posterior score. +With this GP approximation at hand, we define an “acquisition function” +which uses the current GP state to select the most promising scene hypothe- +sis to try in the next iteration j + 1, (S(j+1), R(j+1)). Various active sampling +(or learning) heuristics are proposed in the literature (see e.g. J¨arvenp¨a¨a et al. +(2019); Kandasamy et al. (2015); Snoek et al. (2012)). We adopt the EI acqui- +sition function (Snoek et al., 2012), which chooses the scene hypothesis that is +expected to most improve the current posterior score, given all of the previous +posterior evaluations. At each iteration j of our model, the inference procedure +evaluates a scene hypothesis chosen to optimize the EI acquisition function. EI +uses a parameter, denoted ϵ (set to 330 in our simulations), to trade-off between +how much to weigh the predicted posterior score vs. the uncertainty around +that prediction (notice that the GP-based probabilistic regression provides +both the predicted mean posterior score and variance around that prediction +for the entire range of scene hypotheses). To find the scene hypothesis that +optimizes EI, we generate 100, 000 random scene hypotheses and use the high- +est scoring to initialize further local search (using L-BFGS-B). This procedure +yields the scene hypothesis (Sj, Rj) to be evaluated in the next iteration of +the model. We evaluate the posterior at this scene hypothesis using Eq. 1. +We implemented our inference scheme using the BayesOpt (Nogueira, 2014) +and GPy (GPy, 2012) packages. Code implementing the PbAS model (as well +as our behavioral data and analysis) will be made publicly available before +publication. +Bottom-Up Models +We tested three DCNN architectures: AlexNet (Krizhevsky et al., 2012), +ResNet50 (He et al., 2016), and VGG16 (Simonyan & Zisserman, 2014). These +models provide powerful feature hierarchies that are learned as a result of +training to classify images from the large-scale real-world ImageNet (Deng et +al., 2009) dataset. +Imagesets for Fine-tuning +Imagesets for fine-tuning were derived from the 5 shapes/category × 10 cat- +egories = 50 object shapes. These are the identical set of objects as those +underlying the experimental training trials used to familiarize human partici- +pants with the task. (We note that in our behavioral experiments, we did not + +Springer Nature 2021 LATEX template +26 +Shape perception integrates intuitive physics and analysis-by-synthesis +provide feedback during the training phase and indeed did not find any evi- +dence of learning in our behavioral data; see Fig. S5.) We used 8 imagesets per +occlusion condition, with 500 unique trials in each set giving 500 × 8 = 4000 +image triplets. (We evaluated how the amount of data used for fine-tuning +influenced performance, finding that performance plateaued at 8 imagesets, +compared with alternative groups of 1, 2, 8, 18, 28, and 38 imagesets). For +each triplet, we sampled two objects and randomly rotated, draped, and ren- +dered them using our stimulus generation pipeline. We reserved 2 imagesets +for test and the remaining were used for training. To minimize bias, a set of +8 imagesets was sampled from a larger pool of 54 at the beginning of each +fine-tuning procedure. We fine-tuned each model 32 times for each occlusion +condition and report accuracy averaged over the condition-specific replicas. +Modifying Network Architectures for Fine-tuning +To fine-tune AlexNet and VGGG16, we removed their top classification layer +and replaced it with a linear fully connected layer of size 120. We trained the +linear layer from scratch and fine-tuned the weights of the layer preceding it. +Unlike AlexNet and VGG16, the ResNet-50 model does not contain multi- +ple final fully-connected layers; thus, we used a modified approach to fine-tune +it.We replaced both its top classification layer as well as the preceding Aver- +age Pooling layer with a convolutional layer with kernel size 2, stride 2, and +dilation 2, without zero-padding. This convolutional layer takes as input 2048 +feature maps (the number of output feature maps in the fourth Residual Block +of the ResNet-50 model) each with dimensionality 7 × 7 and outputs 300 fea- +ture maps (each with dimensionality 3 × 3). The ReLU activation function is +applied to the flattened outputs of this convolutional layer, which is followed +by a single linear fully-connected layer of size 120. We trained the weights of +the new convolutional layer as well as the fully-connected layer from scratch, +while keeping all other weights in the network unchanged. +Details of the Training Procedure +To adapt the networks to our visual matching task, we used metric learning +with a triplet margin loss (Schultz & Joachims, 2003). The goal is to adapt the +network’s representational space so that distance in that space reflects the sim- +ilarity structure of our stimuli. Concretely, the distance between an “anchor” +image and a “positive example” should be smaller than the distance between +the anchor and “negative example”. A training triplet has the same structure +as our behavioral match-to-sample task setup: anchor corresponds to the tar- +get item; positive example corresponds to the ground-truth matching test item; +and the negative example corresponds to the distractor test item. (Remember +that the training datasets are crafted differently for each occlusion condition +and different networks are trained for each of these occlusion conditions.) We +fine-tuned each architecture for a total of 200 epochs and used a held-out test +set to make sure the models did not overfit over the course of training. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +27 +We set batch size to 8 and set the triplet loss margin to 2.0. We used +the ADAM optimizer (Kingma & Ba, 2015) with ASMGrad (Reddi, Kale, & +Kumar, 2018) using the following optimization parameters. We set β1 to 0.9, +β2 to 0.999, learning rate to 1.2 × 10−6 and ℓ2 weight decay to 1.8 × 10−3 at +the beginning of training. In an attempt to optimize the performance of the +bottom-up networks, we explored a range of custom learning rate schedules +as well as regularization methods. During training, we scheduled the learning +rate as the following. The learning rate is multiplied by 1.2 from epoch 1 +to epoch 12. From epoch 13 to epoch 161, the learning rate is annealed by +multiplying it with 0.985, after which it was kept constant until epoch 200. In +addition, to avoid overfitting, we employ regularization using a weight decay +strategy and data augmentation. From epoch 13 to 161, we multiply the weight +decay parameter by 1.04 and 1.03 in fine-tuning the occluded and unoccluded +task conditions, respectively. (We observed that without this scheduled weight +decay, models essentially memorized the training image set, giving rise to a +substantial discrepancy between training and test performance.) As a form +of data augmentation, during training, we randomly perturb each image by +adding white noise (variance set to 8.3×10−3) with probability 0.3 (the added +noise was restricted to the foreground pixels). All pixel values were truncated +to ensure that their values lie between 0 and 1. +Evaluation of the bottom-up models on the +object-under-cloth task +The accuracy of the pretrained bottom-up model on a given trial was calculated +using the following procedure. Recall that each trial in the object-under-cloth +task consists of three images: the target item, the matching test item, and the +distractor test item. We compute a feature embedding of each of these three +images from the first fully-connected layer of the network. We define a correct +answer (accuracy=1 for this trial) from the network if the correlation between +the embeddings of the target item and the matching test item (denoted corrm) +is greater than the correlation between the embeddings of the target item and +the distractor test item (denoted corrd). Otherwise, the network got the trial +wrong (accuracy=0). The accuracy levels of the pretrained bottom-up model +underlying Fig. 4D, F are calculated in this way. +In Fig. 5 where we require a continuous covariate per trial from each model +(as opposed to a binary accuracy label), we use Luce’s choice rule (i.e., soft- +max) to transform the above mentioned correlation values to a continuous +score: corrm/(corrm +corrd). Notice that the model predictions in Fig. 5B are +normalized to the range of [0, 1] for all models. +The trial-level accuracy of the fine-tuned model is calculated in a manner +similar to the PbAS model. For a given trial and a fine-tuned network, we select +the test item that is closer to the target item as the network’s guess and report +the fraction of correct guesses (i.e., the closest test item was the matching test +item) across the ensemble of 32 independently fine-tuned networks. + +Springer Nature 2021 LATEX template +28 +Shape perception integrates intuitive physics and analysis-by-synthesis +Behavioral Methods +Participants +A total of 173 participants were recruited from Amazon’s crowdsourcing plat- +form Mechanical Turk. The experiment took about 20 minutes to complete. +Each participant was paid $1.50. A total of 12 subjects were excluded due +to performing at or below chance performance (1 in Unoccluded-1sec; 4 in +Occluded-1sec; 3 in Occluded-2secs; and 4 in Occluded-Unlimited). Approval +for our behavioral study was obtained from the Massachusetts Institute of +Technology Institutional Review Board (the Committee on the Use of Humans +as Experimental Subjects), and we obtained each participant’s informed +consent prior to any experimental session. +Stimuli and Procedure +We used 240 unique ShapeNet meshes from 10 object categories to create the +120 match-to-sample shape pairs in our task. We selected 24 objects from +each category and allocated them evenly between the same-category (target +and distractor from same object category) and different-category conditions, +pairing each shape with another from the same category or a different cate- +gory as appropriate. Pairings were sampled randomly without replacement. We +thus obtained 6 same-category and 6 different-category pairs for each object +category, with no duplicate shapes across trials. +We designed a visual matching experiment based on the object-under-cloth +task. The experiment assigned participants to either the occluded or unoc- +cluded conditions as well as one of three conditions varying presentation time +lengths, for a between-subject design with 2 occlusion × 3 presentation time += 6 conditions. In the 1- and 2-second conditions, the target and test items +were displayed for the indicated period of time and the unlimited time condi- +tion let participants view the items for as long as they wished, i.e. until their +response. Images appeared and disappeared simultaneously. +The spatial organization of the display differed slightly by occlusion con- +dition. In the unoccluded condition, the two test images were placed side by +side, below the target item; for the occluded condition, the test images were +placed side by side but above the target. +Participants completed 10 practice trials before moving on to the 120 exper- +imental trials. Participants were provided with running feedback, seeing their +average task performance at every 5th trial throughout the experiment except +during the practice block (the performance feedback calculation excluded +practice trial accuracy). +Split-Half Correlations +To estimate the data noise ceiling, we used bootstrapped split-half correlations. +We sampled 1000 random splits of our participants in each occlusion condition + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +29 +(only considering the unlimited presentation time condition), each split divid- +ing the participants into two groups of equal size. (Participants were sampled +without replacement for each partition.) For a single division (one random +split) of participants, we computed the average accuracy of each split-half on +each trial, then correlated the group accuracies across all trials. (In essence, +we used the responses of one split of participants to model the responses of +the other half.) We did the same for each of the 1000 random splits, yielding +1000 bootstrap estimates of the behavioral noise ceiling and allowing us to +assess their average value and spread. But because this procedure effectively +halved our participant number, our split-half correlations likely underestimate +the true noise ceiling. +References +Bates, C.J., Yildirim, I., Tenenbaum, J.B., Battaglia, P. (2019). Modeling +human intuitions about liquid flow with particle-based simulation. PLoS +computational biology, 15(7), e1007210. +Bi, W., Shah, A.D., Wong, K.W., Scholl, B., Yildirim, I. (2021). Perception of +soft materials relies on physics-based object representations: Behavioral +and computational evidence. bioRxiv. +Blender Online Community (2015). 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Proceedings of the 38th annual conference +of the cognitive science society. + +Springer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +35 +Yuille, A., & Kersten, D. (2006). Vision as bayesian inference: analysis by +synthesis? Trends in cognitive sciences, 10(7), 301–308. +Acknowledgements: This work was supported by the Center for +Brains, Minds and Machines (CBMM), funded by NSF STC award +CCF-1231216; ONR MURI N00014-13-1-0333 (to J.B.T.); a grant +from Toyota Research Institute (to J.B.T.); and a grant from Mit- +subishi MELCO (to J.B.T.). A high performance computing cluster +(OpenMind) was provided by the McGovern Institute for Brain +Research. We thank Kevin Smith, Bernhard Egger, Kelsey Allen, +Goker Erdogan, Marty Tenenbaum, Nancy Kanwisher, and Vivian +Paulun for their comments on a previous version of this manuscript. +Supplementary Information +Fig. S1 Accuracy levels of the three models we considered including pretrained versions +(top row) and after finetuning (bottom row). AlexNet results are presented in the main text. + +Alexnet (Pretrained) +VGG (Pretrained) +ResNet (Pretrained) +Different +Same +Same +Different +Different +Same +Category +Category +Category +Category +Category +Category +1.0 +1.0 +1.0 +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +0.4 - +0.4 +0.4 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +VGG (Finetuned) +Alexnet (Finetuned) +ResNet (Finetuned) +Different +Same +Different +Same +Different +Same +Category +Category +Category +Category +Category +Category +1.0 +1.0 +1.0- +0.8 +0.8 +0.8 +0.6 +0.6 +0.6 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +oUSpringer Nature 2021 LATEX template +36 +Shape perception integrates intuitive physics and analysis-by-synthesis +Fig. S2 Behavioral results. (A) Average human accuracy in the 3 presentation time con- +ditions pooling data across the occlusion conditions. Overall, participants performed well +above chance under all presentation time conditions. Behavioral accuracy improved with +longer presentation times. (B) Average human accuracy shown separately for each occlusion +and presentation time condition. Participants’ average performance ranged from 73% in the +cloth-occluded condition under 1 sec presentation time to 93% in the unoccluded condition +under unlimited time. The gain in performance was significant within each occlusion condi- +tion, p < .05 for all pairwise comparisons of presentation times, except in the 1 sec vs. 2 secs +comparison in the cloth-occluded condition, p = .07. (C) Average response times (in millisec- +onds; pooling data across the occlusion conditions) lengthen with longer presentation times, +p < .001 for all pairwise comparisons of presentation time conditions. (D) Average response +times shown separately for each occlusion and presentation time condition. Lengthening of +response times is still evident for each occlusion condition (p < .05 for all pairwise com- +parisons of presentation times, except in the 1 sec vs. 2 secs comparison in the unoccluded +condition and 1 secs vs. Unl. comparisons in the cloth-occluded condition). Error bars show +standard deviation. +Fig. S3 Trial-level accuracy correlations in the (A) 2 secs presentation time condition +and (B) 1 sec presentation time conditions. The physics-based analysis-by-synthesis (PbAS) +model correlates well with behavior across all presentation time and occlusion condition time +conditions, relative to the alternatives based on bottom-up features optimized for image +classification (BU: bottom-up network with pretrained weights from ImageNet dataset; FT: +fine-tuned networks, separately fine-tuned for each occlusion conditions) and Pixel-PbAS, +an ablation of PbAS without the bottom-up image encoding modules (using pixels directly +for likelihood computation). Significance convention same as main text. Error bars indicate +bootstrapped 95% confidence intervals. + +A 2 secs +Unoccluded, +Occluded, +Unoccluded, +Occluded, +1.0- +all trials +all trials +1.0 +all trials +all trials +*** +*** +*** +*** +Correlation to behavior +* +*** +n.s. +*** +0.8 +0.8 +*** +*** +*** +*** +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +C +0.0 +0.0 +S +up +.up +PbAS +PbAS +PbAS +tun +tun +ttom- +ottom +ottom +BO +pix +BO +FinetA +B +Unocc. +Occ. +C +D +Unocc. +Occ. +1.0 +** +** +*** +1.0 +*** +Response times +5000 +*** +laccuracy +0.8 ++** +5000 +*** +0.8 +p=,12 +4000 +4000 +p=.09 +0.6 +0.6 +3000 +3000 +Human +0.4 +0.4 +2000 +2000 +0.2 +0.2 +1000 +1000 +0.0 +0.0 +efiSpringer Nature 2021 LATEX template +Shape perception integrates intuitive physics and analysis-by-synthesis +37 +Fig. S4 Trial-level accuracy correlations for the difficult, same category trials in the (A) +Occluded and (B) Unoccluded conditions. Results are arranged by model type and stimu- +lus presentation time. Error bars show bootstrapped 95% confidence intervals. In the easier +unoccluded, shape-category conditions, all three models that use DCNN features to match +images (PbAS, BU, and FT) perform similarly across all presentation times; Pixel-PbAS +performs significantly worse across all presentation times. In the more difficult occluded, +same-category conditions, PbAS clearly outperforms all other models, except for BU which +performs similarly in the shortest (1 sec) presentation time. Notably both pure DCNN +models, BU and FT, consistently correlate less well with human trial-level accuracies as pre- +sentation times increase, while PbAS correlations tend to increase, and FT correlations are +not significantly different from zero in the challenging occluded same-category conditions +(with BU correlations being only barely higher than zero in the 2 sec and unlimited condi- +tions). This overall pattern is consistent with the success of DCNNs at capturing the rapid +feedforward contributions to human object recognition for familiar stimuli viewed under stan- +dard conditions, and strengthens our proposal that more challenging viewing conditions and +longer processing times engage top-down, iterative, generative model based computations of +the form instantiated in PbAS. The combination of physics-based analysis by synthesis with +DCNN features for matching generative model simulations to images, as instantiated in the +full PbAS model but not Pixel-PbAS, is the only model that accounts well (and better than +or equal to any other model) for all stimulus conditions and all presentation times. + +A +Occluded, same category trials +PbAS +Bottom-up +Finetuned +Pixel-PbAS +0.6 +0.5 +Correlation +0.4 +0.3 +0.2 +0.1 +1 sec 2 sec +Unl. +1 sec 2 sec +Unl. +1 sec 2 secUnl. +1 sec 2 sec +Unl. +Duration +B +Unoccluded, same category trials +PbAS +Bottom-up +Finetuned +Pixel-PbAS +0.6 +0.5 +Correlation +0.4 +0.3 +0.2 +0.1 +0.0 +-0.1 +1 sec 2 secUnl. +1 sec 2 sec +Unl. +1 sec 2 sec +Unl. +1 sec 2 sec +Unl. +DurationSpringer Nature 2021 LATEX template +38 +Shape perception integrates intuitive physics and analysis-by-synthesis +Fig. S5 Behavioral learning curves in the two occlusion conditions. We show moving win- +dow averages (window size=10) of human accuracy levels in the two occlusion conditions +(red=UU, green=OU) under the unlimited presentation time condition. We find no evidence +of learning throughout the experiment. Shaded region shows standard error. +Fig. S6 Divergence between the Pixel-PbAS model (i.e., using pixels for likelihood with- +out bottom-up image encoding) and human performance at each model iteration. Colored +lines show ℓ2 distance between this model and human accuracy for all trials in indicated +presentation time condition. Colored triangles indicate the best matching iterations for each +presentation time condition. This model asymptotes at a larger distance to behavior than +the PbAS model. + +1.0 +accuracy +0.8 +0.6 +Behavioral +0.4 +0.2 +0.0 +1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +Time window (bin size = 10 trials)0.6 +Behavior Distance +1 sec +0.5 +Pixel-PbAS to +2 secs +0.4 +Unlimited +0.3 +0.2 +0.1 +0.0 +1 +20 +40 +60 +80100120140160180 +# of iterations in Pixel-PbAS \ No newline at end of file diff --git a/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/load_file.txt b/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c538643f40c45889874421c9bc2bca37bdcfa58 --- /dev/null +++ b/r9E2T4oBgHgl3EQfLQZF/content/tmp_files/load_file.txt @@ -0,0 +1,1581 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf,len=1580 +page_content='Springer Nature 2021 LATEX template 3D Shape Perception Integrates Intuitive Physics and Analysis-by-Synthesis Ilker Yildirim1,2,3*†, Max H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Siegel4,5*†, Amir A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Soltani4,5,6†, Shraman Ray Chaudhari7 and Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Tenenbaum4,5* 1*Department of Psychology, Yale University, New Haven, CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2Department of Statistics & Data Science, Yale University, New Haven, CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' cWu-Tsai Institute, Yale University, New Haven, CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4Department of Brain & Cognitive Sciences, MIT, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5The Center for Brains, Minds, and Machines, MIT, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 6Department of Psychology, Boston College, Chestnut Hill, Massachusetts, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 7Department of Electrical Engineering & Computer Science, MIT, Cambridge, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' E-mail(s): ilker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='yildirim@yale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' maxs@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' jbt@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Abstract Many surface cues support three-dimensional shape perception, but peo- ple can sometimes still see shape when these features are missing – in extreme cases, even when an object is completely occluded, as when covered with a draped cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We propose a framework for 3D shape per- ception that explains perception in both typical and atypical cases as analysis-by-synthesis, or inference in a generative model of image for- mation: the model integrates intuitive physics to explain how shape can be inferred from deformations it causes to other objects, as in cloth- draping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Behavioral and computational studies comparing this account with several alternatives show that it best matches human observers in 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='03711v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='NC] 9 Jan 2023 Springer Nature 2021 LATEX template 2 Shape perception integrates intuitive physics and analysis-by-synthesis both accuracy and response times, and is the only model that corre- lates significantly with human performance on difficult discriminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our results suggest that bottom-up deep neural network models are not fully adequate accounts of human shape perception, and point to how machine vision systems might achieve more human-like robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Introduction For more than a century, vision scientists have studied the many cues that humans or machines use to recover shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Edges or bounding contours, gradi- ents of shading or texture, stereo disparity, and motion parallax are just a few of the cues that can be computed from the visible surface of an object and that can reliably indicate an object’s three-dimensional (3D) shape across different views (Bulthoff, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' When available, surface cues effectively support shape perception in humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' However, a set of recent studies (Lit- tle & Firestone, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Phillips & Fleming, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Wong, Bi, Soltani, Yildirim, & Scholl, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yildirim, Siegel, & Tenenbaum, 2016) present a challenge to the classical cue-based theory of shape perception: Even when a surface is obscured, humans can sometimes perceive shape, without directly seeing the object at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Consider the synthesized images of cloth-covered objects in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1A and B, in which each object is completely occluded by a thin, cotton-like fabric draping over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Although the draped shapes look very different from the comparison objects (shown in randomly chosen orientations), observers can nonetheless pick out which unoccluded airplane or chair matches the 3D shape of the appropriate occluded object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 Here we ask: How can people perceive object shape (and pose, size, cat- egory, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') in these images, when all the classic visual cues are mostly or entirely absent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Even those image cues that are present may be highly mis- leading, as they belong not to the underlying object’s shape but the surface of the occluding cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Somehow we are able to interpret the shape of the cloth as an interaction between the underlying rigid object’s geometry and the way the cloth drapes and deforms upon contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' While sculptors have long exploited this capacity of the visual system to depict human faces and figures, only recently have detailed behavioral studies provided convincing evidence that humans somehow “undo” (Phillips & Fleming, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2022) the effect of the cloth to access the hidden object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Thus far, a computational account of vision that can explain shape perception, even in the absence of surface cues, remains absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' There are at least two ways one might explain how people perceive the shape of draped objects, corresponding to two contemporary frameworks which each seek to advance beyond classical cue-based approaches to 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' One 1The correct shape matches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1A and B are: (A) left airplane on top goes with left on bottom, right on top goes with right on bottom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) left chair on top goes with right on bottom, right on top goes with left on bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 Seeing 3D shape through a cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Bottom row: two different airplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Top row: the same airplanes, draped with cloth and presented in random order and in a random pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) Same as (A) but for two different chairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Despite the variation in viewpoint and complete occlusion of the cloth-draped objects, human observers can still match the cloth-draped and unoccluded object pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (See note in main text for correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') (C-E) Sculptors have long displayed their skill in works that are crafted from a single rigid material but convey both an illusory effect of cloth draping and rich 3D shape for the object under the illusory cloth, as in Giovanni Strazza’s (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1850s) “Veiled Virgin” (C, marble), Gabriel Klasmer’s (2000) “Car in the Sun” (D, fiberglass), or Wendell Castle’s (1985) “Ghost Clock” (E, mahogany, partially bleached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' possibility is that instead of a relatively constrained set of interpretable, mean- ingful cues, often derived from an analysis of the geometry of objects and of image formation processes, the human visual system might engage a much larger set of cues which are obtained through some black-box learning mecha- nism (and are therefore difficult to write down or interpret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Recent computer vision models based on deep convolutional neural networks (DCNNs) have demonstrated learned feature hierarchies which facilitate impressive object recognition capabilities (Krizhevsky, Sutskever, & Hinton, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' LeCun, Ben- gio, & Hinton, 2015) and which are relatively robust to variation in appearance and pose even though the model training objective does not explicitly include these goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Moreover, these same features have been shown to enable many other seemingly disparate visual tasks, including shape perception, with only minor adaptation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' fine-tuning, or adding one or a small number of addi- tional output layers) (Hong, Yamins, Majaj, & DiCarlo, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Perhaps these features are sufficiently robust to generalize across even more extreme image transformations, such as cloth occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A second possibility is that we see 3D shape via “analysis by synthesis”, or inference in a physics-based generative model of how scenes form and give rise to images (Mumford, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yuille & Kersten, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' On this view, shape per- ception is not driven solely or primarily by a fixed, universal set of image cues, computed bottom-up from any image and sufficient for any downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Rather, we infer 3D shape through a top-down interpretation process based A E B CSpringer Nature 2021 LATEX template 4 Shape perception integrates intuitive physics and analysis-by-synthesis on an internal model of how images are formed and the role that shape plays in that model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The generative model approach sees cloth draping as just one exemplar of a potentially unbounded space of atypical presentations of objects, in which some aspects of the physics of scenes and images grossly alter an object’s appearance from its typical form while remaining easily interpretable by humans: consider as other examples viewing an object such as the chair or airplane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 outside in a rainstorm, or under ten feet of cloudy water, or through colored plastic wrap, or in the light of a full moon at night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The open-ended compositionality of the visual world may imply that it is difficult or impossible to specify or learn a single set of bottom-up image cues or fea- tures which reliably and robustly encode an object’s 3D shape even in such atypical, rare conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Instead, a system should model individual, scene- level causes – the physical objects and processes that generate images – and how they combine and yield visual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Then, by reversing their effects, it may recover the original physical scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Thus a visual system could still iden- tify a draped object and even perceive its fine-grained 3D shape if it were able to model and somehow invert cloth physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our goal in this paper is to use the perception of objects under cloth as a case study to evaluate concrete versions of each of these accounts of shape perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The theoretical merits of the pure bottom-up and top- down approaches have been extensively debated in the literature, but it has been difficult to find strong evidence distinguishing the bottom-up cue-based and top-down model-based approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' until recently, neither discriminative classifiers nor Bayesian generative models performed well in realistic visual tasks, so comparisons with biological vision were limited to controlled sce- narios with simplified, non-naturalistic stimuli (Liu, Knill, & Kersten, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Advances in algorithms and computing hardware, however, have led to DCNN and analysis-by-synthesis models that achieve good performance with complex natural images and can now be rigorously evaluated as models of how we per- ceive 3D shape in challenging cases with rich naturalistic stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' They also let us explore various hybrid accounts that to date have received very little direct evaluation in human psychophysics: in particular, we compare human judgments with top-down analysis-by-synthesis models attempting to match images at either the level of raw pixels or intermediate-level representations based on DCNN features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To our knowledge, only one empirical study has studied human percep- tion in light of these improved methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Erdogan and Jacobs (2017) defined a compositional generative model of 3D shape and compared human judgments of shape similarity2 with those derived from bottom-up classifiers and from top-down inference in a 3D generative model, concluding that the latter might underlie shape perception because it correlated better with human responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' But this study, while pioneering, provides only limited evidence for top-down 2Erdogan and Jacobs (2017) elicited graded similarity judgments between different objects (so that the study and target items could be similar but not identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In contrast, we use forced- choice judgments of same or different object identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the correct shape is always a response option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 5 analysis-by-synthesis in perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The best bottom-up model was also quan- titatively predictive of human judgments and performed almost as well as the top-down model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' With more training and improved DCNNs, the quantitative gap between these models might be expected to narrow even further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In addi- tion, neither model improved dramatically over a simple baseline matching test to target images at the pixel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Here, we demonstrate a stronger, qualitative distinction between model classes enabled by our completely occluded cloth- draped stimuli: standard DCNNs and pixel-based observers, unlike people and our generative models, perform no better than chance on harder instances and even with extensive specialized training show little improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our design choices offer several other advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Because we chose uncom- mon stimuli with variable difficulty, we find meaningful variance in human performance and response time, which allows for finer-grained model evalua- tions and comparisons of humans and models on trial-by-trial speed as well as accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The generative model that we consider performs iterative inference with significant stimulus-driven variation in the number of computational steps and therefore can be directly compared with subjects’ reaction times, poten- tially revealing signatures and roles for feedback or recurrence in biological shape perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Results The Object-Under-Cloth Task While in some cases humans can recognize draped objects from a single image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1C, D), we chose as our experimental setting a visual match-to- sample task that allows us to directly address the above three possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We choose this setting as we are primarily interested in how generative models can support online, detailed 3D shape perception, rather than object catego- rization or any kind of memory-based process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The essence of our proposal is that observers may perceive the 3D shape of cloth-covered objects in arbi- trary orientations by approximately simulating in their minds the process of how cloth drapes over the object in three dimensions, and imagining what the resulting 2D image would look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' So we constructed an experimental task that should be directly solvable via this mechanism: We show observers an unoccluded matching object along with a target draped shape (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', the initial matching object rendered in computer graphics under a simulation of cloth draping) and an unoccluded distractor object, in a two-alternative forced choice (2AFC) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We call this the “occluded” condition to contrast with a control “unoccluded” condition (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We chose 10 different everyday object categories (airplane, bicycle, bus, car, chair, guitar, motorcycle, pistol, rifle, table) and sampled object meshes for each category from a large repository of 3D shapes (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We used 24 unique exemplars from each category, yielding 120 visual-matching trials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' each trial used two shapes, and each unique shape appeared in one trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each trial presented an unoccluded target shape, a distractor shape, and Springer Nature 2021 LATEX template 6 Shape perception integrates intuitive physics and analysis-by-synthesis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2 Matching a target shape to one of two unoccluded test objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Left: A trial in the occluded task condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The top image shows a “target item”, the bottom-left image shows a “matching test item” and the bottom-right image shows a “distractor test item”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Right: Trials from the occluded task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each triplet displays, from left-to-right, target item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' matching test item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and distractor test item for one trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We show “easier” trials, with different-category distractor and matching test items, and “harder” trials, where both test items are of the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) Left: A trial from the unoccluded task condition, spatial configuration as in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Right: Each triplet shows a trial from the unoccluded task, showing instances of easier and harder trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the target shape after cloth draping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We varied the similarity between the distractor and matching items (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2A, right) to generate visual-matching triplets ranging in difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In half of the trials, the target and distractor objects were drawn from the same category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' these we term “harder” trials because same-category shapes are generally more difficult to distinguish than different-category objects, which we call “easier”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To create the cloth-occluded stimuli, we simulated cloth draping via a particle-based physics engine (Macklin, M¨uller, Chentanez, & Kim, 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' we chose simulation parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', number of iterations) and the mechani- cal and material properties of the simulated cloth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', stiffness and mass) to enable efficient, stable simulation of natural-looking cotton-like cloth (see Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the unoccluded condition, we use the same objects but show the tar- get shape without cloth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In this version of the task, viewpoint variability and the shape similarity between the matching and distractor test items are the only confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A Different category distractor Same category distractor Occluded (Easier trials) (Harder trials) Which is the same object as above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' B Unoccluded Which is the same object as above?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 7 Physics-Based Analysis-By-Synthesis (PbAS) We formalize the problem of matching a cloth covered object with its unoc- cluded counterpart as approximate Bayesian inference in a causal generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our physics-based analysis-by-synthesis (PbAS) method combines physics and graphics knowledge with statistical inference and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The model consists of three components: a generative model for scenes and images, feature extraction for approximately Bayesian inference (using a pseudo-likelihood approach), and a simulator-in-the-loop inference engine based on Bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' As an account of how people can perceive the shapes of objects under cloth (or other challenging viewing conditions),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' we posit that each of these three components has some analog in the mind and brain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and that they operate and interact in something like the ways we specify here – not precisely as we have implemented them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' but close enough that the speed and accuracy characteristics of the PbAS model can be quan- titatively compared with human behavior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' along with different model variants and alternative accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The generative model in PbAS captures the physical scene variables, includ- ing object shape and pose, cloth properties, and the mechanics of how they interact, which together produce the geometry of the occluding cloth surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' It further includes a model of graphics – how surface geometry, material, and light interact to generate an image (some factors, like optics, are handled implicitly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Koch, Baig, and Zaidi (2018) for an explicit treatment, including modeling, of human representation of visual scene geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Given a hypothesized 3D object shape in a hypothesized pose, the model produces a synthesized or hypothetical image which may be compared with the image actually observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the analysis by synthesis framework, perception requires inverting this process to recover the object shape and pose likely to have given rise to the observed image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Like most generative models, PbAS is too complex to invert exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A ubiquitous approximation algorithm, Markov Chain Monte Carlo (MCMC), iteratively constructs samples from a target distribution like the posterior, but in our case requires far too many iterations to work because each step includes costly physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We sought instead to maximize the posterior using Bayesian optimization (Snoek, Larochelle, & Adams, 2012), which relative to MCMC provides a guided inference scheme where the next scene hypothesis to evaluate is informed by all (instead of only the current) evaluations of the posterior function (Cranmer, Brehmer, & Louppe, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' BayesOpt simulta- neously estimates and optimizes the posterior, providing an algorithm which efficiently samples increasingly more probable hypotheses for object shape and rotation given an input occluded image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (Psychologically, BayesOpt can be seen as implementing a kind of goal-conditioned mental imagery;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Hamrick and Griffiths (2013) for an application in the context of mental rota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') The probability of a scene hypothesis is computed by comparing its corresponding rendered hypothesis image with the input, using a feedforward Springer Nature 2021 LATEX template 8 Shape perception integrates intuitive physics and analysis-by-synthesis feature hierarchy fenc implemented as the first fully-connected layer of a pre- trained DCNN (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' While the goal of inference in our model is posterior probability maximization, the optimization trajectory is also of interest for comparison with human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS can arrive at a reasonable percept rapidly (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3C, D) compared to sampling-based methods like standard MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' It therefore provides a more plausible quantitative standard for understanding average human accu- racy, how accuracy improves with longer viewing time, and stimulus-driven variability in response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Synthesis: Generative Model The generative model consists of (i) latent variables describing the scene: a 3D object shape S and its rotation R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (ii) a forward physics simulator along with cloth parameters: cloth size, position, stiffness, mass, and friction, denoted fΨ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and (iii) a rendering function and lighting parameters, together denoted fΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We set the physics simulation parameters fΨ and renderer parameters fΓ to the same values as used for stimuli generation (see Section “The object-under- cloth task” and Materials & Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' While the model is designed to perceive cloth-covered objects, it applies to unoccluded objects, as in the unoccluded task condition, as a special case by setting fΨ to the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Given an occluded input observation (indicated as “Input” in red frame, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B) and an unoccluded “context object” (in blue frame, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B), we wish to estimate the object shape S and rotation R that best explains the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' More formally, we wish to invert the generative model to find scene hypotheses that explain perceptual input using Bayesian inference, which amounts to finding the posterior Pr(S, R | Iobs) ∝ Pr(Iobs | S, R, fΨ, fΓ)Pru(S)Pr(R)δfΨδfΓ = Pr(Iobs | Ihyp)Pru(S)Pr(R), (1) where Pr(Iobs | S, R, fΨ, fΓ) is a likelihood term induced by the physics engine fΨ and rendering function fΓ, and the delta functions select fixed physics fΨ and rendering fΓ parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For brevity, in the equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 and below we write Ihyp = fΨ(fΓ(S, R)) for the hypothesis image given latent scene parameters and suppress the delta notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We next explain each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The context object allows observers to form a distribution Pru(S) over the possible shape of the draped object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' because the context object is pre- sented as a 2D rendering, its shape is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Even though human observers do not need auxiliary shape information to process cloth-occluded images, this accompanying context object provides a computationally tractable shape hypothesis space for generative modeling (see the Discussion for future direc- tions relaxing this constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We represent this shape uncertainty Pru(S) using a categorical distribution over the K nearest neighbors of the actual con- text object (excluding the context object itself) in a large repository of shapes (the ShapeNet dataset (Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In our simulations Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 9 we take K = 4 and each neighbor is assigned a probability as a function of its distance rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Materials & Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We place a uniform prior over rotations Pr(R) covering the half-sphere centered at canonical pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Executing the physics simulator fΨ with a scene hypothesis (a sampled shape S and its rotation R) results in a draped cloth geometry G (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Passing the resulting scene to the rendering function fΓ in turn yields an image Ihyp = fΓ(G) of the cloth-draped object (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' a hypothesis image which may be compared with an input observed image Iobs to evaluate its likelihood under the scene hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The detailed geometry resulting from cloth simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the partic- ular pattern of wrinkles) can vary significantly with even small changes in the values of random variables (Macklin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' therefore, calculating an accurate likelihood (through marginalization) for any scene hypothesis is computationally intractable (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' As a result, we define a pseudo-likelihood function Pr(Iobs | Ihyp) based on the distance D(Iobs, Ihyp) between the input and hypothesis images in a suitable feature space arising from an encoder fenc(·);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' here, we set D = ℓ1 and adopt the features com- puted by the first fully-connected layer of AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012) as the encoder fenc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The PbAS (pseudo-) likelihood for an input image, given a hypothesis image rendered from a scene proposal, is then Pr(Iobs | Ihyp) ∝ exp (− ∥(fenc(Iobs) − fenc(Ihyp)∥1) With these choices for the prior and likelihood, the posterior Pr(S, R | Iobs) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 depends only on two terms: the discrepancy Pr(Iobs | Ihyp) between the observed image and the rendered latent parameters, and the uncertainty Pru(S) over the shape of the context image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' By measuring the discrepancy between rendered scene hypotheses and observed images in terms of DCNN encoder-based features, the PbAS model as described is an instance of a hybrid top-down/bottom-up (or model-based/cue- based) approach to 3D shape recovery (see also Wang, Mei, Yuille, and Kortylewski (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We also consider a purely top-down analysis-by-synthesis approach which is identical except that image discrepancies are computed in terms of raw pixel deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The likelihood is then simply Pr(Iobsms | Ihyp) ∝ exp � − ∥Iobs − Ihyp∥1 � We refer to this alternative as the “Pixel Likelihood PbAS” model, or “Pixel- PbAS” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Analysis: Inference Using Bayesian Optimization The posterior Pr(S, R | Iobs) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 contains all information that our model extracts from an observed image Iobs, but computing this distribu- tion is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Standard simulation-based inference methods based on MCMC ensure eventual convergence to the full posterior but in practice spend Springer Nature 2021 LATEX template 10 Shape perception integrates intuitive physics and analysis-by-synthesis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3 Overview of Physics-based Analysis-by-Synthesis (PbAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Application of the PbAS model to solve the object-under-cloth task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Given an image triplet, two PbAS models are run in parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' each execution takes as input a test item and the target item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' On each iteration, the two executions of the model are compared to determine how well the target item is explained by each test candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) Interpreting an input cloth-covered image (red frame), with a context unoccluded object (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' blue frame) supplying a prior over object shape Pru(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Bayesian optimization (BayesOpt) efficiently guides inference, improving shape (S) and rotation (R) hypotheses across iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S and R proposals initialize the cloth drap- ing simulation, then are evaluated by computing the distance D between the current scene hypothesis (rendered to a hypothesis image) and the input in a suitable feature encoding space fenc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (C) Visualization of three inference (panel B) trajectories over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Rows are independent runs of PbAS each with input as in A, B (red frame) and show the cumulative best scene hypothesis at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Blocks show hypotheses visualized with (upper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' ”Pre- diction”) and without (lower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' ”Without cloth”) cloth occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Model estimation accuracy improves with increasing iteration number, but some uncertainty remains as the model (like people) cannot in general perfectly identify the shape or pose of a draped object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (D) Evolu- tion of model accuracy averaged across multiple runs in the occluded task condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Model predictions by iteration for 15 same-category “harder” and 15 different-category “easier” trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' too many iterations in low probability regions (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We focus instead on the maximum a posteriori (MAP) setting: finding the best single scene interpretation rather than the full posterior over all possible latent variable settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Following previous work in simulation-based inference (J¨arvenp¨a¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Kandasamy, Schneider, & P´oczos, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Tamura & Hukushima, 2018), we employ Bayesian optimization (or BayesOpt (Snoek et A c Iterations 30 60 90 120 Iterations Run 1 PbAS Prediction Run 2 Which unoccluded object better explains the cloth-covered image?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Run 3 PbAS Run 1 B Without cloth Run 2 Unoccluded context K=4 nearest shapes object, u Pru(S) Run 3 D[fene(Input), BayesOpt fenc(Prediction) Pr(R) D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 Input [S, R} to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 try next 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 Prediction A Different category at iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' j 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 Same category 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1 40 80 120 160 G (draped Model iterations geometry)Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 11 al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' unlike gradient-based algorithms, BayesOpt allows us to optimize functions which include procedures, such as our scene renderer, which do not expose or do not support gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' See Materials & Methods for an overview and details of BayesOpt applied in PbAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Solving the Object-Under-Cloth Task Using the Model Human participants see two unoccluded context objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', test items) and one target object on each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Recall that by its design, the PbAS model interprets a target (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', cloth-draped or unoccluded depending on experiment condition) object in the context of an unoccluded object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Thus, to model a given trial, we form two pairs, each consisting of a context object (either the matching item or the distractor item) and the target object, and apply PbAS to each pair (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each PbAS run aims to explain the same input image, but with different shape hypotheses derived from either the matching object or the distractor object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' At every iteration, we save the current best parameter estimates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', shape and rotation) and the log posterior score for that scene hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Using the odds ratio decision rule, we obtain the model’s best estimate of the underlying shape for each inference step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We ran the PbAS model 32 times on each trial, for 200 iterations each, and treated each of these runs as a simulated participant (although with finer temporal resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' At each of the 200 iterations, we averaged the binary decisions across runs to obtain mean accuracy predictions – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', simulating the accuracy of participants’ average shape choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In our analysis we compare the dynamics of model choice with human decisions sampled at three different time intervals, corresponding to three different presentation durations that varied across experimental conditions (see the Section “Iterative refinement in PbAS explains Human Accuracy and Response Time” for comparisons of models and human behavior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3D shows how the average model performance changes as a function of iteration for a subset of our stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Bottom-Up Models Based On DCNNs To help evaluate the PbAS model and its correspondence with human per- ception, we considered several well-studied bottom-up models as comparisons for human and model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Recent computer vision models based on DCNNs learn powerful visual feature hierarchies achieving state-of-the-art object recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These feature hierarchies are relatively robust to variation in pose and lighting, can predict certain aspects of variance in neural and behavioral data, and are considered the “current best models of the primate visual stream” (Schrimpf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Moreover they are useful for visual tasks beyond object recognition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' these features have been used for a number of other vision problems e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' object localization and pose estimation (Yamins & DiCarlo, 2016), among others, with minor or no modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In testing these pretrained models, our goal is not to establish whether DCNNs, considered as a model class, can perform the object-under-cloth task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' DCNNs are universal function approximators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' with enough data, enough compute, and Springer Nature 2021 LATEX template 12 Shape perception integrates intuitive physics and analysis-by-synthesis the right architecture and optimization procedure, they are likely able to learn to perform our visual-matching task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Instead, our goal is to assess whether the features learned from categorizing objects in natural scenes can suffice to perceive cloth-occluded shapes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Because our synthesized stimuli and task design differ from those used for the pretrained DCNNs, we also test the same networks after fine-tuning them using images similar to our experimental stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We tested the following architectures, each pretrained using ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2009): AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012), ResNet-50 (He, Zhang, Ren, & Sun, 2016), and VGG16 (Simonyan & Zisserman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each DCNN was fine-tuned separately for the cloth-occluded and unoccluded conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The task was the same visual matching problem presented to humans: given an image containing two unoc- cluded test shapes and one target object (a “triplet”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' objects sampled from a total of 50 shapes), determine which test shape corresponds to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We repeated this process 32 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' thus we fine-tuned 32 copies (to match the num- ber of PbAS runs per trial) of each architecture for each occlusion condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We report the average accuracy of these 32 fine-tuned networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' See Materials & Methods for dataset generation, fine-tuning, and evaluation procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For both the pretrained and fine-tuned conditions, we found that no archi- tecture was more accurate than AlexNet (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Therefore, we use both the pretrained AlexNet and our fine-tuned variant in our comparisons of bottom-up models with behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Iterative Refinement in PbAS Explains Human Accuracy and Response Times To evaluate PbAS as a candidate model for human perception, we compared its predictions on the object-under-cloth task with two key behavioral mea- sures: average accuracy and response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We recruited human subjects and assigned them to either the occluded or unoccluded condition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2A, B left panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Participants were also divided into three presentation time con- ditions: the two fixed (1 or 2 second) time conditions and the unlimited time condition, which presents stimuli until subjects respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In total, the exper- iment consisted of 2 occlusion × 3 presentation time = 6 conditions in a between-subjects design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' As is typical in modeling studies, we compared the average accuracy of PbAS and alternative models with that of humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Because accuracy measures alone might simply favor models that are more performant, we also examined how PbAS “response times” – the number of inference iterations used per trial – might explain human response times on the same trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 13 Explaining Human Accuracy Across Presentation Time Conditions We first established that behavioral performance is significantly affected by task setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' While participants performed well above chance across all occlu- sion and presentation time conditions, their performance varied with respect to these design parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Most obviously, human performance was better in the unoccluded setting (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' With longer presentation time, average per- formance significantly improved (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S2A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S2B for results broken down by occlusion condition) and response times increased (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S2C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S2D for results broken down by occlusion condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We also note that there was no learning effect throughout the experiment, with participants’ average performance remaining fairly constant across trials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The design of our behavioral experiment offers a multifaceted view of human performance in terms of presentation time, trial difficulty (defined as whether test items are of same or different category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2), and occlusion condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4A, we present average human accuracy levels for each pre- sentation time, pooled with respect to the two difficulty types (“Different Category” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “Same Category”) and two occlusion conditions (occluded vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' unoccluded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Observers performed significantly above chance even in the most challenging setting with cloth occlusion, same-category distractors, and the briefest presentation time (1 second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Note also that performance improved with longer presentation time in the same-category distractor trials where, unlike the easier different-category distractor case, performance does not reach ceiling even with unlimited presentation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We now ask whether PbAS and bottom-up models can explain these nuanced results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 14 Shape perception integrates intuitive physics and analysis-by-synthesis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4 PbAS explains how human accuracy increases with longer stimulus presentation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Behavioral accuracy for each presentation time, occlusion condition, and difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A trial is said to be hard if the distractor test item is of the same category as the target item, and easy if the distractor test item is of a different category than the target item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') (B) Divergence between model and human performance at each model iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Colored lines show ℓ2 distance between PbAS model and human accuracy levels in indicated presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Human accuracy at each increasing presentation time is best matched by model at correspondingly greater iteration (colored triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (C) Accuracy of the PbAS model at the three iteration numbers chosen to be close to the best matching iterations marked by colored triangles in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (We show results for 50 rather than 48 iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') Evolution of PbAS accuracy levels over these snapshots closely matches human accuracy levels at the corresponding presentation times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' compare (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (D) Performance of the bottom-up network (BU) and the fine-tuned (FT) variants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' FT model reports ensemble average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Unlike humans and the PbAS model, in harder cloth-occluded trials with same- category distractors, the BU and FT models remain close to chance (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (E) Performance of the “Pixel-PbAS” model, which performs a more top-down form of analysis- by-synthesis by attempting to explain input images at the pixel level, by excluding the bottom-up image encoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Relative to the PbAS model, this model requires more iterations to reach human-level performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' more critically, it qualitatively misses a key aspect of behavior by performing equally well across occlusion conditions, specifically in the harder same category trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Error bars in panels A, C and E show standard deviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' significance determined using independent-sample t-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (F) ℓ2 distance between human accuracy (A) and models: PbAS, bottom-up network pretrained (BU) and after fine-tuning (FT), and PbAS without image encoding (using pixels for likelihood computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “Pixel- PbAS”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For the PbAS and Pixel-PbAS models, for each presentation time, we present the distances based on their corresponding best-matching iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Error bars show 95% bootstrapped confidence intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “***”: p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “*”: p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='s.”: not significant (p > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A B Human Behavior distance 1 sec 2 secs Unlimited 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 Same Different Different Same Different Same Category Category Category Category Category Category 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 1 sec *** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 2 secs behavior C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 Unlimited 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 PbAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1 20406080100120140160180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 # of iterations in PbAs un un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' D Physics-based Analysis-by-Synthesis (PbAS) Bottom-up Fine-tuned (BU) (FT) Iteration 50 Iteration 80 Iteration 110 Different Same Different Same Different Same Category Category Category Category Category Different Same Different Same Category Category Category Category Category *** 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 Accuracy Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 ocC ocC Un ocC Un E F Pixel Likelihood PbAS Distance between accuracy Iteration 80 Iteration 110 Iteration 140 levels of models and humans Different Same Different Same Different Same 1 sec 2 secs Unlimited Category Category Category Category Category Category 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 Distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 Un ocC Un ocC Un occ ocC OCCSpringer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 15 We compared average human accuracy levels for each presentation time condition (collapsing over occlusion and difficulty) with PbAS accuracy at each model iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The comparison used the ℓ2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We found that the longer the presentation time, the more model iterations are needed to best match behavior: the fit for 1 second data requires fewer (48) iterations than are needed for the 2 second condition (80), and even more iterations (110) are needed to match the unlimited time data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The performance of the PbAS model at the best-fitting iteration numbers for each presentation time closely matches their corresponding behavioral accuracies (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4A and C, which shows model performance at iterations 50, 80, and 110 for simplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' model accuracy levels at 48 and 50 iterations are essentially identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In particular, the correspondence between PbAS and behavior (measured as the ℓ2 distance between behavioral and model accuracy levels) is stronger than it is for any other model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4F, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05, except PbAS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' FT in the 1 second condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Unlike the PbAS model, the bottom-up features derived from pre-trained DCNNs failed to explain human accuracy levels, nor did they after fine- tuning these networks separately for each occlusion condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4D, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' As expected, the performance of the pretrained bottom-up network declined sub- stantially under occlusion, but it did so even for the easier different category distractor trials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4D, “Bottom-up (BU)”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For the harder cloth-draped, same-category trials, the performance of the bottom-up model reduced to chance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4D, “Bottom-up (BU)”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fine-tuning this network improved its overall performance, but most of this improvement manifested in the different- category trials and indeed its performance remained near chance in the harder cloth-occluded trials with same category distractors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4D, “Fine-tuned (FT)”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These results are reflected in the correspondence between human and network accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In all but one condition, the discrepancy between bottom-up and fine-tuned models, and human behavior, is higher than it is for PbAS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (In the 1-second condition, the fine-tuned model is statistically insepara- ble from PbAS, but it decouples from behavior in finer-grained trial-by-trial analysis, as we explain in the next section (see also Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5, S3, S4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Overall, unlike PbAS, the discrepancy between human and network accu- racy levels increased with presentation time, suggesting the need for additional computations beyond the bottom-up processing implemented in these DCNN models (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These results provide support for the role of top-down computations (the generative model) in the hybrid architecture embodied in PbAS: The DCNN feature hierarchies that alone cannot explain behavior are useful when they guide inference (by defining the likelihood) in the generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Is this bottom-up component necessary to explain behavior?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We evaluated a model that removed the image encoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This ablation – referred to as the Pixel Likelihood PbAS (or “Pixel-PbAS” for short) – computes likelihood in the pixel space, keeping everything else unchanged from PbAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We found that this ablation fails to reproduce an important aspect of behavior: Unlike the Springer Nature 2021 LATEX template 16 Shape perception integrates intuitive physics and analysis-by-synthesis PbAS model and human judgments, the Pixel-PbAS model performs equally well in the harder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', same category) occluded and unoccluded trials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Moreover, it takes longer to reach human level accuracy relative to PbAS, requiring about 30 more iterations for each presentation time condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Finally, this model does not match behavior as well as PbAS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' using its best-fitting iteration numbers, the distance to behavior is greater than that of PbAS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05 for each pairwise comparison, using direct boot- strap hypothesis testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' However, we note that unlike the bottom-up models, the distance from the Pixel-PbAS model to behavior is constant or decreases slightly across presentation time conditions, indicating that the iterative refine- ment of scene hypotheses is still crucial to explain how behavioral performance improves with longer exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These results establish that both top- down and bottom-up components of the PbAS architecture are needed to account for behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Relative to the bottom-up models, PbAS’s superior account of behavior is not merely a result of its better task performance, but is instead due to its making similar perceptual judgments, and errors, as humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The next two sections provide further evidence for these conclusions using fine-grained error and response time analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Explaining Trial-Level Human Accuracy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5 Fine-grained analysis of human accuracy at the level of individual trials in the unlim- ited time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS explains behavior better than alternative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Trial-level average accuracy correlations between models and humans in the unlimited time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS, bottom-up network pretrained (BU) and after fine-tuning (FT), and the PbAS model without image encoding (“Pixel-PbAS”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (The fine-tuned model reports ensemble average of multiple fine-tuned networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') Error bars show bootstrapped 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Statistical comparisons are made using direct bootstrap hypothesis testing (“***”: p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' “n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='s.”: p > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) The hardest, same-category trials reveal that only the PbAS model consistently correlates with behavior in both the unoccluded and occluded conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The x-axis values are normalized to range between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Correlation coefficients are indicated on each scatter plot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' bootstrapped 95% confidence intervals in brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Next, we evaluated the ability of the models to explain average human accu- racy at the level of individual trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the unlimited time condition, we found A Unoccluded, Occluded, B Occluded, same category trials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 all trials all trials r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='44 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='33, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='54] r=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='11 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='03, -19] r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='02 [-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='07, -10] r=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='22 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='31] accuracy *** *** behavior 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' *** 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 Unoccluded, same category trials accuracy :.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='44[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='36,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='51] r=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='33 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='27,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='39] r= .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='46 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='37, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='55] r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='24 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='16, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='32] 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 el 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 BO PbAS Bottom-up Fine-tuned Pixel-PbASSpringer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 17 that the trial-by-trial accuracy of the PbAS model at the best-fitting iteration (iteration 110, marked by the dark blue triangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4B) correlated well with behavior, and did so consistently in both occlusion conditions (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='55 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='62 in unoccluded and cloth-occluded conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the unoccluded condition, PbAS better correlated with behavior relative to the pretrained bottom-up network features (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001, using bootstrap direct hypothesis test- ing), but fine-tuning was effective in closing the gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS and the fine-tuned model showed no difference (p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' However, in the occluded condition, the PbAS model better explained behavior relative to both the pretrained and fine-tuned alternatives (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS also correlated with behav- ior better than the Pixel-PbAS model in both the unoccluded and occluded conditions (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S3 for qualitatively similar results in the other two presentation time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') Despite the superior quantitative account of PbAS, we note that none of the models considered could explain all of the reproducible variance in the behavioral data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Split-half correlations across participants (see Materials & Methods) in the unlimited presentation time condition were around r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='80 for both occlusion conditions, significantly higher (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05) than the correlation achieved by PbAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' What underlies the PbAS model’s ability to consistently account for behavioral accuracy at the trial-level across both occlusion conditions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We hypothesize that both top-down generative knowledge and our bottom-up fea- ture embedding are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To address, we first notice that in the easier, different category trials, humans performance is at ceiling, especially in the unlimited time condition (see the Different Category bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' There is therefore little variance to explain in these easier trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Thus, we focus on the difficult same-category trials where there is appreciable variance in behavioral accuracy across trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We find that in these difficult trials, when compared to the bottom-up models, only PbAS can account for behavior in both occlusion conditions (Unoccluded: r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='44[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='36, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Occluded: r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='44[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='33, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='54], where [l, u] indicates lower/upper 95% confidence intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the regular, unoc- cluded condition, the fine-tuned model (and to some extent the pretrained model) can explain some of these fine-grained behavioral patterns, however, these models, especially the fine-tuned model, decouple from behavior under cloth occlusion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The Pixel-PbAS model also falls short of the perfor- mance of the full PbAS model in both occlusion conditions (p ¡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S4 for qualitatively similar results in the other two presentation time conditions), further demonstrating the necessity of both top-down generative knowledge and the bottom-up image embedding for successful prediction of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Explaining Trial-Level Response Times As Iterative Inference Our analyses have so far focused on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Here, we analyze human response times to ask whether the time course of inference in PbAS can explain the evo- lution of observers’ perceptual decision-making at the level of individual trials – how long they decide to view a stimulus before making their choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Thus, Springer Nature 2021 LATEX template 18 Shape perception integrates intuitive physics and analysis-by-synthesis in the unlimited time condition, we compare the number of iterations required for the model to arrive at a decision on a given trial (in a given experimental condition) with the average human response time for that trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To do so, we devised a simple decision rule in the model that applies to individual trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' At each model iteration, this decision rule compares the average model accuracy to a criterion set to the average participant accuracy within the trial’s con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We record the earliest iteration that PbAS performance exceeds that criterion (or the maximum iteration number, 200, otherwise) and take it as a predictor for that trial’s average response time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This is akin to a drift-diffusion model3 (Usher & McClelland, 2001) where evidence accumulation naturally arises from the iterative refinement of scene hypotheses in the PbAS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The results are response time predictions for each trial of each condition in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 6 Trial-level response time comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Trial-by-trial average human response times (log milliseconds) are explained by the PbAS model (log number of iterations) based on a simple decision threshold (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The PbAS model captures significantly more variance than the ablated Pixel-PbAS model in each occlusion condition (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001, using direct bootstrap hypothesis testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For each comparison, the mean correlation and bootstrapped 95% confidence intervals (in brackets) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Despite the simplicity of this decision rule, we found a remarkable corre- spondence between the number of iterations needed to solve a trial in PbAS and the time humans took to respond on that trial (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the relationship holds for both occlusion conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' No parameters (beyond taking human per- formance as criterion for each condition) were fit to explain response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Because the Pixel-PbAS model also performs iterative inference, we can test its ability to explain response time data as we did with PbAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We found that PbAS gave a better account of response time data than the ablated model in each occlusion condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Discussion We presented evidence for the use of generative model computations in visual perception, in the form of physics-based mental simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our behavioral 3Unlike standard drift diffusion models, the drift rate and other parameters arise from model inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' no parameters are fit save the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Unoccluded Occluded ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='73] r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='67 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='52, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='77] =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='71 67, r = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='55 [52, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='58] r=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='48 [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='35, -56] ms) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 601 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='00- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='75 - 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0- RT 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='50 luman 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='00 工 2 4 0 2 0 2 0 2 4 0 4 PbAS RT Pixel-PbAS RT PbAS RT Pixel-PbAS RT (log # iterations) (log # iterations) (log # iterations) (log # iterations)Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 19 results as well as recent related literature (Little & Firestone, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Phillips & Fleming, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yildirim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2016) raise a fundamental question: How is it possible to perceive the shape of an object when none of the classic visual cues to shape are visible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We proposed that the mind and brain exploit internal representations of the physical processes which form scenes and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our Physics-based Analysis by Synthesis (PbAS) model incorporates knowledge of scene structure and dynamics to explain, through online optimization and physics simulation, why a cloth-covered object appears the way it does – as the result of dropping a cloth on an inferred shape in an inferred pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We tested PbAS in a shape matching task which required subjects to match a cloth- draped object with its unoccluded (and randomly rotated) counterpart, in the presence of a distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The PbAS model predicts not only overall human accuracy in this visual matching task, but also how performance improves with longer stimulus presentation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Crucially, the number of inference steps needed to reach a behaviorally-determined performance threshold predicts, on a trial-by-trial basis, average participant response times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our work adds to the growing literature showing that perception in the brain can be understood as efficient approximate inference in generative mod- els, or analysis-by-synthesis (Echeveste, Aitchison, Hennequin, & Lengyel, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Erdogan & Jacobs, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yildirim, Belledonne, Freiwald, & Tenenbaum, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yuille & Kersten, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Past studies have examined some predictions of this theory, but have not provided quantitative evidence that such rich gener- ative models – incorporating shape, object interaction dynamics, and sensory features – are used online during perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS also differs from previ- ously considered generative models in its focus on scene elements and causal processes, which when composed allow it to interpret images which are out- side typical perceptual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In this way, our work identifies the flexible use of ad hoc dynamic scene properties in perception, such as cloth mechan- ics, that only indirectly influence image formation and are not usually seen as cues to 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Perceiving shape through cloth occlusion highlights how such “nuisance” variables can play a central role in 3D object perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our work argues that the compositional use of generative models provides the best way of understanding how these factors influence perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Bottom-up models based on DCNNs performed poorly both in the object- under-cloth task and in mimicking human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A DCNN that has been fine-tuned on thousands of images of cloth-occluded objects produces behavior with roughly similar average accuracy as humans in our briefest presentation conditions (1 sec), but unlike the PbAS model fails to explain how performance improves with time and does not correlate at all with trial-by-trial accuracy in the most challenging conditions (occluded with cloth and same-category distractors, for all presentation conditions tested).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' DCNNs, as a model class, should in principle be able to learn any mapping from inputs to outputs, but our fine-tuning results show that in practice, the data requirements can be substantial (and likely exceed human experience) and the best results far from human-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Given the broader context of the many atypical, challenging Springer Nature 2021 LATEX template 20 Shape perception integrates intuitive physics and analysis-by-synthesis viewing conditions that the visual system may encounter, these findings under- score the importance of generalization and robustness, ongoing challenges for DCNNs, and illustrate how top-down knowledge can enable perception in dif- ficult novel contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Bottom-up models do, however, play an important role in our framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' relative to the ablated Pixel-PbAS model, the hybrid archi- tecture implemented in PbAS demonstrates that powerful feature hierarchies can usefully facilitate or guide inference in generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This perspec- tive is compatible with much research on “core” object recognition showing the explanatory power of bottom-up models (DiCarlo, Zoccolan, & Rust, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Yamins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Future work should also evaluate continuing developments in DCNNs, trained using alternative loss functions, architectures or datasets (Geirhos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Konkle & Alvarez, 2022), which may show improved generalization to difficult perceptual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The PbAS model suggests that perception of cloth-covered objects in the brain relies on a combination of feedforward, feedback, and recurrent compu- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We believe that this is valuable as, relative to the case of feedforward processing, there is little evidence to constrain or generate hypotheses regard- ing the role of feedback and recurrent computation in visual scene analysis (Gilbert, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' PbAS suggests a new computational goal for feedback and recurrence in the brain, which is in some ways related to pattern theory as expressed in Mumford (1994): Such processing might implement the progres- sive unfolding of one or a number of physical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' It is likely that these forms of neural computation implement multiple computational goals needed for such diverse functions as attention, learning, and perception (Gilbert, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The hypothesis suggested by PbAS – internal simulations of physical processes – is not exclusive of the others and future work should explore their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The present implementation of the PbAS model accounts for behavior in the specific matching task we studied here (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Future work should exploit its modular architecture to address other experimental paradigms and perceptual problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For example, from an image of a single draped object (without a comparison unoccluded object), humans can often infer its category, approximate pose, and partial shape (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1C-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' While evaluating PbAS in this more difficult scenario is beyond the scope of the present paper, the framework readily extends to this setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig 7 for a demonstration from a proof-of-concept implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Our results suggest that shape perception under cloth draping involves mental operations beyond the rapid, bottom-up processing believed character- istic of traditional object recognition (Grill-Spector & Kanwisher, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To what extent are the computations hypothesized by PbAS – 3D shape infer- ence, mental rotation, or mental simulations of physical and image formation processes – also engaged in rapid, automatic visual processing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' And how do they relate to other cognitive mechanisms supporting dynamic processing such as visual routines (Ullman, 1987) and mental imagery (Shepard & Metzler, 1971)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 7 Seeing the shape of a single cloth-draped object, without the aid of unoccluded candidates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' By expanding its shape hypothesis space to contain a large set of category-specific objects (as opposed to the four nearest neighbors of the available context object as in our main model) and removing the unoccluded inference module, PbAS can obtain plausible estimates of 3D pose and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Rows show (from left) input image containing target cloth-covered object and four inferred shape/pose hypotheses under this modified PbAS model, ordered from high to low posterior probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Recent psychophysical work suggests that these computations might be implemented in the visual system as part of spontaneous processing of sen- sory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2022) studied cloth-covered object perception across a battery of visual tasks, finding evidence that scenes are rapidly and auto- matically parsed as the appropriate physical causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In addition to behavioral probes, cognitive neuroscience can address where in the brain the computa- tions specified by the PbAS model might be implemented (see Shams and Beierholm (2022) for a recent review of Bayesian causal models and inference in the brain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For example, fMRI studies (Fischer, Mikhael, Tenenbaum, & Kanwisher, 2016) have identified brain regions supporting intuitive physical judgments in a dorsal frontoparietal network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' it is of significant interest to Springer Nature 2021 LATEX template 22 Shape perception integrates intuitive physics and analysis-by-synthesis answer whether the same or similar brain regions are also recruited during the perception of cloth-draped objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The PbAS framework discussed and supported here may also play a broader role in visual processing beyond our cloth-draped object setting, unifying competencies beyond traditional shape and object perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A common computational engine may therefore support perception of the dynamical prop- erties of objects, such as the relative masses of colliding rigid bodies or single objects reacting to the application of external forces (Sanborn, Mansinghka, & Griffiths, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Schwettmann, Tenenbaum, & Kanwisher, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Wu, Yildirim, Lim, Freeman, & Tenenbaum, 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the stiffness of deformable objects under- going natural transformations (Bi, Shah, Wong, Scholl, & Yildirim, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Paulun & Fleming, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Paulun, Schmidt, van Assen, & Fleming, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' vis- cosity and flow of liquids (Bates, Yildirim, Tenenbaum, & Battaglia, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Kubricht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Van Assen, Barla, & Fleming, 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and in general the perception of the physical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', non-intentional) causal history of an object (Chen & Scholl, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fleming & Schmidt, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Schmidt, Phillips, & Flem- ing, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In each of these cases, it is at least plausible that the brain uses generative models to simulate the physical processes that could have produced the observed scene, and compare the results of these simulations to the sensory input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A better understanding of how the brain supports these abilities could also lead to more robust, and more human-like, machine vision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Materials and Methods Generative model Cloth Simulations We used the FLeX engine, a particle-based physics engine, for cloth physics simulation (Macklin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Simulation parameters as well as the mechanical-material properties of the cloth were chosen so as to achieve fast, stable simulation of natural-looking, cotton-like cloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Simulation parameters were as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Iterations: 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' subiterations: 19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' particle radius: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0078;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' collision distance: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0078;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' shape collision margin: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='00078;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' particle collision margin: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' relaxation mode: default;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' relaxation factor: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' drag: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' damping: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' dis- sipation: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' restitution: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The mechanical-material properties of the cloth were as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Strength stiffness: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' bend stiffness: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' shear stiffness: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' particle mass: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' static friction: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' dynamic friction: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To increase simulation efficiency, we simplified the geometry of the ShapeNet meshes using Blender (Blender Online Community, 2015)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' First, we corrected the surface normals on each mesh by ensuring that they were consistent and pointed outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Second, we used Blender’s “Solidify” mesh modifier with the thickness parameter set to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Finally, we merged faces that were adjacent and approximately coplanar (with surface normals differing by less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='02 rad ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='15°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 23 We initialized simulations by placing a square cloth (represented compu- tationally with 210 × 210 particles) just above the geometric center of the rotated object to be draped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We then ran the simulation for 150 steps, suffi- cient to fully drape all objects we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each cloth simulation took between 3 and 40 seconds on a NVIDIA 2080TI GPU, on the order of 1000 times faster than alternative implementations using CPU-based cloth simulation and unsimplified meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Image Rendering The scene was lit to minimize shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We placed 14 point lights with energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 on a sphere with radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='22 (object radius normalized to 1), with lights distributed approximately equidistant using the Fibonacci sphere algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We rendered these scenes to 224×224 images using Blender’s internal renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To equate the texture appearance of the draped and unoccluded images, we replaced the optical materials associated with the original ShapeNet meshes with a diffuse material (diffuse color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='75 in each RGB channel, diffuse intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='75, and specular intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We used a very similar material to render draped cloths (diffuse color 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 in each RGB channel, diffuse intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8, and specular intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We reasoned that equating the texture appearance in this way would aid the bottom-up neural network models in emphasizing shape over texture (Geirhos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The experimental stimuli underlying the object-under-cloth task are pub- licly available: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='com/CNCLgithub/intuitive-physics-3d-shape perception-stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Approximating Shape Distance Given two shapes from ShapeNet Si and Sj, we define a shape distance met- ric by (1) rendering each object in a standard canonical pose, (2) passing each image through a pretrained AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012) and extracting feature activations at the first fully-connected layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', applying fenc as for pseudo-likelihood evaluation during inference), and (3) evaluating the ℓ1 dis- tance between the feature activations for each shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The resulting measure is similar to that used when calculating the pseudo-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Shape Prior Given an unoccluded context object s0, we modeled the observer’s shape uncertainty Pru(S) as a categorical distribution over the K = 4 shapes near- est to s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Let dsk be the weight of the kth closest shape sk to s0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' then Pru(S = sk) ∝ exp(−dsk) with 1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The Shapenet database forms a sparse approximation to the space of all object shapes, and we found that the distance between an object and its closest neighbors could vary wildly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' one reason is that some object classes have many more exemplars than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Therefore, a prior defined solely using shape distance showed high variance across trials and was unsuitable for our purposes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', it induces arbitrary Springer Nature 2021 LATEX template 24 Shape perception integrates intuitive physics and analysis-by-synthesis bias towards either the distracting or matching object from trial to trial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The unnormalized weights for each nearest shape were instead assigned based on the rank order of their distance to the context object, starting at ds1 = 750 and increasing at increments of 75 so that dsk = 750 + (k − 1)75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The scale of these weights was chosen so that the relative contributions of the prior and likelihood were comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Inference Using Bayesian Optimization In comparison with traditional inference schemes based on random-walk MCMC (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2020), Bayesian optimization provides a more guided or “active” approach to inference, where the next scene hypothesis to evaluate the posterior on is informed by all of the previous evaluations of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In adopting Bayesian optimization, we forego full posterior estimation (which MCMC can provide in principle) in favor of a good point maximum a posteriori (MAP) estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This choice is further motivated by the computational cost of cloth simulation, which is responsible for nearly all of the work our model must do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' BayesOpt requires many fewer iterations, and therefore cloth sim- ulations, than random-walk MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' It trades expensive overhead (compared to other methods) in choosing search candidates for greater search efficiency (Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Following Kandasamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2015), we sought to learn a function from latent scene variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', shape and rotation) to their (unnormalized) log posterior scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' By specifying a tractable Gaussian Process (GP) prior over functions and conditioning on all available data, BayesOpt yields an online strategy for adaptively choosing parameter settings to evaluate and prescribes how the results update the GP posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The uncertainty in the GP approxi- mation of the log posterior score decreases as the number of inference iterations increases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', as more evaluations of the posterior are observed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This prob- abilistic approximation is computationally cheap to evaluate and has support over the entire range of scene hypotheses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', can be evaluated for any scene hypothesis including those that are previously not evaluated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' BayesOpt requires specification of the GP kernel, which encodes prior assumptions about (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g) the smoothness of functions (Rasmussen & Williams, 2006), and an acquisition function which selects the next hypothesis given the results of all previous evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In our work, we used a Mat´ern kernel with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 (the Mat´ern 3/2 kernel) and Automatic Relevance Deter- mination (Rasmussen & Williams, 2006) to learn a probabilistic mapping from latent scene hypothesis onto posterior scores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and we use the Expected Improvement (EI) as our acquisition function, which favors scene hypothe- ses that are expected to most improve the posterior score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each iteration of BayesOpt consists of (i) updating the estimated regression function (from scene hypotheses to posterior scores) and (ii) optimizing the acquisition func- tion to determine which scene hypothesis to evaluate in the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We next describe each of these two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 25 Scene hypotheses are coded specially for BayesOpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We represent rotations using normalized Euler angles R = {Rx, Ry, Rz}, with each orthogonal axis taking values in [0, 1] and together spanning the half-sphere of rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The shape variable is discrete, which we transform to a continuous encoding using vector quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We map the interval [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='25) to shape hypothesis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', nearest neighbor) 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='50) maps to the next nearest neighbor, shape 2, and so on up to shape 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The GP therefore learns a regression function from a 4-dimensional input (three numbers for rotation, one for shape) to a scalar, the log posterior score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' With this GP approximation at hand, we define an “acquisition function” which uses the current GP state to select the most promising scene hypothe- sis to try in the next iteration j + 1, (S(j+1), R(j+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Various active sampling (or learning) heuristics are proposed in the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' J¨arvenp¨a¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Kandasamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We adopt the EI acqui- sition function (Snoek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012), which chooses the scene hypothesis that is expected to most improve the current posterior score, given all of the previous posterior evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' At each iteration j of our model, the inference procedure evaluates a scene hypothesis chosen to optimize the EI acquisition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' EI uses a parameter, denoted ϵ (set to 330 in our simulations), to trade-off between how much to weigh the predicted posterior score vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' the uncertainty around that prediction (notice that the GP-based probabilistic regression provides both the predicted mean posterior score and variance around that prediction for the entire range of scene hypotheses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To find the scene hypothesis that optimizes EI, we generate 100, 000 random scene hypotheses and use the high- est scoring to initialize further local search (using L-BFGS-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This procedure yields the scene hypothesis (Sj, Rj) to be evaluated in the next iteration of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We evaluate the posterior at this scene hypothesis using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We implemented our inference scheme using the BayesOpt (Nogueira, 2014) and GPy (GPy, 2012) packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Code implementing the PbAS model (as well as our behavioral data and analysis) will be made publicly available before publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Bottom-Up Models We tested three DCNN architectures: AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2012), ResNet50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2016), and VGG16 (Simonyan & Zisserman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These models provide powerful feature hierarchies that are learned as a result of training to classify images from the large-scale real-world ImageNet (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 2009) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Imagesets for Fine-tuning Imagesets for fine-tuning were derived from the 5 shapes/category × 10 cat- egories = 50 object shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' These are the identical set of objects as those underlying the experimental training trials used to familiarize human partici- pants with the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (We note that in our behavioral experiments, we did not Springer Nature 2021 LATEX template 26 Shape perception integrates intuitive physics and analysis-by-synthesis provide feedback during the training phase and indeed did not find any evi- dence of learning in our behavioral data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') We used 8 imagesets per occlusion condition, with 500 unique trials in each set giving 500 × 8 = 4000 image triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (We evaluated how the amount of data used for fine-tuning influenced performance, finding that performance plateaued at 8 imagesets, compared with alternative groups of 1, 2, 8, 18, 28, and 38 imagesets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For each triplet, we sampled two objects and randomly rotated, draped, and ren- dered them using our stimulus generation pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We reserved 2 imagesets for test and the remaining were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' To minimize bias, a set of 8 imagesets was sampled from a larger pool of 54 at the beginning of each fine-tuning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We fine-tuned each model 32 times for each occlusion condition and report accuracy averaged over the condition-specific replicas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Modifying Network Architectures for Fine-tuning To fine-tune AlexNet and VGGG16, we removed their top classification layer and replaced it with a linear fully connected layer of size 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We trained the linear layer from scratch and fine-tuned the weights of the layer preceding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Unlike AlexNet and VGG16, the ResNet-50 model does not contain multi- ple final fully-connected layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' thus, we used a modified approach to fine-tune it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='We replaced both its top classification layer as well as the preceding Aver- age Pooling layer with a convolutional layer with kernel size 2, stride 2, and dilation 2, without zero-padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This convolutional layer takes as input 2048 feature maps (the number of output feature maps in the fourth Residual Block of the ResNet-50 model) each with dimensionality 7 × 7 and outputs 300 fea- ture maps (each with dimensionality 3 × 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The ReLU activation function is applied to the flattened outputs of this convolutional layer, which is followed by a single linear fully-connected layer of size 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We trained the weights of the new convolutional layer as well as the fully-connected layer from scratch, while keeping all other weights in the network unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Details of the Training Procedure To adapt the networks to our visual matching task, we used metric learning with a triplet margin loss (Schultz & Joachims, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The goal is to adapt the network’s representational space so that distance in that space reflects the sim- ilarity structure of our stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Concretely, the distance between an “anchor” image and a “positive example” should be smaller than the distance between the anchor and “negative example”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A training triplet has the same structure as our behavioral match-to-sample task setup: anchor corresponds to the tar- get item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' positive example corresponds to the ground-truth matching test item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and the negative example corresponds to the distractor test item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (Remember that the training datasets are crafted differently for each occlusion condition and different networks are trained for each of these occlusion conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') We fine-tuned each architecture for a total of 200 epochs and used a held-out test set to make sure the models did not overfit over the course of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 27 We set batch size to 8 and set the triplet loss margin to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We used the ADAM optimizer (Kingma & Ba, 2015) with ASMGrad (Reddi, Kale, & Kumar, 2018) using the following optimization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We set β1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='9, β2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='999, learning rate to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 × 10−6 and ℓ2 weight decay to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 × 10−3 at the beginning of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In an attempt to optimize the performance of the bottom-up networks, we explored a range of custom learning rate schedules as well as regularization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' During training, we scheduled the learning rate as the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The learning rate is multiplied by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 from epoch 1 to epoch 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' From epoch 13 to epoch 161, the learning rate is annealed by multiplying it with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='985, after which it was kept constant until epoch 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In addition, to avoid overfitting, we employ regularization using a weight decay strategy and data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' From epoch 13 to 161, we multiply the weight decay parameter by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='04 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='03 in fine-tuning the occluded and unoccluded task conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (We observed that without this scheduled weight decay, models essentially memorized the training image set, giving rise to a substantial discrepancy between training and test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') As a form of data augmentation, during training, we randomly perturb each image by adding white noise (variance set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3×10−3) with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 (the added noise was restricted to the foreground pixels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' All pixel values were truncated to ensure that their values lie between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Evaluation of the bottom-up models on the object-under-cloth task The accuracy of the pretrained bottom-up model on a given trial was calculated using the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Recall that each trial in the object-under-cloth task consists of three images: the target item, the matching test item, and the distractor test item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We compute a feature embedding of each of these three images from the first fully-connected layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We define a correct answer (accuracy=1 for this trial) from the network if the correlation between the embeddings of the target item and the matching test item (denoted corrm) is greater than the correlation between the embeddings of the target item and the distractor test item (denoted corrd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Otherwise, the network got the trial wrong (accuracy=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The accuracy levels of the pretrained bottom-up model underlying Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4D, F are calculated in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5 where we require a continuous covariate per trial from each model (as opposed to a binary accuracy label), we use Luce’s choice rule (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', soft- max) to transform the above mentioned correlation values to a continuous score: corrm/(corrm +corrd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Notice that the model predictions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 5B are normalized to the range of [0, 1] for all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The trial-level accuracy of the fine-tuned model is calculated in a manner similar to the PbAS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' For a given trial and a fine-tuned network, we select the test item that is closer to the target item as the network’s guess and report the fraction of correct guesses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', the closest test item was the matching test item) across the ensemble of 32 independently fine-tuned networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 28 Shape perception integrates intuitive physics and analysis-by-synthesis Behavioral Methods Participants A total of 173 participants were recruited from Amazon’s crowdsourcing plat- form Mechanical Turk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The experiment took about 20 minutes to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Each participant was paid $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A total of 12 subjects were excluded due to performing at or below chance performance (1 in Unoccluded-1sec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 4 in Occluded-1sec;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 3 in Occluded-2secs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' and 4 in Occluded-Unlimited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Approval for our behavioral study was obtained from the Massachusetts Institute of Technology Institutional Review Board (the Committee on the Use of Humans as Experimental Subjects), and we obtained each participant’s informed consent prior to any experimental session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Stimuli and Procedure We used 240 unique ShapeNet meshes from 10 object categories to create the 120 match-to-sample shape pairs in our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We selected 24 objects from each category and allocated them evenly between the same-category (target and distractor from same object category) and different-category conditions, pairing each shape with another from the same category or a different cate- gory as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Pairings were sampled randomly without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We thus obtained 6 same-category and 6 different-category pairs for each object category, with no duplicate shapes across trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We designed a visual matching experiment based on the object-under-cloth task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The experiment assigned participants to either the occluded or unoc- cluded conditions as well as one of three conditions varying presentation time lengths, for a between-subject design with 2 occlusion × 3 presentation time = 6 conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the 1- and 2-second conditions, the target and test items were displayed for the indicated period of time and the unlimited time condi- tion let participants view the items for as long as they wished, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' until their response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Images appeared and disappeared simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The spatial organization of the display differed slightly by occlusion con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the unoccluded condition, the two test images were placed side by side, below the target item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' for the occluded condition, the test images were placed side by side but above the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Participants completed 10 practice trials before moving on to the 120 exper- imental trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Participants were provided with running feedback, seeing their average task performance at every 5th trial throughout the experiment except during the practice block (the performance feedback calculation excluded practice trial accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Split-Half Correlations To estimate the data noise ceiling, we used bootstrapped split-half correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We sampled 1000 random splits of our participants in each occlusion condition Springer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 29 (only considering the unlimited presentation time condition), each split divid- ing the participants into two groups of equal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (Participants were sampled without replacement for each partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') For a single division (one random split) of participants, we computed the average accuracy of each split-half on each trial, then correlated the group accuracies across all trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (In essence, we used the responses of one split of participants to model the responses of the other half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=') We did the same for each of the 1000 random splits, yielding 1000 bootstrap estimates of the behavioral noise ceiling and allowing us to assess their average value and spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' But because this procedure effectively halved our participant number, our split-half correlations likely underestimate the true noise ceiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' References Bates, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', & Kersten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Vision as bayesian inference: analysis by synthesis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Trends in cognitive sciences, 10(7), 301–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Acknowledgements: This work was supported by the Center for Brains, Minds and Machines (CBMM), funded by NSF STC award CCF-1231216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' ONR MURI N00014-13-1-0333 (to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' a grant from Toyota Research Institute (to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We thank Kevin Smith, Bernhard Egger, Kelsey Allen, Goker Erdogan, Marty Tenenbaum, Nancy Kanwisher, and Vivian Paulun for their comments on a previous version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Supplementary Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S1 Accuracy levels of the three models we considered including pretrained versions (top row) and after finetuning (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' AlexNet results are presented in the main text.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 oUSpringer Nature 2021 LATEX template 36 Shape perception integrates intuitive physics and analysis-by-synthesis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S2 Behavioral results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (A) Average human accuracy in the 3 presentation time con- ditions pooling data across the occlusion conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Overall, participants performed well above chance under all presentation time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Behavioral accuracy improved with longer presentation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (B) Average human accuracy shown separately for each occlusion and presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Participants’ average performance ranged from 73% in the cloth-occluded condition under 1 sec presentation time to 93% in the unoccluded condition under unlimited time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The gain in performance was significant within each occlusion condi- tion, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05 for all pairwise comparisons of presentation times, except in the 1 sec vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2 secs comparison in the cloth-occluded condition, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (C) Average response times (in millisec- onds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' pooling data across the occlusion conditions) lengthen with longer presentation times, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='001 for all pairwise comparisons of presentation time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' (D) Average response times shown separately for each occlusion and presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Lengthening of response times is still evident for each occlusion condition (p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='05 for all pairwise com- parisons of presentation times, except in the 1 sec vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 2 secs comparison in the unoccluded condition and 1 secs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' comparisons in the cloth-occluded condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Error bars show standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S3 Trial-level accuracy correlations in the (A) 2 secs presentation time condition and (B) 1 sec presentation time conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The physics-based analysis-by-synthesis (PbAS) model correlates well with behavior across all presentation time and occlusion condition time conditions, relative to the alternatives based on bottom-up features optimized for image classification (BU: bottom-up network with pretrained weights from ImageNet dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' FT: fine-tuned networks, separately fine-tuned for each occlusion conditions) and Pixel-PbAS, an ablation of PbAS without the bottom-up image encoding modules (using pixels directly for likelihood computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Significance convention same as main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Error bars indicate bootstrapped 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A 2 secs Unoccluded, Occluded, Unoccluded, Occluded, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0- all trials all trials 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 all trials all trials *** *** *** *** Correlation to behavior *** n.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 1000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 efiSpringer Nature 2021 LATEX template Shape perception integrates intuitive physics and analysis-by-synthesis 37 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S4 Trial-level accuracy correlations for the difficult, same category trials in the (A) Occluded and (B) Unoccluded conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Results are arranged by model type and stimu- lus presentation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Error bars show bootstrapped 95% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the easier unoccluded, shape-category conditions, all three models that use DCNN features to match images (PbAS, BU, and FT) perform similarly across all presentation times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Pixel-PbAS performs significantly worse across all presentation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' In the more difficult occluded, same-category conditions, PbAS clearly outperforms all other models, except for BU which performs similarly in the shortest (1 sec) presentation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Notably both pure DCNN models, BU and FT, consistently correlate less well with human trial-level accuracies as pre- sentation times increase, while PbAS correlations tend to increase, and FT correlations are not significantly different from zero in the challenging occluded same-category conditions (with BU correlations being only barely higher than zero in the 2 sec and unlimited condi- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This overall pattern is consistent with the success of DCNNs at capturing the rapid feedforward contributions to human object recognition for familiar stimuli viewed under stan- dard conditions, and strengthens our proposal that more challenging viewing conditions and longer processing times engage top-down, iterative, generative model based computations of the form instantiated in PbAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' The combination of physics-based analysis by synthesis with DCNN features for matching generative model simulations to images, as instantiated in the full PbAS model but not Pixel-PbAS, is the only model that accounts well (and better than or equal to any other model) for all stimulus conditions and all presentation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' A Occluded, same category trials PbAS Bottom-up Finetuned Pixel-PbAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 secUnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Duration B Unoccluded, same category trials PbAS Bottom-up Finetuned Pixel-PbAS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 Correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 1 sec 2 secUnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1 sec 2 sec Unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' DurationSpringer Nature 2021 LATEX template 38 Shape perception integrates intuitive physics and analysis-by-synthesis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S5 Behavioral learning curves in the two occlusion conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We show moving win- dow averages (window size=10) of human accuracy levels in the two occlusion conditions (red=UU, green=OU) under the unlimited presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' We find no evidence of learning throughout the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Shaded region shows standard error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' S6 Divergence between the Pixel-PbAS model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=', using pixels for likelihood with- out bottom-up image encoding) and human performance at each model iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Colored lines show ℓ2 distance between this model and human accuracy for all trials in indicated presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' Colored triangles indicate the best matching iterations for each presentation time condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' This model asymptotes at a larger distance to behavior than the PbAS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 Behavioral 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1 10 20 30 40 50 60 70 80 90 100 Time window (bin size = 10 trials)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='6 Behavior Distance 1 sec 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='5 Pixel-PbAS to 2 secs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='4 Unlimited 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} +page_content='0 1 20 40 60 80100120140160180 # of iterations in Pixel-PbAS' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E2T4oBgHgl3EQfLQZF/content/2301.03711v1.pdf'} diff --git a/s9E1T4oBgHgl3EQfQQNJ/content/2301.03037v1.pdf b/s9E1T4oBgHgl3EQfQQNJ/content/2301.03037v1.pdf new file mode 100644 index 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b/s9E1T4oBgHgl3EQfQQNJ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a99f718bcebee7d3d593f7476cddf885e7dd8b38fcf987abcac5b2507afda410 +size 131892 diff --git a/stE2T4oBgHgl3EQffQeV/content/tmp_files/2301.03925v1.pdf.txt b/stE2T4oBgHgl3EQffQeV/content/tmp_files/2301.03925v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d7e6897344d31d703d5450224ea8037408df47e --- /dev/null +++ b/stE2T4oBgHgl3EQffQeV/content/tmp_files/2301.03925v1.pdf.txt @@ -0,0 +1,633 @@ +arXiv:2301.03925v1 [cond-mat.stat-mech] 10 Jan 2023 +Another normality is possible. +Distributive transformations and emergent Gaussianity +Massimiliano Giona, Chiara Pezzotti and Giuseppe Procopio∗ +Dipartimento di Ingegneria Chimica, Materiali, +Ambiente La Sapienza Universit`a di Roma +Via Eudossiana 18, 00184 Roma, Italy +( Dated: January 11, 2023) +A distributional route to Gaussianity, associated with the concept of Conservative Mixing Trans- +formations in ensembles of random vector-valued variables, is proposed. This route is completely +different from the additive mechanism characterizing the application of Central Limit Theorem, as +it is based on the iteration of a random transformation preserving the ensemble variance. Gaussian- +ity emerges as a “supergeneric” property of ensemble statistics, in the case the energy constraint +is quadratic in the norm of the variables. This result puts in a different light the occurrence of +equilibrium Gaussian distributions in kinetic variables (velocity, momentum), as it shows mathe- +matically that, in the absence of any other dynamic mechanisms, almost Gaussian distributions +stems from the low-velocity approximations of the physical conservation principles. Whenever, the +energy constraint is not expressed in terms of quadratic functions (as in the relativistic case), the +J¨uttner distribution is recovered from CMT. +The concept of normal distribution is so central in +probability theory and physics that, as observed by Mark +Kac, partially in jest (quoting Henry Poincar´e), it is not +clear in the scientific community whether it comes from +a mathematical property or it is a law of nature [1]. As +well known, the main route to normal distributions stems +from the Central Limit Theorem (CLT) [2, 3]. There is +an entire galaxy of different versions of CLT, either re- +laxing the assumptions and the convergence properties, +or modifying the way the summation is performed [4, 5]. +Similarly, there are parallel extensions of CLT, initiated +by the work of C. Stein, applied and generalized in several +directions, e.g. for sums of dependent random variables +[7]. +Besides these mathematical subtleties and chiselings, +of utmost value in probability theory and statistics, the +core of CLT - finding a widespread application in sta- +tistical physics for interpreting the phenomenological oc- +currence of normal distributions [8] - lies in its simplest +version: +given a sequence of independent and identi- +cally distributed (iid) random variables {Xn}∞ +n=1, pos- +sessing zero mean and bounded variance σ, the random +variable YN = �N +n=1 Xn/σY,N, corresponding to their +sum normalized to unit variance, (as, by independence, +σ2 +YN = N σ2), converges weakly for N → ∞ to the +normal distribution, so that its limit probability density +function (pdf) is given by pY (y) = e−y2/2/ +√ +2 π, (hence- +forth the normal distribution is indicated with the symbol +N(0, 1).). It is important to observe that the weak con- +vergence, i.e., the convergence in distribution, does not +prevent the potential occurrence of anomalies as regards +the higher order moments m(n) +YN = ⟨Y n +N⟩, for n > 2 [9]. +CLT explains in a simple and elegant way the phe- +nomenon of molecular diffusion addressed by A. Ein- +stein in relation to the phenomenon of Brownian motion +[10]: the molecular motion, or the kinematics of micro- +metric particles driven by thermal fluctuations, can be +interpreted as the resultant of the superposition of in- +dependent (uncorrelated) displacements that, provided +the boundedness of their variances, unavoidably leads by +mathematical principles (i.e,. by enforcing CLT) to the +occurrence in the long-time limit of the parabolic diffu- +sion equation. Thus invoking CLT, the diffusion equa- +tion follows, just by randomness, independence and the +smallness of the perturbations. By the internal coherence +of CLT, this interpretation provides also the criteria for +assessing the failure of the parabolic approximation: (i) +lack of independence, leading to subdiffusion, as in the +case of the random motion in disordered systems and +fractal media [11, 12], or (ii) unbounded lower-order mo- +ments, associated with superdiffusive phenomena such +as L´evy walks [13, 14], that are strongly related to the +extended version of CLT (usually referred to as the gen- +eralized CLT) and to the L´evy theory of α-stable distri- +butions [15, 16]. +In its basic “machinery”, CLT can be viewed as a +mathematical superpositional route to N(0, 1), in which +normality is achieved by addition of independent con- +tributions, properly rescaled. In this Letter we want to +present another, completely different, mechanism lead- +ing to normality, in which the Gaussian distribution fol- +lows as a consequence of a distributional dynamics in- +volving the binary interaction of iid random variables +subjected to some conservation principles and possess- +ing an arbitary initial distribution. In the present case, +the conservation principles are the fundamental conser- +vation principles of physics, that apply both in the clas- +sical and in the quantum world, namely the conservation +of momentum and energy, both for matter-matter and +matter-radiation interactions. This finds in the relativis- +tic space-time formalism its maximum compactness and +elegance, as it implies, for any physical interactions, the + +2 +overall 4-momentum pµ, µ = 1, .., 4, pµ = +� +p, E +c +� +is con- +served, where p = (p1, p2, p3) is the momentum, E the +energy and c the speed of light in vacuo [17] +Specifically, we introduce and analyze the distribu- +tional route to normality associated with the concept of +Conservative Mixing Transformation (CMT) of random +ensembles, for which the occurrence of limit Gaussian +distributions is entirely entitled to the energy represen- +tation, as discussed in the analysis of low-velocity vs rel- +ativistic constraints. +CMT’s originate from the stochastic generalization of +the Boltzmannian description of binary collisions in di- +luted gases [18, 19]. Their introduction does not simply +represent a stochastic transposition of the Boltzmannian +kinetic theory. From the statistical analysis of CMT it is +possible to derive the extreme genericity (as explained in +the remainder) of the occurrence of “almost” Gaussian +velocity distributions at equilibrium, and their connec- +tion with the nature and representation of fundamental +conservation principles. +Moreover, they provide a for- +mal stochastic setting amenable to be extended to any +physical impulsive interaction mechanism, and involving +arbitrary conservation principles. In the present Letter, +CMT is analyzed beyond the classical low-velocity condi- +tions in the relativistic case. Although the Boltzmannian +kinetic theory leads to Gaussian distributions at equilib- +rium, no one, to the best of our knowledge, has extended +and generalized the mechanism of binary interactions in +the form of a universal stochastic route to Gaussianity, +which is the scope of CMT’s and of the present Letter. +The structure of this Letter is organized as follows. We +formalize the concept of CMT, the nature of the physical +constraints, and the generic emergence of almost Gaus- +sian distributions. Subsequently, we address how Gaus- +sianity can be broken, by the assumption of an energy +conservation law different from a quadratic one. +Consider an ensemble of N random vector-valued vari- +ables E = {zh}N +h=1, with zh = (zh,1, . . . , zh,d) ∈ Rd, +d = 1, 2, . . ., and let Σd,N the space of all the N- +dimensional ensembles of d-dimensional random variables +over the field R. +Each zh can be referred to as the +state vector of an element of the ensemble. A Mixing +Transformation M : Σd,N → Σd,N (acronym MT) is +a transformation of the ensemble E, into an ensemble +E′ = M(E) = {z′ +h}N +h=1, defined by a binary operation +amongst randomly selected elements of the ensemble. It +can be defined in the following way: +1. let φ(z1, z2; r), ψ(z1, z2; r) : Rd × Rd × ∂Sd → Rd +be two random functions, depending on a random +vector r, |r| = 1 defined on the surface of the d- +dimensional unit sphere ∂Sd, by the pdf g(r); +2. select randomly two elements α, β ∈ (1, . . . , N) +with β ̸= α; +3. the transformed ensemble {z′ +h}N +h=1 is given by +z′ +α = φ(zα, zβ; r) +z′ +β = ψ(zα, zβ; r) +(1) +z′ +h = zh , +for h ̸= α, β +Whenever it is not conceptually necessary, the explicit +dependence of φ and ψ on r will be omitted. +The concept of MT’s so defined is too general for physi- +cal applications, and constraints on the random functions +φ, ψ, should be introduced. +A Conservative Mixing Transformation is a MT, for +which Nc functions fh(z) : Rd → R are defined, such +that the transformations φ(z1, z2) and ψ(z1, z2) satisfy +the constraints +fh(z′ +1) + fh(z′ +2) = fh(z1) + fh(z2) , +h = 1, .., Nc (2) +where z′ +1 = φ(z1, z2), z′ +2 = ψ(z1, z2), for any z1, z2 ∈ Rd, +and r ∈ ∂Sd. +The CMT’s of physical relevance are those satisfying +the Nc = d + 1 conservation laws +fh(z) = zh , +h = 1, . . . , d +(3) +fd+1(z) = e(|z|) +(4) +where e(|z|) is a non negative function solely of the norm +|z| of z, representing, modulo a multiplicative factor, the +energy function (kinetic energy) of the element. Eqs. (3) +and (4) correspond mathematically to the consevation +of momentum and energy. Several observation, follows +from this setting. (i) The concept of random ensembles +involves a finite number N of elements, and should not be +confused with Gibbsian ensembles or with other collec- +tive groupings (such as in the replica method) [20, 21]. +A “random ensemble” in the CMT-theory means sim- +ply a system of random variables, in the sense that (i) +their initial conditions are randomly chosen for each el- +ement, and that their evolution is subjected to random +laws. Correspondingly, the ensemble average of any func- +tion q(z) of the state vector z, is simply expressed by +⟨q(z)⟩ = +1 +N +�N +h=1 g(zh), and consequently a single pdf +pz(z) is associated with the ensemble. (ii) The definition +of MT proposed above is “event-based”, in the mean- +ing that each transformation M involves solely a single +binary event modifing the statistical properties of the en- +semble. If E0 = {z(0) +h }N +h=1 is the initial ensemble, assume +⟨z(0)⟩ = 0, and ⟨e(|z(0))|⟩ = E0 < ∞. +Let (φ−1, ψ−1) be the inverse transformation of (φ, ψ), +i.e., φ−1(φ, ψ) = ψ−1(φ, ψ) = Id., and set +Jφ,ψ(z′ +1, z′ +2)| = +���� +∂ (φ(z1, z2), ψ(z1, z2)) +∂(z1, z2) +���� z1 = φ−1(z′ +1, z′ +2) +z2 = ψ−1(z′ +1, z′ +2) +for the Jacobian determinant of the transformation. The +statistical evolution of a CMT can be described as fol- +lows. If pdf pz(z) is the ensemble pdf for E and E′ = + +3 +M(E), the pdf p′ +z(z′) of E′, averaged over the probability +measure of r, is expressed by +p′ +z(z′) = 1 +2 +� +Rd [π′(z′, z1) + π′(z1, z′)] dz1 +(5) +where +π′(z′ +1, z′ +2) = +� +∂Sn +pz(φ−1(z′ +1, z′ +2; r))pz(ψ−1(z′ +1, z′ +2; r))g(r)dr +|Jφ,ψ(z′ +1, z′ +2)| +(6) +The independence of z1 and z2, expressed by the factor- +ization of the two densities in the integrand at the r.h.s. +of eq. (6) stems from the random selection rule (point +2.) in the definition of a MT. Consider for the energy +function the expression, +e(|z|) = |z|2 +(7) +corresponding to the classical form for the kinetic energy +(for identical particles) In this case, the transformations +φ(z1, z2; r), and φ(z1, z2; r) can be chosen as +� +φ(z1, z2; r) = (1 − λ) z1 + λ z2 + αλ(z1, z2) r +ψ(z1, z2; r) = λ z1 + (1 − λ) z2 − αλ(z1, z2) r +(8) +where λ ∈ [0, 1] is a parameter, and αλ(z1, z2) is defined +to fulfil eqs. (2), (4). For λ = 0, αλ = −(z1 − z2) · r +(where “·” indicates the Euclidean scalar product), for +λ = 1/2, αλ = |z1 − z2|/2. Consider λ = 1, as the other +cases are equivalent. It is easy to check that |Jφ,ψ| = 1, so +that, if p∗ +z(z) is the equilibrium distribution, π′(z′ +1, z′ +2) = +p∗ +z(z′ +1) p∗ +z(z′ +2), and eq. (6) becomes +p∗ +z(z′ +1) p∗ +z(z′ +2) = +� +∂Sd +p∗ +z (z′ +1 − (z′ +1 − z′ +2) · r r) +× p∗ +z (z′ +1 + (z′ +1 − z′ +2) · r r) g(r) dr +(9) +Since, +|z′ +1 − (z′ +1 − z′ +2) · r r|2 + |z′ +1 − (z′ +1 − z′ +2) · r r|2 = |z′ +1|2 + |z′ +2|2 +the solution of eq. (9), is given by the Gaussian +p∗(z) = Ae−β |z|2 +(10) +where the parameter β depends on the initial ensem- +ble variance, and A is the normalization constant. +In +deriving eq. +(10) we have not made use of the prop- +erty of g(r), and therefore, eq. +(10) is valid for any +statistical structure of r, providing that it gives rise to +a steady and unique equilibrium distribution. +There +is only an exception to this property, represented by +non-dispersive random transformations, defined as fol- +lows. +The transformations φ and ψ of a CMT sa- +tisfying eqs. (2)-(3) are non-dispersive if for all the al- +lowable values of r φh(zα, zβ) is either zα,h, or zβ,h, +h = 1, . . . , d (and complementarily for ψ). This means +that no real mixing amongst the entries of the state vari- +ables occur but, componentwise, the two transformations +are either the identify or simply determine an exchange +of values amongst the elements. +An example of non- +dispersive CMT using eq. +(8) with λ = 0 occurs for +d = 2, where r = (cos ϕ, sin ϕ), ϕ ∈ [0, 2 π), in the +case g(r) dr = gϕ(ϕ) dϕ is any atomic distribution of the +form gϕ(ϕ) = (1 − µ) δ(ϕ) + µ δ(ϕ − π/2), µ ∈ (0, 1). +When ϕ = 0, z′ +α = (zβ,1, zα,2), z′ +β = (zα,1, zβ,2), while +for ϕ = π/2, z′ +α = (zα,1, zβ,2), z′ +β = (zβ,1, zα,2). +We can state the following result with reference to the +CMT transformations with energy constraint given by eq. +(8): for any initial distribution with E0 > 0, for any λ ∈ +[0, 1], and for almost all the probability measures of the +random vector r, the ensemble pdf of the CMT converges +in the limit for N → ∞ to a Gaussian distribution. +This stems for λ = 0 from eq. (9), and similarly for +any λ ∈ [0, 1] from the analogous property. The limit for +N → ∞ is here introduced in order to consider an infinite +ensemble, for which a smooth and continuous probabilis- +tic characterization (i.e., the existence of a smooth pdf) +can be applied. For any large but finite N, the result- +ing limit distribution is still accurately approximated by +a Gaussian distribution, apart for the asymptotic tails, +that necessarily vanishes to zero, as for any finite N, the +support of the limit pdf should be compact, simply be- +cause of energy conservation and of the initial assumption +of finite E0. +It is interesting to compare the classical CLT route to +Gaussianity with the emergence of it stemming from the +iterative application of CMT’s. In the CLT route, nor- +mality is an emergent property of the procedure of sum- +ming a large number of independent contributions. The +existence of a limit density stems from a renormaliza- +tion procedure, of rescaling the summation by removing +its mean and normalizing its variance. In this sense the +observation of Jona-Lasinio [22] on the strong analogy +between CLT and the Renormalization Group of quan- +tum field theory is acute and cogent. In the CMT route, +the process is purely distributional within a closed system +(the ensemble) of vector-valued random variables. Using +eq. (7), both the mean and the variance of the ensemble +distribution are conserved in the process. There is no +renormalization procedure in this approach. Moreover, +in finite ensembles the convergence towards the Gaus- +sian distribution is only approximate (albeit arbitrarily +accurate for large N). +This is an important, physical +property, as it resolves the unphysical tails for arbitrar- +ily large |z| in real finite systems. +Moreover, what is remarkable in the distributional +route to normality expressed by CMT is its generic- +ity. +While in CLT the emergent Gaussian behavior +holds for any distribution of iid random variables (with +bounded mean and variance), in the iteration of CMT +it emerges: (i) for generic initial ensemble distributions, +(ii) for generic transformations φ, ψ at least expressed + +4 + 1.8 + 2.2 + 2.6 + 3 + 0 + 5 + 10 + 15 + 20 +κ(t) +t +FIG. 1. Kurtosis κ(t) of the marginal ensemble distribution +(for the first entry z = zh,1 of zh) vs the normalized opera- +tional time t = n/N. The arrow indicates first the d = 2 case +(either with λ = 0 or λ = 1/2, that practically coincide), then +the d = 3 case at λ = 0, and finally the d = 3 case at λ = 1/2. +by eq. (7); (iii) for an almost generic statistical nature +of the random variable r, i.e. apart from the very pecu- +liar (and physically irrelevant) non-dispersive transfor- +mations. For this reason it is legitime to refer to this +qualitative behavior as the “supergenericity” of the CMT +distributional route to Gaussianity, in analogy with the +concept of “superuniversality” coined for phase transi- +tions (see [23] and references therein). +In a physical +perspective, “supergenericity” is the mathematical coun- +terpart of a thermodynamic principle at work in CMT, +determining a statistical emergent behavior that is com- +pletely independent either of the details on the initial +state or of the details on the transformations involved at +a microscopic level. +To make an example, figure 1 depicts the evolution +of the kurtosis κ(t) vs the operational time t = n/N, +(where n is the number of CMT operations, and N is +the ensemble size) for the first ensemble entry zh,1 at its +convergence to the Gaussian limit κ = 3 for d = 2, 3 +and for two different values of λ entering eq. (8). In this +case N = 106, and the entries of the initial ensamble are +uniformly distributed with zero mean and unit variance. +All the stochastic simulations refer to a random pertur- +bation r uniformly distributed in ∂Sd. The equilibrium +probability density for a larger ensemble N = 109, start- +ing from the same initial distribution is depicted in figure +1 at d = 3, for λ = 0, 1/2 . +The emergence of a limit Gaussian density in CMT +is physics-dependent. Assuming the constraints eq. (3), +this entirely depends on the form of the energy constaint +eq. (4). In the presence of energy functions e(|z|) differ- +ent from eq. (7) the stationary pdf for the entries of z is +different from the Gaussian. This can be illustrated by +means of a simple example of physical relevance. Con- +sider the relativistic extension [24–26], and consequently +the energy function e(|z|) given by +e(|z|) = +� +|z| c2 + m2c4 +(11) +corresponding to the relativistic energy of particle of +10-8 +10-6 +10-4 +10-2 +100 +-6 +-4 +-2 + 0 + 2 + 4 + 6 +pz*(z) +z +FIG. 2. +Equilibrium probability density functions for the +third entry z = zh,3 of zh, starting from the same initial con- +ditions as in figure 1 for the CMT dynamics eq. (7). Symbols +refer to stochastic simulations over an ensemble of N = 109 +elements: (◦): λ = 0, (□): λ = 1/2. The solid line represents +the normal distribution. +10-6 +10-4 +10-2 +100 +-15 +-10 +-5 + 0 + 5 + 10 + 15 +a +b +pz*(z) +z +FIG. 3. Equililibrium probability density functions p∗ +z(z) for +the first entry z = zh,1 of z for d = 3 in the presence of the +energy function eq. (11). Symbols are the results of stochastic +simulations of the CMT with E0 = 2 (symbols □), and E0 = 3 +(symbols ◦). +Lines (a) and (b) correspond to the J¨uttner +distribution eq. (12) at two values of the parameter β. +mass m, provided that z represents its momentum, while +keeping the linear constraints eq. (3). Set m = 1, c = 1 +a.u. +CMT transformations can be applied on equal footing +to this case, adopting for the transformations φ, ψ the +functional form eq. (8). The case λ = 0 is considered, +where the group αλ is defined in order to account for +the structure of the energy function eq. +(11). +Figure +3 depicts the asymptotic (equilibrium) densities at d = +3 for a generic entry of zh, at two different values of +E0. Data refer to an ensemble size of N = 107, and the +initial conditions are of impulsive nature. Each initial +z(0) +h,k attains with equal probability the values ±a, where +a = +� +(E2 +0 − 1)/3. +Deviation for the Gaussian behaviour is sensible, and +the simulation data converge to the J¨uttner distribution +p∗ +z(z) = Ae−β +√ +z2 c2+m2 c4 +(12) +where A is the normalization constant, and the parame- +ter β is determined by the initial value E0 of the ensemble + +5 +average of e(|z|). The detailed analysis of the relativis- +tic case is marginal in the present discussion, and it will +thoroughly developed elsewhere. What is significant for +the scope of this Letter is that Gaussianity in CMT’s is +a consequence of the physical assumptions on the energy +constraint. +To conclude, CMT’s provide the physical counterpart +of CLT (which is strictly speaking a mathematical prop- +erty), as regards the statistical characterization of kinetic +variables (velocity, momentum), the dynamics of which +is intrinsically distributional (owing to the conservation +principles) and not additive. It can be stated, in a picto- +rial way, that while “kinematic Gaussianity” stems from +CLT, as in the spatial propagation of Brownian motion, +“dynamic Gaussianity”, as in equilibrium velocity and +momentum distributions, is a consequence of CMT, with +its imbedded supergeneric occurrence. +The definition of CMT finds another major application +in the study of thermalization, and of equilibrium prop- +erties of molecular gases, in which, apart from particle- +particle collisions, quantum effects, related to the struc- +ture of the quantized energy levels of the molecules, +should be necessarility taken into account. This prob- +lem, that is an extension of the work by Einstein [27, 28] +on the momentum transfer by emission and absorption +of radiation, and of the stochastic modeling of radiative +effects [29] will be developed in a forthcoming work. +∗ corresponding author:massimiliano.giona@uniroma1.it +[1] M. Kac, Statistical Independence in Probability, Analysis +& Number Theory, (Dover Publ., Mineola, 2018). +[2] B. V. Gnedenko and A. N. Kolmogorov, Limit Dis- +tributions for Sums of Independent Random Variables, +(Addison-Wesley, Reading, 1954). +[3] V. V. Petrov, Sums of Independent Random Variables, +(Springer-Verlag, New York, 1975). +[4] P. Billingsley, Probability and Measure, (John Wiley & +Sons, New York, 1995). +[5] B. V. Gnedenko and V. Yu. Korolev, Random Summation +(CRC Press, Boca Raton, 1996). +[6] C. Stein, Proc. of the Sixth Berkeley Symp. on Math. +Statist. and Probab., Vol. II: Probability theory, 1972, +583–602. +[7] S. Chatterjee, arXiv:1404.1392, 2014. +[8] M. Kardar, Statistical Physics of Particles (Cambridge +Univ. Press, Cambridge, 2007). +[9] M. Giona, A. Cairoli and R. Klages, J. Phys. A 55 (2022) +475002. +[10] A. Einstein, Investigations on the Theory of Brownian +Movement, (Dover Publ., Mineola, 1956). +[11] S. Havlin and D. Ben-Avraham, Adv. Phys. 36 (1987) +695. +[12] R. Klages, G. Radons and I. M. Sokolov (Eds.), Anoma- +lous transport, (Wiley-VCH Verlag, Weinheim, 2008). +[13] M. F. Shlesinger, B. J. West and J. Klafter J, Phys. Rev. +Lett. 58, (1987) 1100. +[14] V. Zaburdaev, S. Denisov and J. Klafter, Rev. Mod. +Phys. 87, (2015) 483. +[15] P. L´evy, Calcul d´es Probabilit´es, (Gautier-Villars, Paris, +1925). +[16] V. V. Uchaikin and V. M. Zolotarev, Chance and Sta- +bility: Stable Distributions and their Applications, (de +Gruyter, Berlin, 1999). +[17] W. Pauli, Theory of Relativity, (Dover Publ., Mineola, +1981). +[18] S. Chapman and T. G. Cowling, The Mathematical The- +ory of Non-Uniform Gases, (Cambridge University Press, +Cambridge, 1952). +[19] S. Harris, An Introduction to the Theory of the Boltz- +mann Equation, (Dover Publ., Mineola, 2004). +[20] P. Ehrenfest and T Ehrenfest, The Conceptual Founda- +tions of the Statistical Approach in Mechanics, (Dover +Publ., Mineola, 2014). +[21] M. Mezard, G. Parisi and M. A. Virasoro, Sping Glass +Theory and Beyond, (World Scientific, Singapore, 1987). +[22] G. Jona-Lasinio, Il Nuovo Cimento 26 (1975) 99. +[23] P. Goswami and S. Chakravarty, Phys. Rev. B 95 (2017) +075131. +[24] J. Dunkel and P. H¨anggi, Phys. Rep. 471 (2009) 1. +[25] J. Dunkel, P. Talkner and P. H¨anggi, New J. Phys. 9 +(2007) 144. +[26] M. Giona, EPL 126 (2019) 50001. +[27] A. Einstein, Phys. Zeit. 18 (1917) 121. +[28] P. W. Milonni, The Quantum Vacuum. An introduc- +tion to Quantum Electrodynamics, (Academic Press, San +Diego, 1994). +[29] C. Pezzotti and M. Giona, Particle-photon radiative in- +teractions and thermalization, (2022) in preparation. + diff --git a/stE2T4oBgHgl3EQffQeV/content/tmp_files/load_file.txt b/stE2T4oBgHgl3EQffQeV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a91cbd6609bc1a84c732b83f6bcef8fdb7cead7f --- /dev/null +++ b/stE2T4oBgHgl3EQffQeV/content/tmp_files/load_file.txt @@ -0,0 +1,339 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf,len=338 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='03925v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='stat-mech] 10 Jan 2023 Another normality is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Distributive transformations and emergent Gaussianity Massimiliano Giona, Chiara Pezzotti and Giuseppe Procopio∗ Dipartimento di Ingegneria Chimica, Materiali, Ambiente La Sapienza Universit`a di Roma Via Eudossiana 18, 00184 Roma, Italy ( Dated: January 11, 2023) A distributional route to Gaussianity, associated with the concept of Conservative Mixing Trans- formations in ensembles of random vector-valued variables, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This route is completely different from the additive mechanism characterizing the application of Central Limit Theorem, as it is based on the iteration of a random transformation preserving the ensemble variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Gaussian- ity emerges as a “supergeneric” property of ensemble statistics, in the case the energy constraint is quadratic in the norm of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This result puts in a different light the occurrence of equilibrium Gaussian distributions in kinetic variables (velocity, momentum), as it shows mathe- matically that, in the absence of any other dynamic mechanisms, almost Gaussian distributions stems from the low-velocity approximations of the physical conservation principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Whenever, the energy constraint is not expressed in terms of quadratic functions (as in the relativistic case), the J¨uttner distribution is recovered from CMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The concept of normal distribution is so central in probability theory and physics that, as observed by Mark Kac, partially in jest (quoting Henry Poincar´e), it is not clear in the scientific community whether it comes from a mathematical property or it is a law of nature [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' As well known, the main route to normal distributions stems from the Central Limit Theorem (CLT) [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' There is an entire galaxy of different versions of CLT, either re- laxing the assumptions and the convergence properties, or modifying the way the summation is performed [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Similarly, there are parallel extensions of CLT, initiated by the work of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Stein, applied and generalized in several directions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' for sums of dependent random variables [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Besides these mathematical subtleties and chiselings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' of utmost value in probability theory and statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' the core of CLT - finding a widespread application in sta- tistical physics for interpreting the phenomenological oc- currence of normal distributions [8] - lies in its simplest version: given a sequence of independent and identi- cally distributed (iid) random variables {Xn}∞ n=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' pos- sessing zero mean and bounded variance σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' the random variable YN = �N n=1 Xn/σY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' corresponding to their sum normalized to unit variance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' by independence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' σ2 YN = N σ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' converges weakly for N → ∞ to the normal distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' so that its limit probability density function (pdf) is given by pY (y) = e−y2/2/ √ 2 π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (hence- forth the normal distribution is indicated with the symbol N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' It is important to observe that the weak con- vergence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=', the convergence in distribution, does not prevent the potential occurrence of anomalies as regards the higher order moments m(n) YN = ⟨Y n N⟩, for n > 2 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' CLT explains in a simple and elegant way the phe- nomenon of molecular diffusion addressed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Ein- stein in relation to the phenomenon of Brownian motion [10]: the molecular motion, or the kinematics of micro- metric particles driven by thermal fluctuations, can be interpreted as the resultant of the superposition of in- dependent (uncorrelated) displacements that, provided the boundedness of their variances, unavoidably leads by mathematical principles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' by enforcing CLT) to the occurrence in the long-time limit of the parabolic diffu- sion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Thus invoking CLT, the diffusion equa- tion follows, just by randomness, independence and the smallness of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' By the internal coherence of CLT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' this interpretation provides also the criteria for assessing the failure of the parabolic approximation: (i) lack of independence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' leading to subdiffusion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' as in the case of the random motion in disordered systems and fractal media [11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' or (ii) unbounded lower-order mo- ments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' associated with superdiffusive phenomena such as L´evy walks [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' that are strongly related to the extended version of CLT (usually referred to as the gen- eralized CLT) and to the L´evy theory of α-stable distri- butions [15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In its basic “machinery”, CLT can be viewed as a mathematical superpositional route to N(0, 1), in which normality is achieved by addition of independent con- tributions, properly rescaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In this Letter we want to present another, completely different, mechanism lead- ing to normality, in which the Gaussian distribution fol- lows as a consequence of a distributional dynamics in- volving the binary interaction of iid random variables subjected to some conservation principles and possess- ing an arbitary initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In the present case, the conservation principles are the fundamental conser- vation principles of physics, that apply both in the clas- sical and in the quantum world, namely the conservation of momentum and energy, both for matter-matter and matter-radiation interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This finds in the relativis- tic space-time formalism its maximum compactness and elegance, as it implies, for any physical interactions, the 2 overall 4-momentum pµ, µ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='., 4, pµ = � p, E c � is con- served, where p = (p1, p2, p3) is the momentum, E the energy and c the speed of light in vacuo [17] Specifically, we introduce and analyze the distribu- tional route to normality associated with the concept of Conservative Mixing Transformation (CMT) of random ensembles, for which the occurrence of limit Gaussian distributions is entirely entitled to the energy represen- tation, as discussed in the analysis of low-velocity vs rel- ativistic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' CMT’s originate from the stochastic generalization of the Boltzmannian description of binary collisions in di- luted gases [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Their introduction does not simply represent a stochastic transposition of the Boltzmannian kinetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' From the statistical analysis of CMT it is possible to derive the extreme genericity (as explained in the remainder) of the occurrence of “almost” Gaussian velocity distributions at equilibrium, and their connec- tion with the nature and representation of fundamental conservation principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Moreover, they provide a for- mal stochastic setting amenable to be extended to any physical impulsive interaction mechanism, and involving arbitrary conservation principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In the present Letter, CMT is analyzed beyond the classical low-velocity condi- tions in the relativistic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Although the Boltzmannian kinetic theory leads to Gaussian distributions at equilib- rium, no one, to the best of our knowledge, has extended and generalized the mechanism of binary interactions in the form of a universal stochastic route to Gaussianity, which is the scope of CMT’s and of the present Letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The structure of this Letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' We formalize the concept of CMT, the nature of the physical constraints, and the generic emergence of almost Gaus- sian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Subsequently, we address how Gaus- sianity can be broken, by the assumption of an energy conservation law different from a quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Consider an ensemble of N random vector-valued vari- ables E = {zh}N h=1, with zh = (zh,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' , zh,d) ∈ Rd, d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=', and let Σd,N the space of all the N- dimensional ensembles of d-dimensional random variables over the field R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Each zh can be referred to as the state vector of an element of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' A Mixing Transformation M : Σd,N → Σd,N (acronym MT) is a transformation of the ensemble E, into an ensemble E′ = M(E) = {z′ h}N h=1, defined by a binary operation amongst randomly selected elements of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' It can be defined in the following way: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' let φ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r), ψ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) : Rd × Rd × ∂Sd → Rd be two random functions, depending on a random vector r, |r| = 1 defined on the surface of the d- dimensional unit sphere ∂Sd, by the pdf g(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' select randomly two elements α, β ∈ (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' , N) with β ̸= α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' the transformed ensemble {z′ h}N h=1 is given by z′ α = φ(zα, zβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) z′ β = ψ(zα, zβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) (1) z′ h = zh , for h ̸= α, β Whenever it is not conceptually necessary, the explicit dependence of φ and ψ on r will be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The concept of MT’s so defined is too general for physi- cal applications, and constraints on the random functions φ, ψ, should be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' A Conservative Mixing Transformation is a MT, for which Nc functions fh(z) : Rd → R are defined, such that the transformations φ(z1, z2) and ψ(z1, z2) satisfy the constraints fh(z′ 1) + fh(z′ 2) = fh(z1) + fh(z2) , h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='., Nc (2) where z′ 1 = φ(z1, z2), z′ 2 = ψ(z1, z2), for any z1, z2 ∈ Rd, and r ∈ ∂Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The CMT’s of physical relevance are those satisfying the Nc = d + 1 conservation laws fh(z) = zh , h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' , d (3) fd+1(z) = e(|z|) (4) where e(|z|) is a non negative function solely of the norm |z| of z, representing, modulo a multiplicative factor, the energy function (kinetic energy) of the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (3) and (4) correspond mathematically to the consevation of momentum and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Several observation, follows from this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (i) The concept of random ensembles involves a finite number N of elements, and should not be confused with Gibbsian ensembles or with other collec- tive groupings (such as in the replica method) [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' A “random ensemble” in the CMT-theory means sim- ply a system of random variables, in the sense that (i) their initial conditions are randomly chosen for each el- ement, and that their evolution is subjected to random laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Correspondingly, the ensemble average of any func- tion q(z) of the state vector z, is simply expressed by ⟨q(z)⟩ = 1 N �N h=1 g(zh), and consequently a single pdf pz(z) is associated with the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (ii) The definition of MT proposed above is “event-based”, in the mean- ing that each transformation M involves solely a single binary event modifing the statistical properties of the en- semble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' If E0 = {z(0) h }N h=1 is the initial ensemble, assume ⟨z(0)⟩ = 0, and ⟨e(|z(0))|⟩ = E0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Let (φ−1, ψ−1) be the inverse transformation of (φ, ψ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=', φ−1(φ, ψ) = ψ−1(φ, ψ) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=', and set Jφ,ψ(z′ 1, z′ 2)| = ���� ∂ (φ(z1, z2), ψ(z1, z2)) ∂(z1, z2) ���� z1 = φ−1(z′ 1, z′ 2) z2 = ψ−1(z′ 1, z′ 2) for the Jacobian determinant of the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The statistical evolution of a CMT can be described as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' If pdf pz(z) is the ensemble pdf for E and E′ = 3 M(E), the pdf p′ z(z′) of E′, averaged over the probability measure of r, is expressed by p′ z(z′) = 1 2 � Rd [π′(z′, z1) + π′(z1, z′)] dz1 (5) where π′(z′ 1, z′ 2) = � ∂Sn pz(φ−1(z′ 1, z′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r))pz(ψ−1(z′ 1, z′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r))g(r)dr |Jφ,ψ(z′ 1, z′ 2)| (6) The independence of z1 and z2, expressed by the factor- ization of the two densities in the integrand at the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (6) stems from the random selection rule (point 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=') in the definition of a MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Consider for the energy function the expression, e(|z|) = |z|2 (7) corresponding to the classical form for the kinetic energy (for identical particles) In this case, the transformations φ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r), and φ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) can be chosen as � φ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) = (1 − λ) z1 + λ z2 + αλ(z1, z2) r ψ(z1, z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' r) = λ z1 + (1 − λ) z2 − αλ(z1, z2) r (8) where λ ∈ [0, 1] is a parameter, and αλ(z1, z2) is defined to fulfil eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (2), (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' For λ = 0, αλ = −(z1 − z2) · r (where “·” indicates the Euclidean scalar product), for λ = 1/2, αλ = |z1 − z2|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Consider λ = 1, as the other cases are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' It is easy to check that |Jφ,ψ| = 1, so that, if p∗ z(z) is the equilibrium distribution, π′(z′ 1, z′ 2) = p∗ z(z′ 1) p∗ z(z′ 2), and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (6) becomes p∗ z(z′ 1) p∗ z(z′ 2) = � ∂Sd p∗ z (z′ 1 − (z′ 1 − z′ 2) · r r) × p∗ z (z′ 1 + (z′ 1 − z′ 2) · r r) g(r) dr (9) Since, |z′ 1 − (z′ 1 − z′ 2) · r r|2 + |z′ 1 − (z′ 1 − z′ 2) · r r|2 = |z′ 1|2 + |z′ 2|2 the solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (9), is given by the Gaussian p∗(z) = Ae−β |z|2 (10) where the parameter β depends on the initial ensem- ble variance, and A is the normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In deriving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (10) we have not made use of the prop- erty of g(r), and therefore, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (10) is valid for any statistical structure of r, providing that it gives rise to a steady and unique equilibrium distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' There is only an exception to this property, represented by non-dispersive random transformations, defined as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The transformations φ and ψ of a CMT sa- tisfying eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (2)-(3) are non-dispersive if for all the al- lowable values of r φh(zα, zβ) is either zα,h, or zβ,h, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' , d (and complementarily for ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This means that no real mixing amongst the entries of the state vari- ables occur but, componentwise, the two transformations are either the identify or simply determine an exchange of values amongst the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' An example of non- dispersive CMT using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (8) with λ = 0 occurs for d = 2, where r = (cos ϕ, sin ϕ), ϕ ∈ [0, 2 π), in the case g(r) dr = gϕ(ϕ) dϕ is any atomic distribution of the form gϕ(ϕ) = (1 − µ) δ(ϕ) + µ δ(ϕ − π/2), µ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' When ϕ = 0, z′ α = (zβ,1, zα,2), z′ β = (zα,1, zβ,2), while for ϕ = π/2, z′ α = (zα,1, zβ,2), z′ β = (zβ,1, zα,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' We can state the following result with reference to the CMT transformations with energy constraint given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (8): for any initial distribution with E0 > 0, for any λ ∈ [0, 1], and for almost all the probability measures of the random vector r, the ensemble pdf of the CMT converges in the limit for N → ∞ to a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This stems for λ = 0 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (9), and similarly for any λ ∈ [0, 1] from the analogous property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The limit for N → ∞ is here introduced in order to consider an infinite ensemble, for which a smooth and continuous probabilis- tic characterization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=', the existence of a smooth pdf) can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' For any large but finite N, the result- ing limit distribution is still accurately approximated by a Gaussian distribution, apart for the asymptotic tails, that necessarily vanishes to zero, as for any finite N, the support of the limit pdf should be compact, simply be- cause of energy conservation and of the initial assumption of finite E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' It is interesting to compare the classical CLT route to Gaussianity with the emergence of it stemming from the iterative application of CMT’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In the CLT route, nor- mality is an emergent property of the procedure of sum- ming a large number of independent contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The existence of a limit density stems from a renormaliza- tion procedure, of rescaling the summation by removing its mean and normalizing its variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In this sense the observation of Jona-Lasinio [22] on the strong analogy between CLT and the Renormalization Group of quan- tum field theory is acute and cogent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In the CMT route, the process is purely distributional within a closed system (the ensemble) of vector-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (7), both the mean and the variance of the ensemble distribution are conserved in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' There is no renormalization procedure in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Moreover, in finite ensembles the convergence towards the Gaus- sian distribution is only approximate (albeit arbitrarily accurate for large N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This is an important, physical property, as it resolves the unphysical tails for arbitrar- ily large |z| in real finite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Moreover, what is remarkable in the distributional route to normality expressed by CMT is its generic- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' While in CLT the emergent Gaussian behavior holds for any distribution of iid random variables (with bounded mean and variance), in the iteration of CMT it emerges: (i) for generic initial ensemble distributions, (ii) for generic transformations φ, ψ at least expressed 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='6 3 0 5 10 15 20 κ(t) t FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Kurtosis κ(t) of the marginal ensemble distribution (for the first entry z = zh,1 of zh) vs the normalized opera- tional time t = n/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The arrow indicates first the d = 2 case (either with λ = 0 or λ = 1/2, that practically coincide), then the d = 3 case at λ = 0, and finally the d = 3 case at λ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (iii) for an almost generic statistical nature of the random variable r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' apart from the very pecu- liar (and physically irrelevant) non-dispersive transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' For this reason it is legitime to refer to this qualitative behavior as the “supergenericity” of the CMT distributional route to Gaussianity, in analogy with the concept of “superuniversality” coined for phase transi- tions (see [23] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In a physical perspective, “supergenericity” is the mathematical coun- terpart of a thermodynamic principle at work in CMT, determining a statistical emergent behavior that is com- pletely independent either of the details on the initial state or of the details on the transformations involved at a microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' To make an example, figure 1 depicts the evolution of the kurtosis κ(t) vs the operational time t = n/N, (where n is the number of CMT operations, and N is the ensemble size) for the first ensemble entry zh,1 at its convergence to the Gaussian limit κ = 3 for d = 2, 3 and for two different values of λ entering eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In this case N = 106, and the entries of the initial ensamble are uniformly distributed with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' All the stochastic simulations refer to a random pertur- bation r uniformly distributed in ∂Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The equilibrium probability density for a larger ensemble N = 109, start- ing from the same initial distribution is depicted in figure 1 at d = 3, for λ = 0, 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The emergence of a limit Gaussian density in CMT is physics-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Assuming the constraints eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (3), this entirely depends on the form of the energy constaint eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' In the presence of energy functions e(|z|) differ- ent from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (7) the stationary pdf for the entries of z is different from the Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This can be illustrated by means of a simple example of physical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Con- sider the relativistic extension [24–26], and consequently the energy function e(|z|) given by e(|z|) = � |z| c2 + m2c4 (11) corresponding to the relativistic energy of particle of 10-8 10-6 10-4 10-2 100 6 4 2 0 2 4 6 pz*(z) z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Equilibrium probability density functions for the third entry z = zh,3 of zh, starting from the same initial con- ditions as in figure 1 for the CMT dynamics eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Symbols refer to stochastic simulations over an ensemble of N = 109 elements: (◦): λ = 0, (□): λ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The solid line represents the normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 10-6 10-4 10-2 100 15 10 5 0 5 10 15 a b pz*(z) z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Equililibrium probability density functions p∗ z(z) for the first entry z = zh,1 of z for d = 3 in the presence of the energy function eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Symbols are the results of stochastic simulations of the CMT with E0 = 2 (symbols □), and E0 = 3 (symbols ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Lines (a) and (b) correspond to the J¨uttner distribution eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (12) at two values of the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' mass m, provided that z represents its momentum, while keeping the linear constraints eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Set m = 1, c = 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' CMT transformations can be applied on equal footing to this case, adopting for the transformations φ, ψ the functional form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The case λ = 0 is considered, where the group αλ is defined in order to account for the structure of the energy function eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Figure 3 depicts the asymptotic (equilibrium) densities at d = 3 for a generic entry of zh, at two different values of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Data refer to an ensemble size of N = 107, and the initial conditions are of impulsive nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Each initial z(0) h,k attains with equal probability the values ±a, where a = � (E2 0 − 1)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Deviation for the Gaussian behaviour is sensible, and the simulation data converge to the J¨uttner distribution p∗ z(z) = Ae−β √ z2 c2+m2 c4 (12) where A is the normalization constant, and the parame- ter β is determined by the initial value E0 of the ensemble 5 average of e(|z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The detailed analysis of the relativis- tic case is marginal in the present discussion, and it will thoroughly developed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' What is significant for the scope of this Letter is that Gaussianity in CMT’s is a consequence of the physical assumptions on the energy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' To conclude, CMT’s provide the physical counterpart of CLT (which is strictly speaking a mathematical prop- erty), as regards the statistical characterization of kinetic variables (velocity, momentum), the dynamics of which is intrinsically distributional (owing to the conservation principles) and not additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' It can be stated, in a picto- rial way, that while “kinematic Gaussianity” stems from CLT, as in the spatial propagation of Brownian motion, “dynamic Gaussianity”, as in equilibrium velocity and momentum distributions, is a consequence of CMT, with its imbedded supergeneric occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' The definition of CMT finds another major application in the study of thermalization, and of equilibrium prop- erties of molecular gases, in which, apart from particle- particle collisions, quantum effects, related to the struc- ture of the quantized energy levels of the molecules, should be necessarility taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' This prob- lem, that is an extension of the work by Einstein [27, 28] on the momentum transfer by emission and absorption of radiation, and of the stochastic modeling of radiative effects [29] will be developed in a forthcoming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' ∗ corresponding author:massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='giona@uniroma1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content='it [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE2T4oBgHgl3EQffQeV/content/2301.03925v1.pdf'} +page_content=' Kac, Statistical 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X-ray pulsar 1A 0535+262 +X. Hou,1, 2 W. Zhang,3, 4 D.F. Torres,3, 4, 5 L. Ji,6 and J. Li7, 8 +1Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China +2Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, Kunming 650216, +China +3Institute of Space Sciences (ICE, CSIC), Campus UAB, 08193 Barcelona, Spain +4Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain +5Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), E-08010 Barcelona, Spain +6School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China +7CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and +Technology of China, Hefei, China +8School of Astronomy and Space Science, University of Science and Technology of China, Hefei, China +(Accepted for publication in ApJ) +ABSTRACT +Binary systems are a well-established subclass of gamma-ray sources. The high mass +X-ray binary pulsar 1A 0535+262 has been considered to be a possible gamma-ray +emitter for a long time, although former gamma-ray searches using Fermi-LAT and +VERITAS data resulted in upper limits only. We aim at a deep search for gamma- +ray emission and pulsations from 1A 0535+262 using more than 13 years of Fermi- +LAT data. The analysis was performed for both the whole Fermi-LAT data set, as +well as for the X-ray outbursts that 1A 0535+262 has experienced since the launch +of Fermi. Various X-ray observations have been used to generate the ephemeris for +the pulsation search. We also investigate the long-term gamma-ray flux variability and +perform orbital phase-resolved analysis for the outbursts. We did not detect any steady +or pulsed gamma-ray emission from 1A 0535+262 during the whole Fermi-LAT mission +span or its X-ray outbursts. We thus derived the deepest gamma-ray luminosity upper +limits to date at the 95% confidence level to be around (2.3−4.7)×1032 erg s−1 depending +on different spectral indices assumed, which results in a ratio of Lγ to LX (2−150 keV) +being (1.9−3.9)×10−6. +Keywords: stars: neutron — X-rays: binaries — Gamma rays +1. INTRODUCTION +Among the various gamma-ray sources detected in the MeV/GeV and/or even the TeV band, binary +systems are a well-established subclass, although their number is yet small. Interestingly, despite their +Corresponding author: X. Hou, D.F. Torres +xhou@ynao.ac.cn, dtorres@ice.csic.es +arXiv:2301.01423v1 [astro-ph.HE] 4 Jan 2023 + +2 +Hou et al. +small number, several varieties of binaries exist with different gamma-ray emission mechanisms. For +a recent review, see Dubus (2015). First, Fermi-LAT has detected gamma-ray binaries themselves, +usually defined as a subclass of high mass X-ray binaries (HMXBs) with O or B companion star, +with two main features: they emit modulated gamma rays peaking above 1 MeV and present orbital +variability at all frequencies. The spectral energy distribution is usually thought to be powered by +the pulsar/stellar wind interaction, although so far the central compact object has been associated +to known pulsars only in three cases: PSR B1259-63/LS 2883 (see Abdalla et al. 2020; Chernyakova +et al. 2020, and references therein), PSR J2032+4127/MT91 213 (see Coe et al. 2019, and references +therein), and the recent identification of LS I +61 303 (Weng et al. 2022). +Recycled pulsars in binaries, called redbacks & black widow systems, are tight (orbital periods less +than a day), low mass (< 0.1M⊙) gamma-ray binaries with a main sequence degenerate companion. +In these systems the pulsar wind is ablating the companion star, leading to eclipses, radio variability, +X-ray/radio anti-correlation, and pulsar nulling (see, e.g., Bogdanov et al. 2018). Transitional pulsars +are special among these systems, since they exhibit two different states (accretion and rotation +powered) which may interchange in a matter of weeks and persist for years (see e.g., Archibald et al. +2009; Papitto et al. 2013; de Martino et al. 2015). These state changes produce significant gamma-ray +variability (Stappers et al. 2014; Torres et al. 2017). +Other Fermi-LAT detected binaries include microquasars, for which the gamma rays seem to be +associated with a relativistic jet. Notable examples are Cyg X-1 (see e.g., Albert et al. 2007; Zanin +et al. 2016; Zdziarski et al. 2017) and Cyg X-3 (see e.g., Abdo et al. 2009; Tavani et al. 2009; +Zdziarski et al. 2018; Sinitsyna & Sinitsyna 2019). The microquasar SS 433, whose central object is +still undetected, was unexpectedly found to produce gamma rays far from the jet (Abeysekara et al. +2018; Rasul et al. 2019; Xing et al. 2019; Fang et al. 2020) with a GeV source showing variability +with the precession period of the system (Li et al. 2020). +Finally, one finds the accreting millisecond pulsar SAX J1808.4−3658 (de O˜na Wilhelmi et al. 2016): +a “bona fide” accreting system apparently emitting gamma rays, although yet not significantly. This +is the only such system for which a gamma-ray detection was hinted so far. Putting upper limits on +accreting sources or detecting them is a must to understand the different variety. +1A 0535+262 is one of the best studied HMXB accreting pulsars. It was discovered in 1975 by +the Rotation Modulation Collimator on Ariel V, with a pulsation period of 104 s (Rosenberg et al. +1975). The compact object in the system is a highly magnetized neutron star which accretes mass +from the O9.7IIIe companion star (Steele et al. 1998). The orbital period of the system is ∼ 111 +days (Coe et al. 2006) and the eccentricity of the orbit is e = 0.47 ± 0.02 (Finger et al. 1996). +1A 0535+262 is relatively close to Earth with a distance of 1.8 ± 0.1 kpc, as measured by Gaia +(Bailer-Jones et al. 2018). Since its discovery, 1A 0535+262 has exhibited different X-ray outbursts +with peak flux ranging from ∼ 100 mCrab to ∼ 12.5 Crab. In particular, three giant X-ray outbursts +were detected since the launch of the Fermi satellite: in 2009 December (Acciari et al. 2011), 2011 +February (Sartore et al. 2015) and 2020 November (Kong et al. 2022; Mandal & Pal 2022, and +references therein). Figure 1 shows the long-term light curves of 1A 0535+262 in different energy +bands obtained from the Swift/BAT and MAXI /GSC Broadband Transient Monitor1. There was +also a double-peaked outburst (Caballero et al. 2013) just prior to the 2009 giant one, which is, +1 http://sakamotoagu.mydns.jp/bat gsc trans mon/web lc/1 Day.php?name=1A 0535+262 + +3 +however, not included in the monitor database. VLA observations during the 2020 outburst revealed +non-thermal radio emission from the source position (van den Eijnden et al. 2020). +1A 0535+262 was earlier associated with the EGRET unidentified gamma-ray source 3EG +J0542+2610 (Romero et al. 2001) and thus has long been considered as a high-energy (HE; E > 100 +MeV) and very-high-energy (VHE; E > 100 GeV) emitter candidate. The first gamma-ray search for +1A 0535+262 dates back to more than 10 years, during its giant 2009 outburst. On that occasion, the +X-ray outburst in December of 2009 triggered VHE VERITAS observations. Only upper limits have +been derived (Acciari et al. 2011). These authors also did a search for HE gamma-ray emission from +1A 0535+262 with Fermi-LAT in a period spanning the onset of the X-ray outburst to the successive +apastron of the pulsar (2009 November 30 to 2010 February 22). No significant GeV excess was seen +and a flux upper limit of F(> 0.2 GeV) < 1.9 × 10−8 photons cm−2 s−1 at 99% confidence level was +imposed. Recently, Lundy (2021) updated the VERITAS VHE search for 1A 0535+262 during its +2020 giant outburst. Again, only upper limits have been obtained. Also, Harvey et al. (2022) used +12.5 years LAT data to search for gamma-ray emission from 1A 0535+262. They claimed a marginal +persistent gamma-ray excess (3.5σ) at the position of the source and found that the gamma-ray +activity may be correlated with the X-ray outbursts. In addition, they found that essentially all of +the gamma-ray excess is concentrated in the orbital phase bin preceding periastron, thus providing +evidence of the gamma-ray excess originating from this binary system. If real, these hints are rele- +vant, and thus can help to get insights to the particle acceleration and emission process during the +accretion. +In this work, we analyze the three giant outbursts, in 2009, 2011 and 2020, and the previous double- +peaked outburst of 1A 0535+262. The time span of each outburst was defined by investigating the +X-ray light curves presented in the literature (Acciari et al. 2011; Sartore et al. 2015; Mandal & Pal +2022; Kong et al. 2022). We perform a deep search for gamma-ray emission and pulsations from +1A 0535+262 using more than 13 years of Fermi-LAT data and the latest 12-year source catalog. +We use the latest Instrument Response Functions (IRFs) and background diffuse models. Our work +therefore extends the result presented in Harvey et al. (2022). The paper is organized as follows: we +describe the data analysis procedure and results in Section 2, and discuss our findings in Section 3. +We finally conclude in Section 4. +2. DATA ANALYSIS AND RESULTS +2.1. Timing solutions +For the 2009 double-peaked and giant outbursts, we adopted the spin measurements and the orbital +ephemeris of 1A 0535+262 from the Fermi/GBM monitoring2. We used a polynomial function to +describe the spin evolution approximately. +For the 2011 outburst, we used the timing solution +reported in Sartore et al. (2015) derived using INTEGRAL observations. For the recent 2020 outburst, +thanks to the extensive coverage of Insight-HXMT observations (Wang et al. 2022), we used the phase- +connection technique (Deeter et al. 1981) to determine the spin evolution accurately. In practice, for +each 1000 s segment we folded background-subtracted light curves in the energy range of 25−80 keV, +for which the pulse profile shape is relatively stable. The 1000 s was chosen because this is the typical +interval for Insight-HXMT’s good time. The time-of-arrival (TOA) of each segment was estimated +2 https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/lightcurves/a0535.html + +4 +Hou et al. +by cross-correlating these pulse profiles with an averaged template. Then the spin evolution was +determined by using the software Tempo2 (Hobbs et al. 2006). We summarize the timing solutions +for different outbursts used in the following pulsation search (Section 2.5) in Table 2. +2.2. Fermi-LAT data set and reduction +We used the Pass 8 data set (Atwood et al. 2013; Bruel et al. 2018) available at the Fermi Science +Support Center (FSSC)3. This spans 166 months, from 2008 August 4 to 2022 June 9, with recon- +structed energy in the range 0.1−300 GeV. We selected SOURCE class events (Front and Back) with +a zenith angle smaller than 90◦ to avoid the Earth limb contamination. The events were further +filtered based on the criteria ‘‘DATA QUAL>0 && LAT CONFIG==1’’ to get the good time intervals in +which the satellite was working in standard data taking mode and the data quality was good. We +did not apply a Region of Interest (ROI)-based zenith angle cut4. The data set was centered at +1A 0535+262 with coordinates (α, δ) = (84.◦7274, 26.◦3158), with a radius of 10◦. The coordinates of +1A 0535+262 were taken from the SIMBAD5 database and in the J2000 frame. The analysis was +performed using the P8R3 SOURCE V3 IRFs and the latest Fermitools6 (v2.2.0) available at the +FSSC. +2.3. Fermi-LAT spectral analysis +In the spectral analysis, the latest 4FGL-DR37 (gll psc v30.fit) (Fermi-LAT collaboration 2022) +sources within a 20◦ circle around 1A 0535+262 were included to build a complete spatial and spec- +tral source model. We also included the latest Galactic interstellar emission model, “gll iem v07.fits”, +as well as the isotropic emission spectrum “iso P8R3 SOURCE V3 v1.txt”, with the latter taking +into account the extragalactic emission and the residual instrumental background8. Both the normal- +izations and spectral indices of sources within 5◦ around 1A 0535+262 were set free to vary except for +4FGL J0534.5+2201i, which is recommended to be fixed to account for the Inverse Compton Scat- +tering component of the Crab Nebula. Extended sources were modeled using the 12-year templates. +Since the closest source in the model is ∼ 1.6◦ away from 1A 0535+262, we added 1A 0535+262 +manually in the model as a point source with a simple Power Law spectral model. This allows us to +check whether the addition of such source is significant. +Model fitting was performed in a 14◦ × 14◦ ROI using the maximum likelihood method (Mattox +et al. 1996). We followed the binned likelihood procedure outlined in the FSSC using a 0.◦1×0.◦1 pixel +size and thirty logarithmic energy bins over 0.1−300 GeV. Two extra energy bins have been added +to take into account the energy dispersion except for the isotropic component. The significance of +a given source in the model is characterized by the Test Statistic (TS), which is expressed as TS += 2(log L − log L0), where log L and log L0 are the logarithms of the maximum likelihood of the +complete source model and of the background model (i.e. the source model without the given source +included), respectively. +We first performed a global binned likelihood fit to the whole data set by fixing the spectral index +of 1A 0535+262 to 2, 2.3 and 3, respectively. Such spectral indices are chosen to represent possible +emission mechanisms. At the same flux level, if the source were to emit a hard spectrum (as the +3 http://fermi.gsfc.nasa.gov/ssc/ +4 https://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/data preparation.html +5 http://simbad.u-strasbg.fr/simbad/ +6 https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/ +7 https://fermi.gsfc.nasa.gov/ssc/data/access/lat/12yr catalog/ +8 https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html + +5 +assumed 2) across the Fermi-LAT energy regime, it should be easier to detect it due to the diminishing +background at higher energies. Then, using the best-fit source model from the whole data set fit, we +performed binned likelihood fits to the different X-ray outbursts that 1A 0535+262 has experienced +in the past, following the same fitting setup and procedure as for the whole data set. To increase +the detection possibility and statistics, we also stacked all the outbursts together to perform the fit. +1A 0535+262 was not detected in any of these cases and we therefore computed a 95% confidence +level energy flux upper limit accordingly. The fitting results are presented in Table 1. +2.4. Gamma-ray variability +We performed two different types of variability analysis for 1A 0535+262. First, to investigate +the long-term gamma-ray flux variability, we computed light curves with a 180-day binning as in +Harvey et al. (2022) in the energy range of 0.1−300 GeV for all spectral indices used in the spectral +analysis (Figure 1). The full data best-fit source model was used as an input for each time bin and +the normalizations of sources within 3◦ around 1A 0535+262 were set free to vary. Upper limits at +95% confidence level were calculated when 1A 0535+262 had TS < 4 in a given time bin. We did not +see any significant detection in all the time bins when fixing the spectral index to 2 or 2.3. For the +light curve with index fixed to 3, there are two bins with TS being about 10 and 20, corresponding to +approximately 3σ and 4σ. However, these two bins correspond to the period where its X-ray emission +was faint according to the X-ray monitoring of 1A 0535+262 (Figure 1). Thus, we conclude that +no correlation between gamma-ray and X-ray was observed, contrary to what Harvey et al. (2022) +has claimed. Furthermore, the variability significance was computed following the same method used +in Acero et al. (2015). Only the light curve with index fixed to 3 has a non-negligible significance +(1.7σ for 27 degrees of freedom), but this is far from declaring a significant variability, which requires +usually a significance of at least 3σ. Any gamma-ray emission is thus consistent with being steady +on a timescale of a few months based on our analysis. +Since gamma-ray binaries usually exhibit orbital flux variability, we also computed the orbital flux +for 1A 0535+262 with 10 bins per orbit and calculated upper limits at 95% confidence level when +1A 0535+262 had TS < 4 in a given orbital bin, as was done in Harvey et al. (2022). Similar to the +long-term light curve, the full data best-fit source model was used as an input for each orbital bin +and the spectral index was fixed to 2, 2.3 and 3, respectively. No significant orbital variability was +observed. The orbital light curves are shown in Figure 2. +2.5. Gamma-ray pulsation search +We performed a pulsation search using reconstructed LAT photons within 1◦ of 1A 0535+262 in +the energy range of 0.1−300 GeV. The signal significance was qualified using the weighted H-test +developed by Kerr (2011), which is based on the original one proposed by de Jager et al. (1989): +Hmw = max +� +Z2 +iw − c × (i − 1) +� +, +1 ≤ i ≤ m, +(1) +where +Z2 +mw = +2 +�N +i=1 w2 +i +× +m +� +k=1 +(α2 +wk + β2 +wk), +(2) +and +αwk = +N +� +i=1 +wi cos (2πkφi), +βwk = +N +� +i=1 +wi sin (2πkφi). +(3) + +6 +Hou et al. +Here, N is the total photon number, φi is the pulsar rotational phase and wi is the photon weight, +m is the maximum search harmonic and c is the offset for each successive harmonic. We used the +standard value c = 4 and varied m in our analysis. We verified that taking the standard value m = 20 +did not change the result. +We employed the Simple Weights method descried in Bruel (2019) and Smith et al. (2019) to +compute the weight for each photon. This is considered as a proxy for the probability that the +photon comes from 1A 0535+262. Assuming that the target source is faint compared to the diffuse +background and that the background emission is isotropic, for a photon with energy E (in MeV) and +angular distance ∆θ to the target source, the weight is: +w(E, ∆θ) = f(E) × g(E, ∆θ), +(4) +where +f(E) = exp(−2log2 +10(E/Eref)) +(5) +is the weight at the pulsar position, which depends on the pulsar and background spectra and on the +LAT’s energy-dependent Point Spread Function (PSF). The geometrical factor g(E, ∆θ) describes +the angular distribution of the gamma-ray photons emitting from a point source and can be written +as +g(E, ∆θ) = +� +1 + +9∆θ2 +4σ2 +psf(E) +�−2 +, +(6) +where +σpsf(E) = +� +p2 +0(E/100)−2p1 + p2 +2 +(7) +is the PSF 68% containment angle with p0 = 5.445, p1 = 0.848 and p2 = 0.084 for LAT P305 Pass 8 +data (Atwood et al. 2013). +Defining the reference energy Eref = 10µw, the H-test versus µw follows a Gaussian distribution +(Bruel 2019). Thus, searching for pulsations means finding the maximum H-test by scanning over +µw. After searching a thousand radio pulsars for possible gamma-ray emission, Smith et al. (2019) +indicate that in most cases µw = 3.6 is a good choice to yield a significant signal. We adopted +this value in our pulsation search for 1A 0535+262. Considering that this value was found for non- +accreting (and many isolated) radio pulsars that have a power law with an exponential cutoff (PLEC) +spectrum (Abdo et al. 2013), and thus may not be appropriate for an accreting system that could +have a different spectral shape, we also verified that scanning over µw to find the best value does +not affect the result significantly. We used the ephemerides for different observations (Table 2) to +phase-fold the gamma-ray photons of 1A 0535+262, restricting the pulsation search range to the +validity of the corresponding ephemerides. However, no significant pulsation was detected during the +individual outbursts. Although a signal of ∼ 1.5σ was hinted for the 2009 outburst, the significance +is far from the LAT detection threshold of 4σ. +3. DISCUSSION +In general, we obtained very different results compared to Harvey et al. (2022). We had no detection +of either persistent or transient or pulsed gamma-ray emission from 1A 0535+262. Although we +have two time bins with TS∼20 and ∼10, they did not correspond to the X-ray outburst periods. +Therefore, our result does not support their conclusion that the gamma-ray emission is correlated + +7 +with the X-ray outburst of the source and is mostly concentrated in specific orbital bins. +Such +difference may mainly come from the fitting procedure and whether or not considering the energy +dispersion. We normally fitted the sources within 5◦ around 1A 0535+262 in the spectral fit and +orbital flux variability study, and within 3◦ for the long-term light curves, while they fitted only 1◦ +around 1A 0535+262. In addition, the spectral index was fixed to different values to account for +possible emission mechanisms in our study, while they let the index free to vary. Actually, for the +long-term variability, no detailed fitting procedure information was found in Harvey et al. (2022), +making a detailed comparison difficult. +Unlike those cases when a detection is found from an astrophysical source, the possible reasons +for a non-detection are essentially unlimited. Gamma-ray binary systems such as LS I +61 303 are +radio sources, and their X-ray spectra contain a significant non-thermal component. Recently, radio +pulsations were detected from this system (Weng et al. 2022). Thus, it is worth noting that LS I +61 +303 is probably not a significantly accreting system, and thus it likely has a different acceleration and +emission mechanism than that acting in 1A 0535+26, if there is any in the latter. On the other hand, +in 1A 0535+262, the absence of a significant quiescent radio emission that could later be associated +with non-thermal processes may simply indicate that leptons are not sufficiently accelerated there. +This observation had earlier promoted hadronic models. From our results, though, we can also rule +out hadronic production acting according to a mechanism originally proposed by Cheng & Ruderman +(1991), where a proton beam accelerated in a magnetospheric electrostatic gap impacts the transient +accretion disk. This model was applied to 1A 0535+262 by Romero et al. (2001), Anchordoqui et al. +(2003) and Orellana et al. (2007). Particularly in the latter paper the theoretical flux prediction +of 3.8 × 10−8 ph cm−2 (derived by extrapolating the result of Orellana et al. 2007), i.e., a gamma- +ray luminosity of about 1033 erg s−1 at 0.3 TeV at the end of giant outbursts to the Fermi-LAT +energy range (see Acciari et al. 2011) is above our limit which is ∼(2.3−4.7)×1032 erg s−1 by one +order of magnitude (and should have been detected earlier on in the mission) depending on different +spectral indices assumed. Taking the X-ray luminosity (2−150 keV) reported for the largest 2020 +outburst (Kong et al. 2021) to be 1.2×1038 erg s−1, the ratio of Lγ to LX is then calculated to be +(1.9−3.9)×10−6. +It remains to be seen, of course, whether the flux was overestimated but the mechanism is still +viable, or whether a longer integration might lead to low-flux quiescent emission or to an occasional +detection. None of these possibilities has happened in the long integration time we have analyzed. +At least with the current gamma-ray sensitivity, 1A 0535+262 is not a gamma-ray source. With our +current limits, there appears to be no reasonable combination of the hadronic model parameters to +still accommodate a persistent gamma-ray flux. +Another interpretation could simply be that none of the possible shocks in the system is energetic +enough to produce sufficiently accelerated particles able to emit gamma-rays. Or that if they do, +the radiation produced is absorbed due to the matter in the surroundings, see e.g., the discussion in +Orellana et al. (2007). Regarding the latter, it would be reasonable to expect that any absorption +would be quite variable. Thus if the emission is produced in the system at all, it would be unlikely +that it is absorbed all the time. +In addition, secondary electrons (and positrons) from the pair +production process would generate gamma rays at MeV–GeV energies, (see e.g., Bednarek 1997, 2006; +Sierpowska-Bartosik & Torres 2008). These could be attenuated by X-ray photons, but probably not +fully attenuated after the X-ray peak has passed. + +8 +Hou et al. +Giant (or Type II) outbursts, though they are rare and unrelated to the orbital cadence, may have +X-ray luminosities close to the Eddington limit. They are likely associated with the formation of +a transient accretion disk. Recently, van den Eijnden et al. (2020) detected a radio counterpart +during the 2020 outburst. This was the first time that a coupled increase in X-ray and radio flux +was seen in 1A 0535+262 and shows that the radio emission may relate to the accretion state. This +would be similar to the behaviour seen in the transient Be X-ray binary Swift J0243.6+6124 (van +den Eijnden et al. 2018). Bednarek (2009a,b) proposed that it is possible that HMXBs produce +gamma-ray emission during accretion periods. The possibility that particle acceleration can take +place even when mass accretion is going on is supported by some observational results: the hinted +gamma-ray emission from SAX J1808.4−3658 (de O˜na Wilhelmi et al. 2016), the gamma-ray emission +found from the sub-luminous state of the transitional pulsars (as quoted in the introduction) and +interpreted as propeller emission (see e.g., Papitto et al. 2014; Papitto & Torres 2015) or a mini +pulsar wind nebula (Papitto et al. (2019), see also Veledina et al. (2019)) and finally, the discovery +of optical and ultraviolet pulsed emission from the accreting millisecond pulsar SAX J1808.4−3658 +(Ambrosino et al. 2021). However, the neutron star Eddington luminosity (LEdd ∼ 1.8 × 1038 erg +s−1) is many orders of magnitude above our upper limits, pointing to a very inefficient mechanism, +if at play at all. Similarly to the case of other transients, for instance, the Be X-Ray Binary 4U +1036−56 (RX J1037.5−5647), which could be associated to AGILE transients, we cannot discard +that a low level of gamma-ray flux is emitted at lower energies, below 100 MeV (see Li et al. 2012, +and in particular their figure 6 for an associated discussion). This remains to be tested with the +advent of MeV missions such as AMEGO (McEnery et al. 2019), e-ASTROGAM (de Angelis et al. +2018), or COSI (Beechert et al. 2022). +4. CONCLUSIONS +We have searched for gamma-ray emission and pulsations from 1A 0535+262 using more than +13 years of Fermi-LAT data. Neither persistent nor transient nor pulsed gamma-ray emission has +been detected significantly in our study for the whole data set or during the X-ray outbursts that +1A 0535+262 has experienced since the launch of the Fermi satellite. Upper limits on the luminosity +at the 95% confidence level were derived to be around (2.3−4.7)×1032 erg s−1 depending on different +spectral indices assumed, the deepest ones to date. The emission of 1A 0535+262 is considered to be +consistent with being steady on a timescale of a few months. Although two time bins in the long-time +light curve hint to have gamma-ray emission at roughly 3 and 4σ, they occurred when the source was +faint in X-rays. Thus, no correlation between gamma-ray and X-ray activities was observed based +on our result. In addition, we did not see any significant orbital gamma-ray variation. We conclude +that 1A 0535+262 is not a gamma-ray emitter at the level of the current gamma-ray sensitivity. + +9 +ACKNOWLEDGMENTS +The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies +and institutes that have supported both the development and the operation of the LAT, as well as +scientific data analysis. These include the National Aeronautics and Space Administration and the +Department of Energy in the United States; the Commissariat `a l’Energie Atomiqueand and the +Centre National de la Recherche Scientifique/Institut National de Physique Nucl´eaire et de Physique +des Particules in France; the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare +in Italy; the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy +Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in +Japan; and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish Na- +tional Space Board in Sweden. Additional support for science analysis during the operations phase is +gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National +d’´Etudes Spatiales in France. +This work was performed in part under DOE Contract DE-AC02-76SF00515. The authors are +supported by the National Natural Science Foundation of China through grants U1938103, 12041303, +12173103, U2038101, 11733009. WZ and DFT work have been supported by the grant PID2021- +124581OB-I00 funded by MCIN/AEI/10.13039/501100011033 and by the Spanish program Unidad +de Excelencia Mar´ıa de Maeztu CEX2020-001058-M. DFT acknowledges as well USTC and the +Chinese Academy of Sciences International Presidential Fellowship Initiative 2021VMA0001. 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A., Malyshev, D., Dubus, G., et al. +2018, MNRAS, 479, 4399, +doi: 10.1093/mnras/sty1618 + +12 +Hou et al. +0 +2 +ph/cm2/s +GSC 2-4 keV +0 +2 +4 +ph/cm2/s +GSC 4-10 keV +0 +2 +ph/cm2/s +GSC 10-20 keV +0.0 +0.2 +cts/s/det +BAT 14-24 keV +0.0 +0.1 +cts/s/det +BAT 24-50 keV +0.00 +0.05 +cts/s/det +BAT 50-100 keV +55000 +56000 +57000 +58000 +59000 +Time (MJD) +0.00 +0.02 +cts/s/det +BAT 100-195 keV +0.4 +0.6 +0.8 +1.0 +1.2 +Flux (ph cm +2 s +1) +1e +8 += 2 +55000 +56000 +57000 +58000 +59000 +Time (MJD) +0.0 +0.5 +1.0 +1.5 +TS +1.0 +1.5 +2.0 +2.5 +Flux (ph cm +2 s +1) +1e +8 += 2.3 +55000 +56000 +57000 +58000 +59000 +Time (MJD) +0.0 +0.2 +0.4 +0.6 +0.8 +TS +0.2 +0.4 +0.6 +0.8 +1.0 +Flux (ph cm +2 s +1) +1e +7 += 3 +55000 +56000 +57000 +58000 +59000 +Time (MJD) +0 +5 +10 +15 +20 +TS +Figure 1. Upper panel: Long-term X-ray light curves of 1A 0535+262 in different energy bands observed +by Swift/BAT and MAXI /GSC as obtained from http://sakamotoagu.mydns.jp/bat gsc trans mon/web lc/ +1 Day.php?name=1A 0535+262. Low three panels: Long-term gamma-ray light curves of 1A 0535+262 with +a time binning of 180 days. Spectral index was fixed to 2, 2.3 and 3, respectively. Upper limits at 95% +confidence level are computed for bins with TS<4. + +13 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +1e +12 +Eflux (erg cm +2 s +1) += 2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Orbital Phase +0.0 +0.2 +0.4 +0.6 +TS +3 +4 +5 +1e +12 +Eflux (erg cm +2 s +1) += 2.3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Orbital Phase +0.00 +0.05 +0.10 +0.15 +0.20 +TS +4 +5 +6 +7 +8 +9 +1e +12 +Eflux (erg cm +2 s +1) += 3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +Orbital Phase +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +TS +Figure 2. Orbital energy flux variation of 1A 0535+262 with 10 bins per orbit. Spectral index was fixed +to 2, 2.3 and 3, respectively. Upper limits at 95% confidence level are computed for bins with TS<4. + +14 +Hou et al. +Table 1. Fermi-LAT spectral analysis results +Perioda +Time Range +Spectral Index +TS +Energy Flux Upper Limitb +(MJD) +(10−12 erg cm−2 s−1) +Whole dataset +full +54682-59739 +2.0 +0.0 +0.6 +full +54682-59739 +2.3 +0.0 +0.7 +full +54682-59739 +3.0 +0.0 +1.2 +Stacked outbursts +Rising+Falling +55040-59207 +2.0 +0.0 +11.9 +Rising+Falling +55040-59207 +2.3 +0.0 +9.3 +Rising+Falling +55040-59207 +3.0 +0.0 +20.9 +The 2009 double-peaked outburst +Rising+Falling +55040-55070 +2.0 +0.4 +34.9 +Rising+Falling +55040-55070 +2.3 +0.4 +27.3 +Rising+Falling +55040-55070 +3.0 +0.0 +21.9 +The 2009 giant outburst +ALL +55165.9-55249.6 +2.0 +0.0 +9.9 +ALL +55165.9-55249.6 +2.3 +0.0 +10.5 +ALL +55165.9-55249.6 +3.0 +0.0 +16.5 +Rising+Falling +55165.9-55193.6 +2.0 +0.0 +27.9 +Rising+Falling +55165.9-55193.6 +2.3 +0.0 +24.7 +Rising+Falling +55165.9-55193.6 +3.0 +0.0 +28.4 +Rising +55165.9-55177.6 +2.0 +0.0 +49.3 +Rising +55165.9-55177.6 +2.3 +0.0 +45.8 +Rising +55165.9-55177.6 +3.0 +0.0 +45.4 +Falling +55178.4-55193.6 +2.0 +0.0 +39.4 +Falling +55178.4-55193.6 +2.3 +0.0 +31.7 +Falling +55178.4-55193.6 +3.0 +0.0 +29.5 +Apastron +55199.4-55216.6 +2.0 +0.0 +23.6 +Apastron +55199.4-55216.6 +2.3 +0.0 +18.1 +Apastron +55199.4-55216.6 +3.0 +0.0 +18.6 +Periastron +55230.4-55249.6 +2.0 +0.0 +44.9 +Periastron +55230.4-55249.6 +2.3 +0.0 +39.3 +Periastron +55230.4-55249.6 +3.0 +0.0 +34.5 +The 2011 giant outburst +Rising+Falling +55600-55645 +2.0 +0.0 +14.6 +Rising+Falling +55600-55645 +2.3 +0.0 +13.4 +Rising+Falling +55600-55645 +3.0 +0.0 +19.6 +Rising +55600-55617 +2.0 +0.0 +33.0 +Rising +55600-55617 +2.3 +0.0 +30.7 +Rising +55600-55617 +3.0 +0.0 +40.0 +Falling +55618-55645 +2.0 +0.0 +30.9 +Falling +55618-55645 +2.3 +0.0 +30.6 +Falling +55618-55645 +3.0 +0.0 +20.0 +The 2020 giant outburst +Rising+Falling +59159-59207 +2.0 +0.0 +33.4 +Rising+Falling +59159-59207 +2.3 +0.0 +27.2 +Rising+Falling +59159-59207 +3.0 +0.0 +31.4 +Rising +59159-59172.5 +2.0 +4.2 +131.5 +Rising +59159-59172.5 +2.3 +3.2 +91.8 +Rising +59159-59172.5 +3.0 +0.5 +90.0 +Falling +59173-59207 +2.0 +0.0 +24.2 +Falling +59173-59207 +2.3 +0.0 +22.0 +Falling +59173-59207 +3.0 +0.0 +30.5 +a Full: the whole dataset used in this work; ALL: the dataset including the rising, falling, +apastron and periastron portions of the outburst. +b The upper limits are given at the 95% confidence level in the energy range of 0.1−300 GeV. + +15 +Table 2. Parameters of the spin evolution of 1A 0535+026. νi is the ith order derivative of the frequency. +Parameters +2009 double outbursta +2009 outbursta +2011 outburstb +2020 outburstc +Epoch (MJD) +55040 +55166.99 +55616.202 +59170 +Tstart (MJD) +55040 +55166.99 +55608 +59159.15 +Tstop (MJD) +55070 +55201 +55637 +59207.92 +ν0(10−3 Hz) +9.66041(4) +9.6618(1) +9.6793(1) +9.66045(2) +ν1(10−12 Hz s−1) +0.67(3) +-4.6(6) +6.43(5) +19.27(4) +ν2(10−17 Hz s−2) +3.3(0.3) +0.121(7) +1.17(4) +ν3(10−23 Hz s−3) +-4.8(6) +-1.43(9) +-5.8(2) +ν4(10−29 Hz s−4) +3(1) +1.67(5) +-2.8(9) +ν5(10−36 Hz s−5) +-8(6) +737(7) +ν6(10−39 Hz s−6) +-1.6(2) +ν7(10−45 Hz s−6) +-5(2) +ν8(10−50 Hz s−6) +3(1) +ν9(10−56 Hz s−6) +-8(2) +ν10(10−62 Hz s−6) +7(1) +a: Derived form the Fermi/GBM monitoring (see text). +b: Adopted from Sartore et al. (2015). +c: Derived from Insight-HXMT observations (Wang et al. 2022, see text). + diff --git a/wdAzT4oBgHgl3EQfdfz3/content/tmp_files/load_file.txt b/wdAzT4oBgHgl3EQfdfz3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb2fb9da231b7cf8f7b2665a7f8f2362b3a20082 --- /dev/null +++ b/wdAzT4oBgHgl3EQfdfz3/content/tmp_files/load_file.txt @@ -0,0 +1,1197 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf,len=1196 +page_content='Draft version January 5, 2023 Typeset using LATEX preprint style in AASTeX63 Deep search for gamma-ray emission from the accreting X-ray pulsar 1A 0535+262 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Hou,1, 2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Zhang,3, 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Torres,3, 4, 5 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Ji,6 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Li7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 8 1Yunnan Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Kunming 650216,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' China 2Key Laboratory for the Structure and Evolution of Celestial Objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Kunming 650216,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' China 3Institute of Space Sciences (ICE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Campus UAB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 08193 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Spain 4Institut d’Estudis Espacials de Catalunya (IEEC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 08034 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Spain 5Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' E-08010 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Spain 6School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Sun Yat-Sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Zhuhai 519082,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' China 7CAS Key Laboratory for Research in Galaxies and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' China 8School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' China (Accepted for publication in ApJ) ABSTRACT Binary systems are a well-established subclass of gamma-ray sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The high mass X-ray binary pulsar 1A 0535+262 has been considered to be a possible gamma-ray emitter for a long time, although former gamma-ray searches using Fermi-LAT and VERITAS data resulted in upper limits only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We aim at a deep search for gamma- ray emission and pulsations from 1A 0535+262 using more than 13 years of Fermi- LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The analysis was performed for both the whole Fermi-LAT data set, as well as for the X-ray outbursts that 1A 0535+262 has experienced since the launch of Fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Various X-ray observations have been used to generate the ephemeris for the pulsation search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We also investigate the long-term gamma-ray flux variability and perform orbital phase-resolved analysis for the outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We did not detect any steady or pulsed gamma-ray emission from 1A 0535+262 during the whole Fermi-LAT mission span or its X-ray outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We thus derived the deepest gamma-ray luminosity upper limits to date at the 95% confidence level to be around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7)×1032 erg s−1 depending on different spectral indices assumed, which results in a ratio of Lγ to LX (2−150 keV) being (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9)×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Keywords: stars: neutron — X-rays: binaries — Gamma rays 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' INTRODUCTION Among the various gamma-ray sources detected in the MeV/GeV and/or even the TeV band, binary systems are a well-established subclass, although their number is yet small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Interestingly, despite their Corresponding author: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Hou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Torres xhou@ynao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='cn, dtorres@ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='es arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='01423v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='HE] 4 Jan 2023 2 Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' small number, several varieties of binaries exist with different gamma-ray emission mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' For a recent review, see Dubus (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' First, Fermi-LAT has detected gamma-ray binaries themselves, usually defined as a subclass of high mass X-ray binaries (HMXBs) with O or B companion star, with two main features: they emit modulated gamma rays peaking above 1 MeV and present orbital variability at all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The spectral energy distribution is usually thought to be powered by the pulsar/stellar wind interaction, although so far the central compact object has been associated to known pulsars only in three cases: PSR B1259-63/LS 2883 (see Abdalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Chernyakova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2020, and references therein), PSR J2032+4127/MT91 213 (see Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2019, and references therein), and the recent identification of LS I +61 303 (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Recycled pulsars in binaries, called redbacks & black widow systems, are tight (orbital periods less than a day), low mass (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1M⊙) gamma-ray binaries with a main sequence degenerate companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In these systems the pulsar wind is ablating the companion star, leading to eclipses, radio variability, X-ray/radio anti-correlation, and pulsar nulling (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Bogdanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Transitional pulsars are special among these systems, since they exhibit two different states (accretion and rotation powered) which may interchange in a matter of weeks and persist for years (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Archibald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Papitto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' de Martino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' These state changes produce significant gamma-ray variability (Stappers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Other Fermi-LAT detected binaries include microquasars, for which the gamma rays seem to be associated with a relativistic jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Notable examples are Cyg X-1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Zanin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2017) and Cyg X-3 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Tavani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Sinitsyna & Sinitsyna 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The microquasar SS 433, whose central object is still undetected, was unexpectedly found to produce gamma rays far from the jet (Abeysekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Rasul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2020) with a GeV source showing variability with the precession period of the system (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Finally, one finds the accreting millisecond pulsar SAX J1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4−3658 (de O˜na Wilhelmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2016): a “bona fide” accreting system apparently emitting gamma rays, although yet not significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This is the only such system for which a gamma-ray detection was hinted so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Putting upper limits on accreting sources or detecting them is a must to understand the different variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1A 0535+262 is one of the best studied HMXB accreting pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' It was discovered in 1975 by the Rotation Modulation Collimator on Ariel V, with a pulsation period of 104 s (Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The compact object in the system is a highly magnetized neutron star which accretes mass from the O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7IIIe companion star (Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The orbital period of the system is ∼ 111 days (Coe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2006) and the eccentricity of the orbit is e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='02 (Finger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1A 0535+262 is relatively close to Earth with a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1 kpc, as measured by Gaia (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Since its discovery, 1A 0535+262 has exhibited different X-ray outbursts with peak flux ranging from ∼ 100 mCrab to ∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 Crab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In particular, three giant X-ray outbursts were detected since the launch of the Fermi satellite: in 2009 December (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2011), 2011 February (Sartore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2015) and 2020 November (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Mandal & Pal 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Figure 1 shows the long-term light curves of 1A 0535+262 in different energy bands obtained from the Swift/BAT and MAXI /GSC Broadband Transient Monitor1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' There was also a double-peaked outburst (Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2013) just prior to the 2009 giant one, which is, 1 http://sakamotoagu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='mydns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='jp/bat gsc trans mon/web lc/1 Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='name=1A 0535+262 3 however, not included in the monitor database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' VLA observations during the 2020 outburst revealed non-thermal radio emission from the source position (van den Eijnden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1A 0535+262 was earlier associated with the EGRET unidentified gamma-ray source 3EG J0542+2610 (Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2001) and thus has long been considered as a high-energy (HE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' E > 100 MeV) and very-high-energy (VHE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' E > 100 GeV) emitter candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The first gamma-ray search for 1A 0535+262 dates back to more than 10 years, during its giant 2009 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' On that occasion, the X-ray outburst in December of 2009 triggered VHE VERITAS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Only upper limits have been derived (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' These authors also did a search for HE gamma-ray emission from 1A 0535+262 with Fermi-LAT in a period spanning the onset of the X-ray outburst to the successive apastron of the pulsar (2009 November 30 to 2010 February 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' No significant GeV excess was seen and a flux upper limit of F(> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 GeV) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9 × 10−8 photons cm−2 s−1 at 99% confidence level was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Recently, Lundy (2021) updated the VERITAS VHE search for 1A 0535+262 during its 2020 giant outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Again, only upper limits have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Also, Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022) used 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 years LAT data to search for gamma-ray emission from 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' They claimed a marginal persistent gamma-ray excess (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5σ) at the position of the source and found that the gamma-ray activity may be correlated with the X-ray outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In addition, they found that essentially all of the gamma-ray excess is concentrated in the orbital phase bin preceding periastron, thus providing evidence of the gamma-ray excess originating from this binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' If real, these hints are rele- vant, and thus can help to get insights to the particle acceleration and emission process during the accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In this work, we analyze the three giant outbursts, in 2009, 2011 and 2020, and the previous double- peaked outburst of 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The time span of each outburst was defined by investigating the X-ray light curves presented in the literature (Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Sartore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Mandal & Pal 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We perform a deep search for gamma-ray emission and pulsations from 1A 0535+262 using more than 13 years of Fermi-LAT data and the latest 12-year source catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We use the latest Instrument Response Functions (IRFs) and background diffuse models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Our work therefore extends the result presented in Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The paper is organized as follows: we describe the data analysis procedure and results in Section 2, and discuss our findings in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We finally conclude in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' DATA ANALYSIS AND RESULTS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Timing solutions For the 2009 double-peaked and giant outbursts, we adopted the spin measurements and the orbital ephemeris of 1A 0535+262 from the Fermi/GBM monitoring2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We used a polynomial function to describe the spin evolution approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' For the 2011 outburst, we used the timing solution reported in Sartore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2015) derived using INTEGRAL observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' For the recent 2020 outburst, thanks to the extensive coverage of Insight-HXMT observations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022), we used the phase- connection technique (Deeter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1981) to determine the spin evolution accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In practice, for each 1000 s segment we folded background-subtracted light curves in the energy range of 25−80 keV, for which the pulse profile shape is relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The 1000 s was chosen because this is the typical interval for Insight-HXMT’s good time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The time-of-arrival (TOA) of each segment was estimated 2 https://gammaray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='nsstc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='gov/gbm/science/pulsars/lightcurves/a0535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='html 4 Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' by cross-correlating these pulse profiles with an averaged template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Then the spin evolution was determined by using the software Tempo2 (Hobbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We summarize the timing solutions for different outbursts used in the following pulsation search (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5) in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Fermi-LAT data set and reduction We used the Pass 8 data set (Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Bruel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018) available at the Fermi Science Support Center (FSSC)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This spans 166 months, from 2008 August 4 to 2022 June 9, with recon- structed energy in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1−300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We selected SOURCE class events (Front and Back) with a zenith angle smaller than 90◦ to avoid the Earth limb contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The events were further filtered based on the criteria ‘‘DATA QUAL>0 && LAT CONFIG==1’’ to get the good time intervals in which the satellite was working in standard data taking mode and the data quality was good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We did not apply a Region of Interest (ROI)-based zenith angle cut4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The data set was centered at 1A 0535+262 with coordinates (α, δ) = (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='◦7274, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='◦3158), with a radius of 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The coordinates of 1A 0535+262 were taken from the SIMBAD5 database and in the J2000 frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The analysis was performed using the P8R3 SOURCE V3 IRFs and the latest Fermitools6 (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0) available at the FSSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Fermi-LAT spectral analysis In the spectral analysis, the latest 4FGL-DR37 (gll psc v30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='fit) (Fermi-LAT collaboration 2022) sources within a 20◦ circle around 1A 0535+262 were included to build a complete spatial and spec- tral source model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We also included the latest Galactic interstellar emission model, “gll iem v07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='fits”, as well as the isotropic emission spectrum “iso P8R3 SOURCE V3 v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='txt”, with the latter taking into account the extragalactic emission and the residual instrumental background8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Both the normal- izations and spectral indices of sources within 5◦ around 1A 0535+262 were set free to vary except for 4FGL J0534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5+2201i, which is recommended to be fixed to account for the Inverse Compton Scat- tering component of the Crab Nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Extended sources were modeled using the 12-year templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Since the closest source in the model is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6◦ away from 1A 0535+262, we added 1A 0535+262 manually in the model as a point source with a simple Power Law spectral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This allows us to check whether the addition of such source is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Model fitting was performed in a 14◦ × 14◦ ROI using the maximum likelihood method (Mattox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We followed the binned likelihood procedure outlined in the FSSC using a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='◦1×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='◦1 pixel size and thirty logarithmic energy bins over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1−300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Two extra energy bins have been added to take into account the energy dispersion except for the isotropic component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The significance of a given source in the model is characterized by the Test Statistic (TS), which is expressed as TS = 2(log L − log L0), where log L and log L0 are the logarithms of the maximum likelihood of the complete source model and of the background model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' the source model without the given source included), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We first performed a global binned likelihood fit to the whole data set by fixing the spectral index of 1A 0535+262 to 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Such spectral indices are chosen to represent possible emission mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' At the same flux level, if the source were to emit a hard spectrum (as the 3 http://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='gov/ssc/ 4 https://fermi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='nasa.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='gov/ssc/data/access/lat/BackgroundModels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='html 5 assumed 2) across the Fermi-LAT energy regime, it should be easier to detect it due to the diminishing background at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Then, using the best-fit source model from the whole data set fit, we performed binned likelihood fits to the different X-ray outbursts that 1A 0535+262 has experienced in the past, following the same fitting setup and procedure as for the whole data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' To increase the detection possibility and statistics, we also stacked all the outbursts together to perform the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 1A 0535+262 was not detected in any of these cases and we therefore computed a 95% confidence level energy flux upper limit accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The fitting results are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Gamma-ray variability We performed two different types of variability analysis for 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' First, to investigate the long-term gamma-ray flux variability, we computed light curves with a 180-day binning as in Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022) in the energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1−300 GeV for all spectral indices used in the spectral analysis (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The full data best-fit source model was used as an input for each time bin and the normalizations of sources within 3◦ around 1A 0535+262 were set free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Upper limits at 95% confidence level were calculated when 1A 0535+262 had TS < 4 in a given time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We did not see any significant detection in all the time bins when fixing the spectral index to 2 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' For the light curve with index fixed to 3, there are two bins with TS being about 10 and 20, corresponding to approximately 3σ and 4σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' However, these two bins correspond to the period where its X-ray emission was faint according to the X-ray monitoring of 1A 0535+262 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Thus, we conclude that no correlation between gamma-ray and X-ray was observed, contrary to what Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022) has claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Furthermore, the variability significance was computed following the same method used in Acero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Only the light curve with index fixed to 3 has a non-negligible significance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7σ for 27 degrees of freedom), but this is far from declaring a significant variability, which requires usually a significance of at least 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Any gamma-ray emission is thus consistent with being steady on a timescale of a few months based on our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Since gamma-ray binaries usually exhibit orbital flux variability, we also computed the orbital flux for 1A 0535+262 with 10 bins per orbit and calculated upper limits at 95% confidence level when 1A 0535+262 had TS < 4 in a given orbital bin, as was done in Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Similar to the long-term light curve, the full data best-fit source model was used as an input for each orbital bin and the spectral index was fixed to 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' No significant orbital variability was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The orbital light curves are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Gamma-ray pulsation search We performed a pulsation search using reconstructed LAT photons within 1◦ of 1A 0535+262 in the energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1−300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The signal significance was qualified using the weighted H-test developed by Kerr (2011), which is based on the original one proposed by de Jager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (1989): Hmw = max � Z2 iw − c × (i − 1) � , 1 ≤ i ≤ m, (1) where Z2 mw = 2 �N i=1 w2 i × m � k=1 (α2 wk + β2 wk), (2) and αwk = N � i=1 wi cos (2πkφi), βwk = N � i=1 wi sin (2πkφi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (3) 6 Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Here, N is the total photon number, φi is the pulsar rotational phase and wi is the photon weight, m is the maximum search harmonic and c is the offset for each successive harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We used the standard value c = 4 and varied m in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We verified that taking the standard value m = 20 did not change the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We employed the Simple Weights method descried in Bruel (2019) and Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2019) to compute the weight for each photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This is considered as a proxy for the probability that the photon comes from 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Assuming that the target source is faint compared to the diffuse background and that the background emission is isotropic, for a photon with energy E (in MeV) and angular distance ∆θ to the target source, the weight is: w(E, ∆θ) = f(E) × g(E, ∆θ), (4) where f(E) = exp(−2log2 10(E/Eref)) (5) is the weight at the pulsar position, which depends on the pulsar and background spectra and on the LAT’s energy-dependent Point Spread Function (PSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The geometrical factor g(E, ∆θ) describes the angular distribution of the gamma-ray photons emitting from a point source and can be written as g(E, ∆θ) = � 1 + 9∆θ2 4σ2 psf(E) �−2 , (6) where σpsf(E) = � p2 0(E/100)−2p1 + p2 2 (7) is the PSF 68% containment angle with p0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='445, p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='848 and p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='084 for LAT P305 Pass 8 data (Atwood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Defining the reference energy Eref = 10µw, the H-test versus µw follows a Gaussian distribution (Bruel 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Thus, searching for pulsations means finding the maximum H-test by scanning over µw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' After searching a thousand radio pulsars for possible gamma-ray emission, Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2019) indicate that in most cases µw = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 is a good choice to yield a significant signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We adopted this value in our pulsation search for 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Considering that this value was found for non- accreting (and many isolated) radio pulsars that have a power law with an exponential cutoff (PLEC) spectrum (Abdo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2013), and thus may not be appropriate for an accreting system that could have a different spectral shape, we also verified that scanning over µw to find the best value does not affect the result significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We used the ephemerides for different observations (Table 2) to phase-fold the gamma-ray photons of 1A 0535+262, restricting the pulsation search range to the validity of the corresponding ephemerides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' However, no significant pulsation was detected during the individual outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Although a signal of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5σ was hinted for the 2009 outburst, the significance is far from the LAT detection threshold of 4σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' DISCUSSION In general, we obtained very different results compared to Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We had no detection of either persistent or transient or pulsed gamma-ray emission from 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Although we have two time bins with TS∼20 and ∼10, they did not correspond to the X-ray outburst periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Therefore, our result does not support their conclusion that the gamma-ray emission is correlated 7 with the X-ray outburst of the source and is mostly concentrated in specific orbital bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Such difference may mainly come from the fitting procedure and whether or not considering the energy dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We normally fitted the sources within 5◦ around 1A 0535+262 in the spectral fit and orbital flux variability study, and within 3◦ for the long-term light curves, while they fitted only 1◦ around 1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In addition, the spectral index was fixed to different values to account for possible emission mechanisms in our study, while they let the index free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Actually, for the long-term variability, no detailed fitting procedure information was found in Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2022), making a detailed comparison difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Unlike those cases when a detection is found from an astrophysical source, the possible reasons for a non-detection are essentially unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Gamma-ray binary systems such as LS I +61 303 are radio sources, and their X-ray spectra contain a significant non-thermal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Recently, radio pulsations were detected from this system (Weng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Thus, it is worth noting that LS I +61 303 is probably not a significantly accreting system, and thus it likely has a different acceleration and emission mechanism than that acting in 1A 0535+26, if there is any in the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' On the other hand, in 1A 0535+262, the absence of a significant quiescent radio emission that could later be associated with non-thermal processes may simply indicate that leptons are not sufficiently accelerated there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This observation had earlier promoted hadronic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' From our results, though, we can also rule out hadronic production acting according to a mechanism originally proposed by Cheng & Ruderman (1991), where a proton beam accelerated in a magnetospheric electrostatic gap impacts the transient accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This model was applied to 1A 0535+262 by Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2001), Anchordoqui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2003) and Orellana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Particularly in the latter paper the theoretical flux prediction of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 × 10−8 ph cm−2 (derived by extrapolating the result of Orellana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2007), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', a gamma- ray luminosity of about 1033 erg s−1 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 TeV at the end of giant outbursts to the Fermi-LAT energy range (see Acciari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2011) is above our limit which is ∼(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7)×1032 erg s−1 by one order of magnitude (and should have been detected earlier on in the mission) depending on different spectral indices assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Taking the X-ray luminosity (2−150 keV) reported for the largest 2020 outburst (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2021) to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2×1038 erg s−1, the ratio of Lγ to LX is then calculated to be (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9)×10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' It remains to be seen, of course, whether the flux was overestimated but the mechanism is still viable, or whether a longer integration might lead to low-flux quiescent emission or to an occasional detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' None of these possibilities has happened in the long integration time we have analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' At least with the current gamma-ray sensitivity, 1A 0535+262 is not a gamma-ray source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' With our current limits, there appears to be no reasonable combination of the hadronic model parameters to still accommodate a persistent gamma-ray flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Another interpretation could simply be that none of the possible shocks in the system is energetic enough to produce sufficiently accelerated particles able to emit gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Or that if they do, the radiation produced is absorbed due to the matter in the surroundings, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', the discussion in Orellana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Regarding the latter, it would be reasonable to expect that any absorption would be quite variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Thus if the emission is produced in the system at all, it would be unlikely that it is absorbed all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In addition, secondary electrons (and positrons) from the pair production process would generate gamma rays at MeV–GeV energies, (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Bednarek 1997, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Sierpowska-Bartosik & Torres 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' These could be attenuated by X-ray photons, but probably not fully attenuated after the X-ray peak has passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 8 Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Giant (or Type II) outbursts, though they are rare and unrelated to the orbital cadence, may have X-ray luminosities close to the Eddington limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' They are likely associated with the formation of a transient accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Recently, van den Eijnden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2020) detected a radio counterpart during the 2020 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This was the first time that a coupled increase in X-ray and radio flux was seen in 1A 0535+262 and shows that the radio emission may relate to the accretion state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This would be similar to the behaviour seen in the transient Be X-ray binary Swift J0243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6+6124 (van den Eijnden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Bednarek (2009a,b) proposed that it is possible that HMXBs produce gamma-ray emission during accretion periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The possibility that particle acceleration can take place even when mass accretion is going on is supported by some observational results: the hinted gamma-ray emission from SAX J1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4−3658 (de O˜na Wilhelmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2016), the gamma-ray emission found from the sub-luminous state of the transitional pulsars (as quoted in the introduction) and interpreted as propeller emission (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=', Papitto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Papitto & Torres 2015) or a mini pulsar wind nebula (Papitto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2019), see also Veledina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2019)) and finally, the discovery of optical and ultraviolet pulsed emission from the accreting millisecond pulsar SAX J1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4−3658 (Ambrosino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' However, the neutron star Eddington luminosity (LEdd ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 × 1038 erg s−1) is many orders of magnitude above our upper limits, pointing to a very inefficient mechanism, if at play at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Similarly to the case of other transients, for instance, the Be X-Ray Binary 4U 1036−56 (RX J1037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5−5647), which could be associated to AGILE transients, we cannot discard that a low level of gamma-ray flux is emitted at lower energies, below 100 MeV (see Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2012, and in particular their figure 6 for an associated discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This remains to be tested with the advent of MeV missions such as AMEGO (McEnery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2019), e-ASTROGAM (de Angelis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2018), or COSI (Beechert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' CONCLUSIONS We have searched for gamma-ray emission and pulsations from 1A 0535+262 using more than 13 years of Fermi-LAT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Neither persistent nor transient nor pulsed gamma-ray emission has been detected significantly in our study for the whole data set or during the X-ray outbursts that 1A 0535+262 has experienced since the launch of the Fermi satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Upper limits on the luminosity at the 95% confidence level were derived to be around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7)×1032 erg s−1 depending on different spectral indices assumed, the deepest ones to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The emission of 1A 0535+262 is considered to be consistent with being steady on a timescale of a few months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Although two time bins in the long-time light curve hint to have gamma-ray emission at roughly 3 and 4σ, they occurred when the source was faint in X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Thus, no correlation between gamma-ray and X-ray activities was observed based on our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' In addition, we did not see any significant orbital gamma-ray variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' We conclude that 1A 0535+262 is not a gamma-ray emitter at the level of the current gamma-ray sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 9 ACKNOWLEDGMENTS The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT, as well as scientific data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' These include the National Aeronautics and Space Administration and the Department of Energy in the United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' the Commissariat `a l’Energie Atomiqueand and the Centre National de la Recherche Scientifique/Institut National de Physique Nucl´eaire et de Physique des Particules in France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' and the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Wallenberg Foundation, the Swedish Research Council, and the Swedish Na- tional Space Board in Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’´Etudes Spatiales in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This work was performed in part under DOE Contract DE-AC02-76SF00515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' The authors are supported by the National Natural Science Foundation of China through grants U1938103, 12041303, 12173103, U2038101, 11733009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' WZ and DFT work have been supported by the grant PID2021- 124581OB-I00 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='13039/501100011033 and by the Spanish program Unidad de Excelencia Mar´ıa de Maeztu CEX2020-001058-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' DFT acknowledges as well USTC and the Chinese Academy of Sciences International Presidential Fellowship Initiative 2021VMA0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' REFERENCES Abdalla, H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 0 2 ph/cm2/s GSC 2-4 keV 0 2 4 ph/cm2/s GSC 4-10 keV 0 2 ph/cm2/s GSC 10-20 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 cts/s/det BAT 14-24 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1 cts/s/det BAT 24-50 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='05 cts/s/det BAT 50-100 keV 55000 56000 57000 58000 59000 Time (MJD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='02 cts/s/det BAT 100-195 keV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 Flux (ph cm 2 s 1) 1e 8 = 2 55000 56000 57000 58000 59000 Time (MJD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 TS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 Flux (ph cm 2 s 1) 1e 8 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 55000 56000 57000 58000 59000 Time (MJD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 TS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 Flux (ph cm 2 s 1) 1e 7 = 3 55000 56000 57000 58000 59000 Time (MJD) 0 5 10 15 20 TS Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Upper panel: Long-term X-ray light curves of 1A 0535+262 in different energy bands observed by Swift/BAT and MAXI /GSC as obtained from http://sakamotoagu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='mydns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='jp/bat gsc trans mon/web lc/ 1 Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='name=1A 0535+262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Low three panels: Long-term gamma-ray light curves of 1A 0535+262 with a time binning of 180 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Spectral index was fixed to 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Upper limits at 95% confidence level are computed for bins with TS<4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1e 12 Eflux (erg cm 2 s 1) = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 Orbital Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 TS 3 4 5 1e 12 Eflux (erg cm 2 s 1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 Orbital Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='20 TS 4 5 6 7 8 9 1e 12 Eflux (erg cm 2 s 1) = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 Orbital Phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 TS Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Orbital energy flux variation of 1A 0535+262 with 10 bins per orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Spectral index was fixed to 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Upper limits at 95% confidence level are computed for bins with TS<4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 14 Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Fermi-LAT spectral analysis results Perioda Time Range Spectral Index TS Energy Flux Upper Limitb (MJD) (10−12 erg cm−2 s−1) Whole dataset full 54682-59739 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6 full 54682-59739 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='7 full 54682-59739 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='2 Stacked outbursts Rising+Falling 55040-59207 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9 Rising+Falling 55040-59207 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 Rising+Falling 55040-59207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9 The 2009 double-peaked outburst Rising+Falling 55040-55070 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='9 Rising+Falling 55040-55070 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='4 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 Rising+Falling 55040-55070 3.' metadata={'source': 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Falling 59173-59207 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 Falling 59173-59207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='0 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='5 a Full: the whole dataset used in this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' ALL: the dataset including the rising, falling, apastron and periastron portions of the outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' b The upper limits are given at the 95% confidence level in the energy range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='1−300 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 15 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Parameters of the spin evolution of 1A 0535+026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' νi is the ith order derivative of the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' Parameters 2009 double outbursta 2009 outbursta 2011 outburstb 2020 outburstc Epoch (MJD) 55040 55166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='99 55616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='202 59170 Tstart (MJD) 55040 55166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='99 55608 59159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='15 Tstop (MJD) 55070 55201 55637 59207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='92 ν0(10−3 Hz) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='66041(4) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6618(1) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6793(1) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='66045(2) ν1(10−12 Hz s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='67(3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6(6) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='43(5) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='27(4) ν2(10−17 Hz s−2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='121(7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='17(4) ν3(10−23 Hz s−3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8(6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='43(9) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8(2) ν4(10−29 Hz s−4) 3(1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='67(5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='8(9) ν5(10−36 Hz s−5) 8(6) 737(7) ν6(10−39 Hz s−6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content='6(2) ν7(10−45 Hz s−6) 5(2) ν8(10−50 Hz s−6) 3(1) ν9(10−56 Hz s−6) 8(2) ν10(10−62 Hz s−6) 7(1) a: Derived form the Fermi/GBM monitoring (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' b: Adopted from Sartore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' c: Derived from Insight-HXMT observations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} +page_content=' 2022, see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdAzT4oBgHgl3EQfdfz3/content/2301.01423v1.pdf'} diff --git a/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/2301.04831v1.pdf.txt b/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/2301.04831v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7dba08b77d8b09e21160f7266e49e5198f5e7d12 --- /dev/null +++ b/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/2301.04831v1.pdf.txt @@ -0,0 +1,996 @@ +arXiv:2301.04831v1 [astro-ph.GA] 12 Jan 2023 +Draft version January 13, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Discovery of the Tadpole Molecular Cloud near the Galactic Nucleus +Miyuki Kaneko,1 Tomoharu Oka,1, 2 Hiroki Yokozuka,2 Rei Enokiya,2 Shunya Takekawa,3 Yuhei Iwata,4, 5 and +Shiho Tsujimoto1 +1School of Fundamental Science and Technology, Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, +Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan +2Department of Physics, Institute of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa +223-8522, Japan +3Faculty of Engineering, Kanagawa University 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa 221-8686. Japan +4Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +5Center for Astronomy, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan +(Received 2022 August 18; Revised 2022 November 23; Accepted 2022 November 25) +Submitted to ApJ +ABSTRACT +In this paper, we report the discovery of an isolated, peculiar compact cloud with a steep velocity +gradient at 2.′6 northwest of Sgr A*. This “Tadpole” molecular cloud is unique owing to its charac- +teristic head-tail structure in the position-velocity space. By tracing the CO J=3–2 intensity peak +in each velocity channel, we noticed that the kinematics of the Tadpole can be well reproduced by +a Keplerian motion around a point-like object with a mass of 1×105 M⊙. Changes in line intensity +ratios along the orbit are consistent with the Keplerian orbit model. The spatial compactness of the +Tadpole and absence of bright counterparts in other wavelengths indicate that the object could be an +intermediate-mass black hole. +Keywords: galaxies: nuclei — Galaxy: center — ISM: clouds — ISM: molecules +1. INTRODUCTION +It is widely accepted that large galaxies host a +central supermassive black hole (SMBH) with mil- +lions to billions times the mass of the Sun (e.g., +Kormendy & Richstone 1995; Kormendy & Ho 2013). +A potential scenario for SMBH formation is based on +intermediate-mass black holes (IMBHs), which have +masses of 102–5 M⊙ (e.g., Mezcua et al. 2017). +Thus, +detecting and studying IMBHs in detail are essential for +understanding the formation and evolution of galactic +nuclei. Numerous IMBH candidates have been identi- +fied in centers of globular clusters (e.g., Kızıltan et al. +2017), in nuclei of dwarf galaxies (e.g., Reines et al. +2013; Baldassare et al. 2015), or as ultraluminous X-ray +sources in extragalaxies (e.g., Farrell et al. 2009). +In the central molecular zone (CMZ) of our Galaxy, +a number of compact (d < 5 pc) clouds with extraor- +Corresponding author: Miyuki Kaneko +miyukikaneko@keio.jp +dinary broad velocity width (∆V > 50 km s−1) have +been detected (e.g., Oka et al. 1998, 1999, 2012, 2022). +These peculiar clouds, namely, high velocity-dispersion +compact clouds (HVCCs), have been assumed to be +accelerated by supernova explosions, protostellar out- +flows, and/or cloud-to-cloud collisions (e.g., Oka et al. +2022). Subsequently, it was determined that the kine- +matics of CO–0.40–0.22, which is one of the most en- +ergetic HVCCs, can be well reproduced by a cloud +being gravitationally kicked by a point-like mass of +∼ 105 M⊙ (Oka et al. 2016, 2017). +The absence of +any bright object near the point-like mass suggest that +it may be an IMBH. Subsequently, it was also sug- +gested that HVCCs HCN–0.009–0.044 (Takekawa et al. +2019a), HCN–0.085–0.094 (Takekawa et al. 2020), and +CO–0.31+0.11 (Takekawa et al. 2019b) were driven by +an IMBH. These discoveries yielded a new method of +finding non-luminous massive objects, such as inactive +and wandering BHs. To date, five IMBH candidates, in- +cluding IRS13E (Tsuboi et al. 2019), have been reported +in the Galactic CMZ. + +2 +Kaneko et al. +1000 +2000 +3000 +0 +–0.05 +–0.1 +0 +–0.05 +Galactic Longitude [deg] +Galactic Latitude [deg] +Tadpole +CO 0.02–0.02 +Sgr A* +NLE +C1 +250:3000@250 +I–200to+200 + 5 I–150to–80 +Figure 1. Map of velocity-integrated CO J=3–2 emission. White contours are drawn at 250 K km s−1 intervals from 250 K +km s−1. The integrated intensity was calculated by +� 200 +−200 TMBdV +5 +� −80 +−150 TMBdV to emphasize the Tadpole, which appears at +(l, b)=(−0.◦090, −0.◦014). CO 0.02–0.02 (Oka et al. 1999, 2008), the C1 cloud (Oka et al. 2011; Takekawa et al. 2017), and the +negative longitude extension of the circumnuclear disk (NLE; Oka et al. 2011; Takekawa et al. 2017) also appear in this map. +The white rectangle indicates the area presented in Figures 2–5 +When searching for gravitationally kicked gas in the +CMZ, we noticed an isolated HVCC in the CO J=3– +2 data obtained with the James Clerk Maxwell Tele- +scope (JCMT; Parsons et al. 2018; Eden et al. 2020). +It appears as an isolated compact cloud at (l, b) ≃ +(−0.◦090, −0.◦014), which corresponds to ∼ 2.′6 Galac- +tic northwest of Sgr A* (Figure 1), with LSR ve- +locities between −140 km s−1 and −90 km s−1. +It +stands out with its peculiar appearance and very high +CO J=3–2/CO J=1–0 intensity ratio (R3–2/1–0 = 1.8 +Oka et al. 2022), which exceeds double of the CMZ av- +erage (R3–2/1–0 ∼0.7 Oka et al. 2007, 2012). In this pa- +per, we report the discovery of the so-called “Tadpole” +molecular cloud, which is listed as id.75 in the catalog +of Oka et al. (2022). Throughout this paper, the dis- +tance to the Galactic center is assumed to be D = 8.3 +kpc (Gravity Collaboration 2018). +2. DATA +We first discovered the Tadpole in the JCMT CO +J=3–2 data, and confirmed it in the CO J=1–0 and +CS J=2–1 line data obtained with the Nobeyama Radio +Observatory (NRO) 45 m telescope. These data sets are +briefly described below. +2.1. CO J=3–2 Line +The 12CO J=3–2 line (345.795990 GHz) observations +of the CMZ were performed using the JCMT from 2013 +July to 2014 July (Parsons et al. 2018). The Heterodyne +Array Receiver Program (HARP; Buckle et al. 2009) +and autocorrelation spectral imaging system (ACSIS) +were used during these observations. +The half-power +beam width (HPBW) of the telescope was approxi- +mately 14′′ at 345 GHz. The ACSIS was operated in the +1 GHz bandwidth (976.56 kHz resolution) mode. During +these observations, the system noise temperature (Tsys) +ranged between 100–200 K. The rms noise level of the +image cubes was between 0.4 K and 0.84 K. The details +of the CO J=3–2 data and JCMT observations are pre- +sented in Parsons et al. (2018). We use the data after +resampling onto a 7.′′5×7.′′5×1 km s−1 grid. +2.2. CO J=1–0 Lines +The CO J=1–0 observations of the CMZ were per- +formed using the Nobeyama Radio Observatory (NRO) +45 m telescope (Tokuyama et al. 2019). The 12CO J=1– +0 (115.27120 GHz) line data were obtained from 2011 +January 19 to 29 using the 25 beam array receiver sys- +tem (BEARS; Sunada et al. 2000). As the receiver back- +end, the AC45 spectrometer system (Sorai et al. 2000) +was employed in the 500 MHz bandwidth (0.5 MHz res- +olution) mode. The 13CO J=1–0 (110.20135 GHz) line +data were obtained from 2016 February to March us- +ing the four-beam receiver system on the 45 m telescope +(FOREST; Minamidani et al. 2016). The spectral anal- +ysis machine on the 45 m telescope (SAM45; Kuno et al. +2011; Kamazaki et al. 2012) was operated in the 1 GHz +(244.14 kHz resolution) mode. The HPBW of the tele- +scope was approximately 15′′ at 115 and 110 GHz. The +typical Tsys was ∼800 K and 150–300 K during the 12CO + +Discovery of the “Tadpole” Molecular Cloud +3 +10 +8 +6 +4 +2 +40 +30 +20 +10 +150 +120 +90 +60 +30 +–0.07 +–0.08 +–0.09 +–0.10 +–0.11 +0.00 +–0.01 +–0.02 +–0.03 +–0.04 +–100 +–150 +Galactic Latitude [deg] +LSR Velocity [km s–1] +–80 +( b ) +( a ) CO J=3–2 +12CO J=1–0 +13CO J=1–0 +200 +150 +100 +50 +( c ) +( d ) +( e ) +–110 +–140 +–90 +–120 +–130 +CS J=2–1 +10 +6 +2 +8 +4 +Galactic Longitude [deg] +1 pc +–0.07 +–0.08 +–0.09 +–0.10 +–0.11 +Galactic Longitude [deg] +HPBW +HPBW +HPBW +HPBW +Figure 2. (a) Map of velocity-integrated CO J=3–2 emission. The velocity range for integration is VLSR =−140 to −80 km s−1. +The intensity unit is K km s−1. (b) Longitude-velocity map of CO J=3–2 emission at b = −0.◦014 (the magenta line in panel +(a)). (c) Map of velocity-integrated 12CO J=1–0 emission. (d) Map of velocity-integrated 13CO J=1–0 emission. (e) Map of +velocity-integrated CS J=2–1 emission. The integration ranges for these lines are the same as that of the CO J=3–2 line. +and 13CO observations, respectively. The data were re- +sampled onto a 7.′′5×7.′′5×2 km s−1 grid. The rms noise +levels of the resultant 12CO and 13CO data cubes were +1.0 K and 0.2 K in main-beam temperature (TMB), re- +spectively. +2.3. CS J=2–1 Line +The CS J=2–1 (97.98096 GHz) line observations of +the CMZ were performed during the NRO 45 m Tele- +scope Large Program through 2019 January–May and +2020 January–April. +The mapping area was set to +−1.◦5 ≤ l ≤ +1.◦5 and −0.◦25 ≤ b ≤ +0.◦25. The FOR- +EST receiver and SAM45 spectrometer were used. The +SAM45 was operated in the 1 GHz bandwidth (244.14 +kHz resolution) mode. The HPBW of the telescope was +≃ 19′′ at 86 GHz. +The Tsys ranged from 150–300 K +during the CS J=2–1 line observations. The data were +resampled onto a 7.′′5×7.′′5×2 km s−1 grid. The rms noise +level of the resultant CS J=2–1 data cube was 0.14 K +in TMB. +The details of the CS data and NRO 45 m +observations will be presented in the forthcoming paper +(Takekawa et al. 2022, in preparation). +20 +10 +0 +–10 +–20 +Intensity [K] +CO J=3–2 +12CO J=1–0 +CS J=2–1 × 3 +–30 +0 +–50 +–100 +–150 +–200 +LSR Velocity [km s–1] +50 +100 +150 +13CO J=1–0 × 2 +200 +Tadpole +–40 +–50 +Figure 3. +Observed line spectra at (–0 .◦091, –0 .◦016) at +where the CO J=3–2 profile of the Tadpole is widest. Black, +red, blue and green lines show CO J=3–2, 12CO J=1–0, 13CO +J=1–0 and CS J=2–1 line spectra, respectively. + +4 +Kaneko et al. +3. RESULTS +3.1. Spatial and Velocity Structure +Figure 2 shows the spatial and velocity structure of the +Tadpole in various molecular lines. The Tadpole appears +as a compact, well-defined clump in velocity-integrated +maps (Figure 2a, c–e). In the CO J=3–2 integrated map +(Figure 2a), the full width of half maximum (FWHM) +angular diameter of the Tadpole is 28′′. This angular +diameter is 2 times the angular resolution (14′′), cor- +responding to a distance of 1.1 pc from the Galactic +center. In the 13CO J=1–0 and CS J=2–1 maps, the +Tadpole has angular sizes similar to that depicted in +the CO J=3–2 map, while the 12CO J=1–0 map depicts +an appearance that is larger by a factor of 2 compared +with those depicted in the other maps. +The velocity extent of the Tadpole is from VLSR ≃ +−135 km s−1 to −90 km s−1 (Figure 2b), indicating that +it may be in the CMZ. The longitude-velocity behavior +of the Tadpole is characterized by a “head-tail” struc- +ture, from which its name is derived. It depicts a steep +velocity gradient of |∆V/∆l|∼16 km s−1pc−1. The tail +is obscure in the 13CO J=1–0 and CS J=2–1 data sets. +We show the spectra of CO and CS lines in Figure 3. +We learned that CO J=3–2 line can trace the kinemat- +ics of the Tadpole best because of its high intensity and +well-defined appearance in the l–b–V space. Figure 4 +shows the velocity channel maps of CO J=3–2 emission. +The Tadpole appears with an elliptical shape in each +velocity channel, changing its angular size with respect +to velocity. It has the largest angular size of 36′′×24′′ +in the VLSR = −130 to −125 km s−1 channel, while the +smallest size of ∼20′′ is detected at both velocity ends. +These angular sizes, which are comparable to the tele- +scope’s HPBWs, indicate that the spatial structure of +the Tadpole is not well resolved using these single-dish +observations; thus, the actual angular size must be far +smaller than those observed. +The CO J=3–2 line emission shows a maximum inten- +sity of 64 K km s−1 in the −130 to −125 km s−1 channel. +We identified that the intensity maximum pixel in the +Tadpole at each velocity channel continuously changes +with respect to velocity. Note that these intensity max- +imum pixels trace an arc in the plane of the sky (Figure +4, bottom right panel). This behavior will be analyzed +in more detail in the discussion section (§4.1). +3.2. Physical Parameters +The physical size of the Tadpole was evaluated as +the size parameter; defined by S ≡ D tan +�√σlσb +� +to +be 2.2 pc. +The velocity dispersion was calculated to +be σV = 22 km s−1. +These give a size-linewidth co- +efficient (σV /S0.5) of 15 km s−1pc−0.5 and virial theo- +rem mass (MVT ≡ 8.7Sσ2 +V /G) of 4.7 × 105 M⊙. +The +molecular gas mass (Mgas) was estimated by summing +the CO J=3–2 line integrated intensity and using the +CO(3–2)-to-H2 conversion factor (XCO3–2 = 1.4×1020 +[cm−2 (K km s−1)−1] Oka et al. 2022) to be 6.6×102 M⊙. +The XCO3–2 we employed is close to 1×1020 that esti- +mated from an LVG calculation for n(H2)=1×104 cm−3, +Tk =60 K, and N(CO)/dV =1×1017 cm−2 (km s−1)−1. +Using the physical parameters described above, we de- +rived the dynamical timescale (tdyn ≡ S/σV ), kinetic +energy (Ekin ≡ 1.5Mgasσ2 +V ), and kinetic power (Pkin ≡ +Ekin/tdyn) of the Tadpoe as 2.2×104 yr, 9.4×1048 erg, +and 1.4×1038 erg s−1, respectively. The kinetic power of +the Tadpole is equal to 3.7 × 104 L⊙ which is far greater +than those of molecular outflows from massive YSOs +(0.01–100 L⊙; Maud et al. 2015). +The virial theorem mass and molecular gas mass of +the Tadpole yield a significantly high virial parameter, +MVT/Mgas ∼ 700. This indicates that the Tadpole is +not in gravitational equilibrium. Following the method +described in Stark et al. (1989), we estimated the self- +gravity of the Tadpole and tidal force by the Galactic +potential. The self-gravity of the cloud estimated here +(∼10−10 m s−2) is 2.5 orders of magnitude weaker than +the tidal force (∼ 4×10−8 m s−2). These assessments +clearly demonstrate that the Tadpole cannot be bound +by its self-gravity. This strongly indicates to the pres- +ence of an object with a mass comparable to the virial +mass (4.7×105 M⊙) inside the Tadpole. +3.3. Multiwavelength View +To search for the driving source behind the Tad- +pole, we referred to the 1.284 GHz data obtained using +MeerKAT (Heywood et al. 2022), mid-infrared data ob- +tained using the Spitzer Space Telescope (Ram´ırez et al. +2008; Churchwell et al. 2009), and X-ray data obtained +at the Chandra X-ray Observatory (Muno et al. 2009). +No radio source brighter than 0.4 mJy beam−1 at +1.284 GHz was detected near the angular extent of the +Tadpole, while a faint filament was seen in the Galactic +south (Figure 5a). In the mid-infrared image (Figure +5b), we identified a faint triangular rim which may be +the irradiated surface of the Tadpole. Considering the +angular resolution of the Spitzer (∼2′′), the mid-infrared +rim defines the angular extent of the Tadpole better than +the CO J=3–2 appearance. We also checked the AT- +LASGAL (870µm) and Hi-GAL (70, 160, 250, 350 and +500 µm) images (Contreras et al. 2013; Molinari et al. +2016), and found no separate feature toward the Tad- +pole above the sea of intense dust emission. A bright +point-like source near the Galactic southwestern edge +of the Tadpole is the long-period variable star V4872 + +Discovery of the “Tadpole” Molecular Cloud +5 +–110 to –105 +–95 to –90 +–100 to –95 +–115 to –110 +–120 to –115 +–125 to –120 +–135 to –130 +–140 to –135 + –0.09 + –0.08 + –0.07 + –0.11 + –0.10 + –0.02 + –0.03 + –0.04 + 0.00 + –0.01 +Galactic Longitude [deg] +Galactic Latitude [deg] +–90 to –85 +–130 to –125 +–105 to –100 +0 +30 +40 +20 +10 +50 +1 pc +HPBW +Figure 4. Velocity channel maps of the CO J=3–2 line from VLSR =−140 to −85 km s−1 near the Tadpole. The angular area +is the same as that depicted in Figure 2a, c–e. The intensity unit is K km s−1. The velocity range for integration is indicated at +the bottom right corner of each panel. The magenta cross denotes the position of the maximum intensity pixel in each panel. +The distribution of the maximum intensity pixels is shown in the bottom right panel. +Sgr (Matsunaga et al. 2009). In addition, we also iden- +tify dozens of point-like, mid-infrared sources toward the +Tadpole. The X-ray image also shows numerous faint +point-like sources toward the Tadpole (Figure 5c). Al- +though the nature of these mid-infrared/X-ray point-like +sources is unclear, we will refer to them in the discus- +sion section (§4.5). In short, the multiwavelength view +confirms the absence of any energetic objects that can +drive the Tadpole. +4. DISCUSSION +4.1. Tracing Molecular Gas Kinematics +The continuous change in the CO J=3–2 intensity +maximum pixel along the arc with velocity (§3.1) sug- +gests that the bulk of warm molecular gas belongs to +a certain trajectory, such as a closed orbit. To deter- +mine the orbit trajectory accurately, we constructed a +CO J=3–2 data cube with a 2′′×2′′×1 km s−1 grid. Then, +we calculated the accurate intensity peak position as the +CO emission center of gravity using 5×5 pixels around +the intensity maximum pixel in the Tadpole for each 1 +km s−1 width velocity channel. +Figure 6 shows the l–b–V and l–b distributions of the +intensity peak position in the Tadpole for each velocity +channel. The arc-shape apparent in the l–b distribution +became clearer than that depicted in Figure 4. +The +l–b–V distribution demonstrates the striking continuity +of peak positions in the velocity direction, indicating +that the bulk of warm molecular gas in the Tadpole may +predominantly follow a certain closed orbit. +4.2. Keplerian Orbit Fitting +The arc-shaped spatial structure and continuous ve- +locity change along the arc suggest that a rotational mo- +tion exists. Thus, we expect that the l–b–V behavior of +warm molecular gas in the Tadpole can be reproduced by +a Keplerian orbit around a point mass1. We performed +the fitting of a Keplerian orbit to the intensity peak +1 Here we do not consider parabollic or hyperbolic orbits, although +the orbit could not be an elliptical. + +6 +Kaneko et al. +–0.07 +–0.08 +–0.09 +–0.10 +–0.11 +0.00 +–0.01 +–0.02 +–0.03 +–0.04 +Galactic Latitude [deg] +( a ) +Galactic Longitude [deg] +0.00 +–0.01 +–0.02 +–0.03 +–0.04 +( b ) +0.00 +–0.01 +–0.02 +–0.03 +–0.04 +( c ) +MeerKAT +1.284 GHz +Chandra +4.7~8.0 kev +3.3~4.7 kev +0.2~3.3 kev +Spitzer +4.5 μm +8.0 μm +5.8 μm +0.2 +0.4 +0.6 +0.8 +0.0 +[ mJy beam–1 ] +Figure 5. Multiwavelength view near the Tadpole. White +contours show the CO J=3–2 integrated intensity with a 50 +K km s−1 interval. (a) Radio continuum image at 1.284 GHz +obtained with the MeerKAT. (b) Composite mid-infrared im- +age obtained with the Spitzer Space Telescope. The image +depicts 8.0 µm band flux in red, 5.8 µm in green, and 4.5 +µm in blue. (c) Composite X-ray image obtained with the +Chandra X-ray Observatory. The image depicts 0.2–3.3 keV +count rate in red, 3.3–4.7 keV in green, and 4.7–8.0 keV in +blue. +positions (Figure 6) following the method described in +Zhao et al. (2009). +We determined nine parameters. +Five of these parameters are three-dimensional orbital +parameters; namely, the semimajor axis (a), eccentric- +ity (e), longitude of ascending node (Ω), argument of +–0.005 +VLSR [km s–1] + –90 + –140 + –130 + –120 + –110 + –100 +–0.020 +–0.025 +–0.015 +–0.010 +Galactic Latitude [deg] +–0.090 +–0.095 +–0.085 +–0.105 +Galactic Longitude [deg] +–0.100 + –150 + –80 +Figure 6. The l–b–V distribution of CO J=3–2 intensity +peak positions in the Tadpole in each 1 km s−1 width velocity +channel. The head and tail of the Tadpole are denoted by +blue and green, respectively. +The l–b distribution is also +shown in the base plane with CO J=3–2 contours. +Grey +lines represent the loci of the best-fit Keplerian orbit (Table +1). The black and red filled circles denote the positions of +the dynamical center and pericenter, respectively. The black +arrow indicates the direction of rotation. +Table 1. Parameters of the best-fit Keplerian Orbit +Parameter +Value +Mass (Mdyn) +(1.01 ± 0.05) × 105 M⊙ +Semimajor axis (a) +0.75 ± 0.01 pc +Eccentricity (e) +0.35 ± 0.03 +Inclination (i) +112◦ ± 1◦ +Argument of pericenter (ω) +82◦ ± 5◦ +Longitude of ascending node (Ω) +−35◦ ± 1◦ +Longitude offset (l0) +−0.◦0946 ± 0.◦0001 +Latitude offset (b0) +−0.◦0146 ± 0.◦0001 +Velocity offset (V0) +−114 ± 1 km s−1 +pericenter (ω), and inclination angle (i). The remaining +parameters are the mass (Mdyn), line-of-sight velocity +(Vdyn) and l–b position (l0, b0) of the dynamical center. +The distance to the Tadpole is assumed to be same as +that to the Galactic center. The χ2 minimization ap- +proach was utilized in the l–b–V space. The χ2 consists +of two terms, namely, the spatial (χ2 +s) and velocity terms +(χ2 +v). The spatial term is defined by the sum of the or- +thogonal distances between the peak position and the +modeled orbit divided by the square of the positional +uncertainty, which was set to 0.15 pc (3.′′6). The veloc- +ity term is, similarly, the sum of the velocity deviations +between those of the peak positions and nearest posi- + +Discovery of the “Tadpole” Molecular Cloud +7 +tions in the modeled orbit divided by a square of the +velocity uncertainty, which was set to 1 km s−1. +The best-fit parameters with 1σ uncertainties are +listed in Table 1. The l–b–V locus of the orbit is pre- +sented in Figure 6. +Note that the best-fit solution is +bivalent because the orbits with i → (180◦ − i) and +Ω→(Ω + 180◦) yield the same l–b–V locus. The fitting +result indicates the presence of a point mass of 1.0×105 +M⊙ in the northwestern periphery of the Tadpole (Fig- +ure 6). +0.8 +0.6 +0.4 +0.2 +0.0 +0.4 +0.3 +0.2 +0.1 +0.0 +acceleration [ 10–7 m s–2 ] +0.5 +diameter of cloud [ pc ] +1.0 +self gravitation +tidal force at pericenter +tidal force at apocenter +Figure 7. Plots of the tidal force due to the point mass +(solid line) and self-gravity of the cloud (dashed lines) work- +ing on the head versus the head diameter. A central mass +of 105 M⊙ and head mass of 660 M⊙ were assumed. The +thick solid line indicates self-gravity at the pericenter dis- +tance (0.49 pc) while the thin solid line indicates that at the +apocenter (1.01 pc). +4.3. Tidal Stability of the Head +We examined the tidal stability of the head, assuming +the Newtonian potential of a point mass, following the +method described in Stark et al. (1989). The tidal force +caused by the point mass and self-gravity doing work +on the head were calculated as functions of the head +diameter (Figure 7). A central mass of 1×105 M⊙ and +head mass of 660 M⊙ were assumed. The curves of the +two forces at the pericenter of the best-fit orbit (0.5 pc) +intersect at d = 0.14 pc, while those at the apocenter +intersect at d = 0.35 pc. +Because the observed head +diameter (∼ 1 pc) is larger than the critical value (0.14 +pc) by a factor of seven, the Tadpole head is tidally +unstable at the pericenter. If the head includes a self- +gravitating core smaller than 0.14 pc, it may be able to +survive several turns. It is possibly that the Tadpole +has been trapped by the gravitational potential of the +105 M⊙ point mass, being stretched by the strong tidal +force to form the characteristic head-tail structure. This +situation is similar to the simulation of a cloud tidally +disrupted by a super massive black hole (Saitoh et al. +2014). Note that the Tadpole must have been lost its +angular momentum to be captured during the encounter +with the point mass. +0 +90 +180 +270 +0 +1 +2 +3 +4 +5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +5 +10 +15 +φ [degrees] +TMB [K] +R3–2/1–0 +RCS/CO +(a) +CO 1–0 +CO 3–2 +CS 2–1 +(b) +Figure 8. (a) Plots of CO J=3–2 (filled circles), CO J=1–0 +(crosses), and CS J=2–1 (open circles) main-beam tempera- +tures at the CO J=3–2 intensity peaks versus orbital phase. +(b) Plots of R3–2/1–0 (filled circles) and RCS/CO (open cir- +cles) at the CO J=3–2 intensity peaks versus orbital phase. +The phase ranges of the head and tail of the Tadpole are +depicted in blue and green shades, respectively. +4.4. Intensity Ratios along the Orbit +Here, we refer to the CO J=1–0, CO J=3–2, and CS +J=2–1 line intensities, as well as CO J=3–2/CO J=1– +0 (R3–2/1–0) and CS J=2–1/CO J=1–0 (RCS/CO) line +intensity ratios to examine the Keplerian orbit scenario. +The critical densities of the CO J=1–0, CO J=3–2, and +CS J=2–1 transitions are approximately 102.5, 104, and +105 cm−3, respectively, while their upper state energies +are 5.6, 33, and 7.1 K, respectively. Thus, RCS/CO is +sensitive to the variations in density while R3–2/1–0 is +affected by both the temperature and density. Figure 8 +shows plots of line intensities and intensity ratios along +the best-fit orbit. The motion along the orbit increases +the orbital phase (φ) at all times, where the pericenter +of the orbit corresponds to φ=0◦. + +8 +Kaneko et al. +Both ratios exhibit higher values at the head, indicat- +ing that both the density and temperature are enhanced +there. At the beginning of the tail, φ ∼ 170◦, R3–2/1–0 +demonstrates a prominent peak while RCS/CO does not. +This suggest that the temperature is enhanced at the +beginning of the tail. +Then, both ratios decrease to- +ward the tip of the tail. +The higher temperature in +the head may have been caused by the shock occurring +at the first encounter with the point mass. This shock +may be caused by collisions of gas clumps at the peri- +center, where adjacent orbits are very close. In a fluid +mechanical treatment, the same process is described as +the “tidal compression”(Saitoh et al. 2014). The gas ac- +cumulation near the apocenter and/or the presence on +a self-gravitating core can cause the higher density in +the head. The tail may be really the low density, tidally +stretched tail of the head. +The temperature increase +at the beginning of the tail could be caused by another +shock at the congestion near the apocenter. Thus, the +behaviors of intensity ratios are consistent with the Ke- +plerian orbit scenario for the Tadpole. +–0.090 +–0.095 +–0.100 +–0.010 +–0.015 +–0.020 +Galactic Latitude [deg] +Galactic Longitude [deg] +Figure 9. Distribution of X-ray point-like sources (open yel- +low circles: Muno et al. 2009, filled yellow circles: Zhu et al. +2018) superimposed on the composite mid-infrared image +(color; same as Fig.5b). The red cross denotes the dynamical +center of the best-fit Keplerian orbit, and the green dotted +circle has a 5′′ radius. Black contours highlight the CO J=3– +2 integrated intensity with a 50 K km s−1 interval. +4.5. What is a Point Mass? +The Keplerian orbit model requires the presence of +a huge point mass at the dynamical center. +What is +this point-like massive object? The mass of 105 M⊙ is +larger than the Arches or Quintuplet clusters (∼104 M⊙ +Figer et al. 1999a,b). According to the best-fit model, +a mass of 105 M⊙ must be concentrated within a ra- +dius significantly smaller than 0.5 pc (the pericenter dis- +tance), resulting in an average mass density higher than +that of the Arches cluster (ρ∼2×105 M⊙ pc−3 Espinoza +2009). +Anyway, the absence of a bright infrared +counterpart toward the Tadpole (Ram´ırez et al. 2008; +Churchwell et al. 2009; Molinari et al. 2011) clearly +rules out a stellar cluster from being a candidate for +the point mass. +The most promising candidate for the point-like mas- +sive object in the Tadpole may be an IMBH. We +searched for a counterpart for this point mass refer- +ring to X-ray images (§3.3). Figure 9 shows the posi- +tions of point-like sources in the X-ray images super- +imposed on the composite mid-infrared images around +(l, b)=(l0, b0). The small statistical uncertainties in the +l–b position of the dynamical center (Table 1) should +be considered with some caution because the entire gas +mass in the Tadpole is not confined within the single +closed orbit. Notice that three point-like sources in the +X-ray and mid-infrared reside within a 5′′ (∼10 times of +the positional ambiguity of the dynamical center) angu- +lar distance from (l0, b0) (Table 2). One source, SSTGC +489898, is apparent in both X-ray and mid-infrared im- +ages. The X-ray detected sources, SSTGC 489898 and +CXOGC 174526.9–290124, exhibit hard spectra, indi- +cating that they may be at the same distance to the +Galactic center. No time variation has been detected +from these sources (Zhu et al. 2018). If we assume the +standard accretion disk model (L = ˙Mc2/12), their X- +ray luminosities correspond to +˙M ∼ 1×10−14 M⊙yr−1. +This very low mass accretion rate could be a challenge +to our interpretation. Anyway, we suppose these three +point-like sources are candidates for the luminous coun- +terpart of the point mass. Although we currently have +very little knowledge on these point-like sources, they +should be considered as candidates for an IMBH in fu- +ture studies. +5. CONCLUSIONS +We have discovered the so-called Tadpole, which is an +isolated, peculiar compact cloud with an extraordinary +velocity width and very high CO J=3–2/J=1–0 intensity +ratio, at 2.′6 northwest of Sgr A*. Our main conclusions +can be summarized as follows: +1. The Tadpole molecular cloud has a size of ∼1 pc at +the distance to the Galactic center and a velocity +width of ∼50 km s−1. +2. It demonstrates the characteristic “head-tail” +structure in position-velocity space, having a steep +velocity gradient of 16 km s−1pc−1. + +Discovery of the “Tadpole” Molecular Cloud +9 +Table 2. Properties of X-ray and mid-infrared sources near the point mass +l +b +F2–10keV +a +[3.6 µm]b +[4.5 µm]b +[5.8 µm]b +[8.0 µm]b +ID +(deg) +(deg) +(10−18 W/m2) +(mag) +(mag) +(mag) +(mag) +SSTGC 489898b +–0.09513 +–0.01486 +1.66 +10.284 +· · · +· · · +· · · +SSTGC 490125b +–0.09384 +–0.01444 +· · · +9.154 +9.003 +8.422 +8.217 +CXOGC 174526.9–290124c +–0.09400 +–0.01346 +16.72 +· · · +· · · +· · · +· · · +Note— +a The X-ray (2–10 keV) energy flux quoted from Zhu et al. (2018). +b The source IDs and mid-infrared magnitudes are quoted from Ram´ırez et al. (2008). +c The source ID defined in Muno et al. (2009). +3. Its kinematics is well reproduced by a Keplerian +motion around a point-like object with a mass of +1×105 M⊙. +4. It is plausible that the Tadpole has been trapped +by the gravitational potential of the huge point +mass, now being stretched by the strong tidal +force. +5. The behaviors of line intensity ratios (R3–2/1–0 +and RCS/CO) are consistent with the Keplerian or- +bit scenario. +6. The absence of bright objects near the putative +point-like object suggests that the object is an in- +active intermediate-mass black hole (IMBH). +These results are based on molecular line maps with +14′′–19′′ resolutions obtained using single-dish tele- +scopes. Future aperture synthesis observations of molec- +ular lines with millimeter/submillimeter arrays will be +able to delineate the orbit stream directly, increasing +the reliability of the Keplerian orbit model for the Tad- +pole. +The results presented in this paper are based on +data obtained using the Nobeyama Radio Observatory +(NRO) 45-m telescope and James Clerk Maxwell Tele- +scope (JCMT). The NRO 45-m radio telescope is oper- +ated by the Nobeyama Radio Observatory, a division of +the National Astronomical Observatory of Japan. +The James Clerk Maxwell Telescope is operated by +the East Asian Observatory on behalf of The National +Astronomical Observatory of Japan, Academia Sinica +Institute of Astronomy and Astrophysics, Korea Astron- +omy and Space Science Institute, National Astronomical +Research Institute of Thailand, and Center for Astro- +nomical Mega-Science (as well as the National Key R& +D Program of China with No. 2017YFA0402700). Ad- +ditional funding support is provided by the Science and +Technology Facilities Council of the United Kingdom, +and participating universities and organizations in the +United Kingdom and Canada. +We are grateful to the NRO staff and all members of +the JCMT team for operating the telescope. T.O. ac- +knowledges the financial support of JSPS Grant-in-Aid +for Scientific Research (A) No. +20H00178. S. Ta ac- +knowledges support from JSPS Grant-in-Aid for Early- +Career Scientists Grant Number JP19K14768. +REFERENCES +Baldassare, V. F., Reines, A. E., Gallo, E., & Greene, J. E. +2015, ApJL, 809, L14 +Buckle, J. V., Hills, R. E., Smith, H., et al. 2009, MNRAS, +399, 1026 +Churchwell, E., Babler, B. L., Meade, M. R., et al. 2009, +PASP, 121, 213 +Contreras, Y., Schuller, F., Urquhart, J. S., et al. 2013, +A&A, 549, A45 +Eden, D. J., Moore, T. J. T., Currie, M. J., et al. 2020, +MNRAS, 498, 5936 +Espinoza, P., Selman, F. J., & Melnick, J. 2009, A&A, 501, +563 +Farrell S. A., Webb N. A., Barret D., Godet O., & +Rodrigues J. M. 2009, Nature, 460, 73 +Figer, D. F., Kim, S. S., Morris, M., et al. 1999a, ApJ, 525, +750 + +10 +Kaneko et al. +Figer, D. F., McLean, I. S., & Morris, M. 1999b, ApJ, 514, +202 +Gravity Collaboration, Abuter, R., Amorim, A., et al. 2018, +A&A, 615, L15 +Heywood, I., Rammala, I., Camilo, F., et al. 2022, ApJ, +925, 165 +Kamazaki, T., Okumura, S. 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R. 2018, ApJS, 235, 26 + diff --git a/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/load_file.txt b/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0b7b465f1754dc1eb2cf340c73239e9b6da4cc2 --- /dev/null +++ b/wtE3T4oBgHgl3EQf_AuU/content/tmp_files/load_file.txt @@ -0,0 +1,852 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf,len=851 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04831v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='GA] 12 Jan 2023 Draft version January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2023 Typeset using LATEX twocolumn style in AASTeX63 Discovery of the Tadpole Molecular Cloud near the Galactic Nucleus Miyuki Kaneko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 Tomoharu Oka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2 Hiroki Yokozuka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 Rei Enokiya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 Shunya Takekawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 Yuhei Iwata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 5 and Shiho Tsujimoto1 1School of Fundamental Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Graduate School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3-14-1 Hiyoshi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kohoku-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kanagawa 223-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Japan 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Keio University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3-14-1 Hiyoshi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kohoku-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kanagawa 223-8522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Japan 3Faculty of Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kanagawa University 3-27-1 Rokkakubashi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kanagawa-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Yokohama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kanagawa 221-8686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Japan 4Division of Science, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 5Center for Astronomy, Ibaraki University, 2-1-1 Bunkyo, Mito, Ibaraki 310-8512, Japan (Received 2022 August 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Revised 2022 November 23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Accepted 2022 November 25) Submitted to ApJ ABSTRACT In this paper, we report the discovery of an isolated, peculiar compact cloud with a steep velocity gradient at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′6 northwest of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This “Tadpole” molecular cloud is unique owing to its charac- teristic head-tail structure in the position-velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' By tracing the CO J=3–2 intensity peak in each velocity channel, we noticed that the kinematics of the Tadpole can be well reproduced by a Keplerian motion around a point-like object with a mass of 1×105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Changes in line intensity ratios along the orbit are consistent with the Keplerian orbit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The spatial compactness of the Tadpole and absence of bright counterparts in other wavelengths indicate that the object could be an intermediate-mass black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Keywords: galaxies: nuclei — Galaxy: center — ISM: clouds — ISM: molecules 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' INTRODUCTION It is widely accepted that large galaxies host a central supermassive black hole (SMBH) with mil- lions to billions times the mass of the Sun (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Kormendy & Richstone 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' A potential scenario for SMBH formation is based on intermediate-mass black holes (IMBHs), which have masses of 102–5 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Mezcua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Thus, detecting and studying IMBHs in detail are essential for understanding the formation and evolution of galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Numerous IMBH candidates have been identi- fied in centers of globular clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Kızıltan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017), in nuclei of dwarf galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Reines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Baldassare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2015), or as ultraluminous X-ray sources in extragalaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In the central molecular zone (CMZ) of our Galaxy, a number of compact (d < 5 pc) clouds with extraor- Corresponding author: Miyuki Kaneko miyukikaneko@keio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='jp dinary broad velocity width (∆V > 50 km s−1) have been detected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 1998, 1999, 2012, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These peculiar clouds, namely, high velocity-dispersion compact clouds (HVCCs), have been assumed to be accelerated by supernova explosions, protostellar out- flows, and/or cloud-to-cloud collisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Subsequently, it was determined that the kine- matics of CO–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='40–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='22, which is one of the most en- ergetic HVCCs, can be well reproduced by a cloud being gravitationally kicked by a point-like mass of ∼ 105 M⊙ (Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2016, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The absence of any bright object near the point-like mass suggest that it may be an IMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Subsequently, it was also sug- gested that HVCCs HCN–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='009–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='044 (Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2019a), HCN–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='085–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='094 (Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2020), and CO–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='11 (Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2019b) were driven by an IMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These discoveries yielded a new method of finding non-luminous massive objects, such as inactive and wandering BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' To date, five IMBH candidates, in- cluding IRS13E (Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2019), have been reported in the Galactic CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2 Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 1000 2000 3000 0 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='05 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 0 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='05 Galactic Longitude [deg] Galactic Latitude [deg] Tadpole CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 Sgr A* NLE C1 250:3000@250 I–200to+200 + 5 I–150to–80 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Map of velocity-integrated CO J=3–2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' White contours are drawn at 250 K km s−1 intervals from 250 K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The integrated intensity was calculated by � 200 −200 TMBdV +5 � −80 −150 TMBdV to emphasize the Tadpole, which appears at (l, b)=(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦090, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' CO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 (Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 1999, 2008), the C1 cloud (Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017), and the negative longitude extension of the circumnuclear disk (NLE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017) also appear in this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The white rectangle indicates the area presented in Figures 2–5 When searching for gravitationally kicked gas in the CMZ, we noticed an isolated HVCC in the CO J=3– 2 data obtained with the James Clerk Maxwell Tele- scope (JCMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Eden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It appears as an isolated compact cloud at (l, b) ≃ (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦090, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦014), which corresponds to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′6 Galac- tic northwest of Sgr A* (Figure 1), with LSR ve- locities between −140 km s−1 and −90 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It stands out with its peculiar appearance and very high CO J=3–2/CO J=1–0 intensity ratio (R3–2/1–0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2022), which exceeds double of the CMZ av- erage (R3–2/1–0 ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7 Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2007, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In this pa- per, we report the discovery of the so-called “Tadpole” molecular cloud, which is listed as id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='75 in the catalog of Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Throughout this paper, the dis- tance to the Galactic center is assumed to be D = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 kpc (Gravity Collaboration 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' DATA We first discovered the Tadpole in the JCMT CO J=3–2 data, and confirmed it in the CO J=1–0 and CS J=2–1 line data obtained with the Nobeyama Radio Observatory (NRO) 45 m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These data sets are briefly described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' CO J=3–2 Line The 12CO J=3–2 line (345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='795990 GHz) observations of the CMZ were performed using the JCMT from 2013 July to 2014 July (Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Heterodyne Array Receiver Program (HARP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Buckle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009) and autocorrelation spectral imaging system (ACSIS) were used during these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The half-power beam width (HPBW) of the telescope was approxi- mately 14′′ at 345 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The ACSIS was operated in the 1 GHz bandwidth (976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='56 kHz resolution) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' During these observations, the system noise temperature (Tsys) ranged between 100–200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The rms noise level of the image cubes was between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='84 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The details of the CO J=3–2 data and JCMT observations are pre- sented in Parsons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We use the data after resampling onto a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×1 km s−1 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' CO J=1–0 Lines The CO J=1–0 observations of the CMZ were per- formed using the Nobeyama Radio Observatory (NRO) 45 m telescope (Tokuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The 12CO J=1– 0 (115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='27120 GHz) line data were obtained from 2011 January 19 to 29 using the 25 beam array receiver sys- tem (BEARS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Sunada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' As the receiver back- end, the AC45 spectrometer system (Sorai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2000) was employed in the 500 MHz bandwidth (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 MHz res- olution) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The 13CO J=1–0 (110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='20135 GHz) line data were obtained from 2016 February to March us- ing the four-beam receiver system on the 45 m telescope (FOREST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Minamidani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The spectral anal- ysis machine on the 45 m telescope (SAM45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kuno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Kamazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2012) was operated in the 1 GHz (244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 kHz resolution) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The HPBW of the tele- scope was approximately 15′′ at 115 and 110 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The typical Tsys was ∼800 K and 150–300 K during the 12CO Discovery of the “Tadpole” Molecular Cloud 3 10 8 6 4 2 40 30 20 10 150 120 90 60 30 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='07 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='08 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='00 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04 –100 –150 Galactic Latitude [deg] LSR Velocity [km s–1] –80 ( b ) ( a ) CO J=3–2 12CO J=1–0 13CO J=1–0 200 150 100 50 ( c ) ( d ) ( e ) –110 –140 –90 –120 –130 CS J=2–1 10 6 2 8 4 Galactic Longitude [deg] 1 pc –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='07 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='08 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='11 Galactic Longitude [deg] HPBW HPBW HPBW HPBW Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (a) Map of velocity-integrated CO J=3–2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The velocity range for integration is VLSR =−140 to −80 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The intensity unit is K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (b) Longitude-velocity map of CO J=3–2 emission at b = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦014 (the magenta line in panel (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (c) Map of velocity-integrated 12CO J=1–0 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (d) Map of velocity-integrated 13CO J=1–0 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (e) Map of velocity-integrated CS J=2–1 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The integration ranges for these lines are the same as that of the CO J=3–2 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' and 13CO observations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The data were re- sampled onto a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×2 km s−1 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The rms noise levels of the resultant 12CO and 13CO data cubes were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 K in main-beam temperature (TMB), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' CS J=2–1 Line The CS J=2–1 (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='98096 GHz) line observations of the CMZ were performed during the NRO 45 m Tele- scope Large Program through 2019 January–May and 2020 January–April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The mapping area was set to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦5 ≤ l ≤ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦5 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦25 ≤ b ≤ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The FOR- EST receiver and SAM45 spectrometer were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The SAM45 was operated in the 1 GHz bandwidth (244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 kHz resolution) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The HPBW of the telescope was ≃ 19′′ at 86 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Tsys ranged from 150–300 K during the CS J=2–1 line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The data were resampled onto a 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′5×2 km s−1 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The rms noise level of the resultant CS J=2–1 data cube was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 K in TMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The details of the CS data and NRO 45 m observations will be presented in the forthcoming paper (Takekawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2022, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 20 10 0 –10 –20 Intensity [K] CO J=3–2 12CO J=1–0 CS J=2–1 × 3 –30 0 –50 –100 –150 –200 LSR Velocity [km s–1] 50 100 150 13CO J=1–0 × 2 200 Tadpole –40 –50 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Observed line spectra at (–0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦091, –0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦016) at where the CO J=3–2 profile of the Tadpole is widest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Black, red, blue and green lines show CO J=3–2, 12CO J=1–0, 13CO J=1–0 and CS J=2–1 line spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4 Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Spatial and Velocity Structure Figure 2 shows the spatial and velocity structure of the Tadpole in various molecular lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Tadpole appears as a compact, well-defined clump in velocity-integrated maps (Figure 2a, c–e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In the CO J=3–2 integrated map (Figure 2a), the full width of half maximum (FWHM) angular diameter of the Tadpole is 28′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This angular diameter is 2 times the angular resolution (14′′), cor- responding to a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 pc from the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In the 13CO J=1–0 and CS J=2–1 maps, the Tadpole has angular sizes similar to that depicted in the CO J=3–2 map, while the 12CO J=1–0 map depicts an appearance that is larger by a factor of 2 compared with those depicted in the other maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The velocity extent of the Tadpole is from VLSR ≃ −135 km s−1 to −90 km s−1 (Figure 2b), indicating that it may be in the CMZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The longitude-velocity behavior of the Tadpole is characterized by a “head-tail” struc- ture, from which its name is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It depicts a steep velocity gradient of |∆V/∆l|∼16 km s−1pc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The tail is obscure in the 13CO J=1–0 and CS J=2–1 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We show the spectra of CO and CS lines in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We learned that CO J=3–2 line can trace the kinemat- ics of the Tadpole best because of its high intensity and well-defined appearance in the l–b–V space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Figure 4 shows the velocity channel maps of CO J=3–2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Tadpole appears with an elliptical shape in each velocity channel, changing its angular size with respect to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It has the largest angular size of 36′′×24′′ in the VLSR = −130 to −125 km s−1 channel, while the smallest size of ∼20′′ is detected at both velocity ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These angular sizes, which are comparable to the tele- scope’s HPBWs, indicate that the spatial structure of the Tadpole is not well resolved using these single-dish observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' thus, the actual angular size must be far smaller than those observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The CO J=3–2 line emission shows a maximum inten- sity of 64 K km s−1 in the −130 to −125 km s−1 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We identified that the intensity maximum pixel in the Tadpole at each velocity channel continuously changes with respect to velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Note that these intensity max- imum pixels trace an arc in the plane of the sky (Figure 4, bottom right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This behavior will be analyzed in more detail in the discussion section (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Physical Parameters The physical size of the Tadpole was evaluated as the size parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' defined by S ≡ D tan �√σlσb � to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The velocity dispersion was calculated to be σV = 22 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These give a size-linewidth co- efficient (σV /S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5) of 15 km s−1pc−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 and virial theo- rem mass (MVT ≡ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7Sσ2 V /G) of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7 × 105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The molecular gas mass (Mgas) was estimated by summing the CO J=3–2 line integrated intensity and using the CO(3–2)-to-H2 conversion factor (XCO3–2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4×1020 [cm−2 (K km s−1)−1] Oka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2022) to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='6×102 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The XCO3–2 we employed is close to 1×1020 that esti- mated from an LVG calculation for n(H2)=1×104 cm−3, Tk =60 K, and N(CO)/dV =1×1017 cm−2 (km s−1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Using the physical parameters described above, we de- rived the dynamical timescale (tdyn ≡ S/σV ), kinetic energy (Ekin ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5Mgasσ2 V ), and kinetic power (Pkin ≡ Ekin/tdyn) of the Tadpoe as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2×104 yr, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4×1048 erg, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4×1038 erg s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The kinetic power of the Tadpole is equal to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7 × 104 L⊙ which is far greater than those of molecular outflows from massive YSOs (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01–100 L⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Maud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The virial theorem mass and molecular gas mass of the Tadpole yield a significantly high virial parameter, MVT/Mgas ∼ 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This indicates that the Tadpole is not in gravitational equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Following the method described in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (1989), we estimated the self- gravity of the Tadpole and tidal force by the Galactic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The self-gravity of the cloud estimated here (∼10−10 m s−2) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 orders of magnitude weaker than the tidal force (∼ 4×10−8 m s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These assessments clearly demonstrate that the Tadpole cannot be bound by its self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This strongly indicates to the pres- ence of an object with a mass comparable to the virial mass (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7×105 M⊙) inside the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Multiwavelength View To search for the driving source behind the Tad- pole, we referred to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='284 GHz data obtained using MeerKAT (Heywood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2022), mid-infrared data ob- tained using the Spitzer Space Telescope (Ram´ırez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Churchwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009), and X-ray data obtained at the Chandra X-ray Observatory (Muno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' No radio source brighter than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 mJy beam−1 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='284 GHz was detected near the angular extent of the Tadpole, while a faint filament was seen in the Galactic south (Figure 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In the mid-infrared image (Figure 5b), we identified a faint triangular rim which may be the irradiated surface of the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Considering the angular resolution of the Spitzer (∼2′′), the mid-infrared rim defines the angular extent of the Tadpole better than the CO J=3–2 appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We also checked the AT- LASGAL (870µm) and Hi-GAL (70, 160, 250, 350 and 500 µm) images (Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2016), and found no separate feature toward the Tad- pole above the sea of intense dust emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' A bright point-like source near the Galactic southwestern edge of the Tadpole is the long-period variable star V4872 Discovery of the “Tadpole” Molecular Cloud 5 –110 to –105 –95 to –90 –100 to –95 –115 to –110 –120 to –115 –125 to –120 –135 to –130 –140 to –135 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='08 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='07 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='11 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='00 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 Galactic Longitude [deg] Galactic Latitude [deg] –90 to –85 –130 to –125 –105 to –100 0 30 40 20 10 50 1 pc HPBW Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Velocity channel maps of the CO J=3–2 line from VLSR =−140 to −85 km s−1 near the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The angular area is the same as that depicted in Figure 2a, c–e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The intensity unit is K km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The velocity range for integration is indicated at the bottom right corner of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The magenta cross denotes the position of the maximum intensity pixel in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The distribution of the maximum intensity pixels is shown in the bottom right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Sgr (Matsunaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In addition, we also iden- tify dozens of point-like, mid-infrared sources toward the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The X-ray image also shows numerous faint point-like sources toward the Tadpole (Figure 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Al- though the nature of these mid-infrared/X-ray point-like sources is unclear, we will refer to them in the discus- sion section (§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In short, the multiwavelength view confirms the absence of any energetic objects that can drive the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Tracing Molecular Gas Kinematics The continuous change in the CO J=3–2 intensity maximum pixel along the arc with velocity (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1) sug- gests that the bulk of warm molecular gas belongs to a certain trajectory, such as a closed orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' To deter- mine the orbit trajectory accurately, we constructed a CO J=3–2 data cube with a 2′′×2′′×1 km s−1 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Then, we calculated the accurate intensity peak position as the CO emission center of gravity using 5×5 pixels around the intensity maximum pixel in the Tadpole for each 1 km s−1 width velocity channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Figure 6 shows the l–b–V and l–b distributions of the intensity peak position in the Tadpole for each velocity channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The arc-shape apparent in the l–b distribution became clearer than that depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The l–b–V distribution demonstrates the striking continuity of peak positions in the velocity direction, indicating that the bulk of warm molecular gas in the Tadpole may predominantly follow a certain closed orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Keplerian Orbit Fitting The arc-shaped spatial structure and continuous ve- locity change along the arc suggest that a rotational mo- tion exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Thus, we expect that the l–b–V behavior of warm molecular gas in the Tadpole can be reproduced by a Keplerian orbit around a point mass1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We performed the fitting of a Keplerian orbit to the intensity peak 1 Here we do not consider parabollic or hyperbolic orbits, although the orbit could not be an elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 6 Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='07 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='08 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='10 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='00 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04 Galactic Latitude [deg] ( a ) Galactic Longitude [deg] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='00 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04 ( b ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='00 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='02 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='04 ( c ) MeerKAT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='284 GHz Chandra 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 kev 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7 kev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 kev Spitzer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 μm 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 [ mJy beam–1 ] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Multiwavelength view near the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' White contours show the CO J=3–2 integrated intensity with a 50 K km s−1 interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (a) Radio continuum image at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='284 GHz obtained with the MeerKAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (b) Composite mid-infrared im- age obtained with the Spitzer Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The image depicts 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 µm band flux in red, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 µm in green, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 µm in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (c) Composite X-ray image obtained with the Chandra X-ray Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The image depicts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 keV count rate in red, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7 keV in green, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='7–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 keV in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' positions (Figure 6) following the method described in Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We determined nine parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Five of these parameters are three-dimensional orbital parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' namely, the semimajor axis (a), eccentric- ity (e), longitude of ascending node (Ω), argument of –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='005 VLSR [km s–1] –90 –140 –130 –120 –110 –100 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='020 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='025 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='015 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='010 Galactic Latitude [deg] –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='090 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='095 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='085 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='105 Galactic Longitude [deg] –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='100 –150 –80 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The l–b–V distribution of CO J=3–2 intensity peak positions in the Tadpole in each 1 km s−1 width velocity channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The head and tail of the Tadpole are denoted by blue and green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The l–b distribution is also shown in the base plane with CO J=3–2 contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Grey lines represent the loci of the best-fit Keplerian orbit (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The black and red filled circles denote the positions of the dynamical center and pericenter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The black arrow indicates the direction of rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Parameters of the best-fit Keplerian Orbit Parameter Value Mass (Mdyn) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='05) × 105 M⊙ Semimajor axis (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 pc Eccentricity (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='03 Inclination (i) 112◦ ± 1◦ Argument of pericenter (ω) 82◦ ± 5◦ Longitude of ascending node (Ω) −35◦ ± 1◦ Longitude offset (l0) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦0946 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦0001 Latitude offset (b0) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦0146 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='◦0001 Velocity offset (V0) −114 ± 1 km s−1 pericenter (ω), and inclination angle (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The remaining parameters are the mass (Mdyn), line-of-sight velocity (Vdyn) and l–b position (l0, b0) of the dynamical center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The distance to the Tadpole is assumed to be same as that to the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The χ2 minimization ap- proach was utilized in the l–b–V space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The χ2 consists of two terms, namely, the spatial (χ2 s) and velocity terms (χ2 v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The spatial term is defined by the sum of the or- thogonal distances between the peak position and the modeled orbit divided by the square of the positional uncertainty, which was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='15 pc (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′′6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The veloc- ity term is, similarly, the sum of the velocity deviations between those of the peak positions and nearest posi- Discovery of the “Tadpole” Molecular Cloud 7 tions in the modeled orbit divided by a square of the velocity uncertainty, which was set to 1 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The best-fit parameters with 1σ uncertainties are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The l–b–V locus of the orbit is pre- sented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Note that the best-fit solution is bivalent because the orbits with i → (180◦ − i) and Ω→(Ω + 180◦) yield the same l–b–V locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The fitting result indicates the presence of a point mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0×105 M⊙ in the northwestern periphery of the Tadpole (Fig- ure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 acceleration [ 10–7 m s–2 ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 diameter of cloud [ pc ] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 self gravitation tidal force at pericenter tidal force at apocenter Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Plots of the tidal force due to the point mass (solid line) and self-gravity of the cloud (dashed lines) work- ing on the head versus the head diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' A central mass of 105 M⊙ and head mass of 660 M⊙ were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The thick solid line indicates self-gravity at the pericenter dis- tance (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='49 pc) while the thin solid line indicates that at the apocenter (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Tidal Stability of the Head We examined the tidal stability of the head, assuming the Newtonian potential of a point mass, following the method described in Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The tidal force caused by the point mass and self-gravity doing work on the head were calculated as functions of the head diameter (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' A central mass of 1×105 M⊙ and head mass of 660 M⊙ were assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The curves of the two forces at the pericenter of the best-fit orbit (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 pc) intersect at d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 pc, while those at the apocenter intersect at d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='35 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Because the observed head diameter (∼ 1 pc) is larger than the critical value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 pc) by a factor of seven, the Tadpole head is tidally unstable at the pericenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' If the head includes a self- gravitating core smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='14 pc, it may be able to survive several turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It is possibly that the Tadpole has been trapped by the gravitational potential of the 105 M⊙ point mass, being stretched by the strong tidal force to form the characteristic head-tail structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This situation is similar to the simulation of a cloud tidally disrupted by a super massive black hole (Saitoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Note that the Tadpole must have been lost its angular momentum to be captured during the encounter with the point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 0 90 180 270 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 0 5 10 15 φ [degrees] TMB [K] R3–2/1–0 RCS/CO (a) CO 1–0 CO 3–2 CS 2–1 (b) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (a) Plots of CO J=3–2 (filled circles), CO J=1–0 (crosses), and CS J=2–1 (open circles) main-beam tempera- tures at the CO J=3–2 intensity peaks versus orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (b) Plots of R3–2/1–0 (filled circles) and RCS/CO (open cir- cles) at the CO J=3–2 intensity peaks versus orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The phase ranges of the head and tail of the Tadpole are depicted in blue and green shades, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Intensity Ratios along the Orbit Here, we refer to the CO J=1–0, CO J=3–2, and CS J=2–1 line intensities, as well as CO J=3–2/CO J=1– 0 (R3–2/1–0) and CS J=2–1/CO J=1–0 (RCS/CO) line intensity ratios to examine the Keplerian orbit scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The critical densities of the CO J=1–0, CO J=3–2, and CS J=2–1 transitions are approximately 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5, 104, and 105 cm−3, respectively, while their upper state energies are 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='6, 33, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='1 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Thus, RCS/CO is sensitive to the variations in density while R3–2/1–0 is affected by both the temperature and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Figure 8 shows plots of line intensities and intensity ratios along the best-fit orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The motion along the orbit increases the orbital phase (φ) at all times, where the pericenter of the orbit corresponds to φ=0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 8 Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Both ratios exhibit higher values at the head, indicat- ing that both the density and temperature are enhanced there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' At the beginning of the tail, φ ∼ 170◦, R3–2/1–0 demonstrates a prominent peak while RCS/CO does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This suggest that the temperature is enhanced at the beginning of the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Then, both ratios decrease to- ward the tip of the tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The higher temperature in the head may have been caused by the shock occurring at the first encounter with the point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This shock may be caused by collisions of gas clumps at the peri- center, where adjacent orbits are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' In a fluid mechanical treatment, the same process is described as the “tidal compression”(Saitoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The gas ac- cumulation near the apocenter and/or the presence on a self-gravitating core can cause the higher density in the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The tail may be really the low density, tidally stretched tail of the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The temperature increase at the beginning of the tail could be caused by another shock at the congestion near the apocenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Thus, the behaviors of intensity ratios are consistent with the Ke- plerian orbit scenario for the Tadpole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='090 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='095 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='100 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='010 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='015 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='020 Galactic Latitude [deg] Galactic Longitude [deg] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Distribution of X-ray point-like sources (open yel- low circles: Muno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009, filled yellow circles: Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2018) superimposed on the composite mid-infrared image (color;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The red cross denotes the dynamical center of the best-fit Keplerian orbit, and the green dotted circle has a 5′′ radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Black contours highlight the CO J=3– 2 integrated intensity with a 50 K km s−1 interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' What is a Point Mass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Keplerian orbit model requires the presence of a huge point mass at the dynamical center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' What is this point-like massive object?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The mass of 105 M⊙ is larger than the Arches or Quintuplet clusters (∼104 M⊙ Figer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 1999a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' According to the best-fit model, a mass of 105 M⊙ must be concentrated within a ra- dius significantly smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 pc (the pericenter dis- tance), resulting in an average mass density higher than that of the Arches cluster (ρ∼2×105 M⊙ pc−3 Espinoza 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Anyway, the absence of a bright infrared counterpart toward the Tadpole (Ram´ırez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Churchwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Molinari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2011) clearly rules out a stellar cluster from being a candidate for the point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The most promising candidate for the point-like mas- sive object in the Tadpole may be an IMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We searched for a counterpart for this point mass refer- ring to X-ray images (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Figure 9 shows the posi- tions of point-like sources in the X-ray images super- imposed on the composite mid-infrared images around (l, b)=(l0, b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The small statistical uncertainties in the l–b position of the dynamical center (Table 1) should be considered with some caution because the entire gas mass in the Tadpole is not confined within the single closed orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Notice that three point-like sources in the X-ray and mid-infrared reside within a 5′′ (∼10 times of the positional ambiguity of the dynamical center) angu- lar distance from (l0, b0) (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' One source, SSTGC 489898, is apparent in both X-ray and mid-infrared im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The X-ray detected sources, SSTGC 489898 and CXOGC 174526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='9–290124, exhibit hard spectra, indi- cating that they may be at the same distance to the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' No time variation has been detected from these sources (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' If we assume the standard accretion disk model (L = ˙Mc2/12), their X- ray luminosities correspond to ˙M ∼ 1×10−14 M⊙yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' This very low mass accretion rate could be a challenge to our interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Anyway, we suppose these three point-like sources are candidates for the luminous coun- terpart of the point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Although we currently have very little knowledge on these point-like sources, they should be considered as candidates for an IMBH in fu- ture studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' CONCLUSIONS We have discovered the so-called Tadpole, which is an isolated, peculiar compact cloud with an extraordinary velocity width and very high CO J=3–2/J=1–0 intensity ratio, at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='′6 northwest of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Our main conclusions can be summarized as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The Tadpole molecular cloud has a size of ∼1 pc at the distance to the Galactic center and a velocity width of ∼50 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It demonstrates the characteristic “head-tail” structure in position-velocity space, having a steep velocity gradient of 16 km s−1pc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Discovery of the “Tadpole” Molecular Cloud 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Properties of X-ray and mid-infrared sources near the point mass l b F2–10keV a [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='6 µm]b [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='5 µm]b [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='8 µm]b [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='0 µm]b ID (deg) (deg) (10−18 W/m2) (mag) (mag) (mag) (mag) SSTGC 489898b –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09513 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01486 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='284 · · · · · · SSTGC 490125b –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09384 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01444 · · 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='154 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='003 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='422 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='217 CXOGC 174526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='9–290124c –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='09400 –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='01346 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='72 · · · · · · · · Note— a The X-ray (2–10 keV) energy flux quoted from Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' b The source IDs and mid-infrared magnitudes are quoted from Ram´ırez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' c The source ID defined in Muno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Its kinematics is well reproduced by a Keplerian motion around a point-like object with a mass of 1×105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' It is plausible that the Tadpole has been trapped by the gravitational potential of the huge point mass, now being stretched by the strong tidal force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The behaviors of line intensity ratios (R3–2/1–0 and RCS/CO) are consistent with the Keplerian or- bit scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The absence of bright objects near the putative point-like object suggests that the object is an in- active intermediate-mass black hole (IMBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' These results are based on molecular line maps with 14′′–19′′ resolutions obtained using single-dish tele- scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Future aperture synthesis observations of molec- ular lines with millimeter/submillimeter arrays will be able to delineate the orbit stream directly, increasing the reliability of the Keplerian orbit model for the Tad- pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The results presented in this paper are based on data obtained using the Nobeyama Radio Observatory (NRO) 45-m telescope and James Clerk Maxwell Tele- scope (JCMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The NRO 45-m radio telescope is oper- ated by the Nobeyama Radio Observatory, a division of the National Astronomical Observatory of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan, Academia Sinica Institute of Astronomy and Astrophysics, Korea Astron- omy and Space Science Institute, National Astronomical Research Institute of Thailand, and Center for Astro- nomical Mega-Science (as well as the National Key R& D Program of China with No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017YFA0402700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' Ad- ditional funding support is provided by the Science and Technology Facilities Council of the United Kingdom, and participating universities and organizations in the United Kingdom and Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' We are grateful to the NRO staff and all members of the JCMT team for operating the telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' ac- knowledges the financial support of JSPS Grant-in-Aid for Scientific Research (A) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 20H00178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2015, MNRAS, 453, 645 Mezcua, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2017, IJMPD, 26, 1730021 Minamidani, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Nishimura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Miyamoto, Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Tsuboi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', & Miyazaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 1998, ApJS, 118, 455 Oka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', White, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=', Hasegawa, T.' metadata={'source': 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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} +page_content=' 2018, ApJS, 235, 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtE3T4oBgHgl3EQf_AuU/content/2301.04831v1.pdf'} diff --git a/x9FST4oBgHgl3EQfSzhz/content/tmp_files/2301.13767v1.pdf.txt b/x9FST4oBgHgl3EQfSzhz/content/tmp_files/2301.13767v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ea657ccbdf11ff0e5376ad244ad5f67ab056c40 --- /dev/null +++ b/x9FST4oBgHgl3EQfSzhz/content/tmp_files/2301.13767v1.pdf.txt @@ -0,0 +1,2492 @@ +Multicalibration as Boosting for Regression +Ira Globus-Harris +Declan Harrison +Michael Kearns +Aaron Roth +Jessica Sorrell +February 1, 2023 +Abstract +We study the connection between multicalibration and boosting for squared error regression. First we +prove a useful characterization of multicalibration in terms of a “swap regret” like condition on squared +error. Using this characterization, we give an exceedingly simple algorithm that can be analyzed both +as a boosting algorithm for regression and as a multicalibration algorithm for a class H that makes use +only of a standard squared error regression oracle for H. We give a weak learning assumption on H that +ensures convergence to Bayes optimality without the need to make any realizability assumptions — giving +us an agnostic boosting algorithm for regression. We then show that our weak learning assumption on H +is both necessary and sufficient for multicalibration with respect to H to imply Bayes optimality. We also +show that if H satisfies our weak learning condition relative to another class C then multicalibration with +respect to H implies multicalibration with respect to C. Finally we investigate the empirical performance +of our algorithm experimentally using an open source implementation that we make available on GitHub1. +1 +Introduction +We revisit the problem of boosting for regression, and develop a new agnostic regression boosting algorithm +via a connection to multicalibration. In doing so, we shed additional light on multicalibration, a recent +learning objective that has emerged from the algorithmic fairness literature [Hébert-Johnson et al., 2018]. In +particular, we characterize multicalibration in terms of a “swap-regret” like condition, and use it to answer +the question “what property must a collection of functions H have so that multicalibration with respect to H +implies Bayes optimality?”, giving a complete answer to problem asked by Burhanpurkar et al. [2021]. Using +our swap-regret characterization, we derive an especially simple algorithm for learning a multicalibrated +predictor for a class of functions H by reduction to a standard squared-error regression algorithm for H. The +same algorithm can also be analyzed as a boosting algorithm for squared error regression that makes calls +to a weak learner for squared error regression on subsets of the original data distribution without the need +to relabel examples (in contrast to Gradient Boosting as well as existing multicalibration algorithms). This +lets us specify a weak learning condition that is sufficient for convergence to the Bayes optimal predictor +(even if the Bayes optimal predictor does not have zero error), avoiding the kinds of realizability assumptions +that are implicit in analyses of boosting algorithms that converge to zero error. We conclude that ensuring +multicalibration with respect to H corresponds to boosting for squared error regression in which H forms +the set of weak learners. Finally we define a weak learning condition for H relative to a constrained class +of functions C (rather than with respect to the Bayes optimal predictor). We show that multicalibration +with respect to H implies multicalibration with respect to C if H satisfies the weak learning condition with +respect to C, which in turn implies accuracy at least that of the best function in C. +Multicalibration +Consider a distribution D P ∆Z defined over a domain Z “ X ˆ R of feature vectors +x P X paired with real valued labels y. Informally, a regression function f : X Ñ R is calibrated if for every +v in the range of f, Epx,yq„Dry|fpxq “ vs “ v. In other words, fpxq must be an unbiased estimator of y, +1Our code repository can be found at https://github.com/Declancharrison/Level-Set-Boosting +1 +arXiv:2301.13767v1 [cs.LG] 31 Jan 2023 + +even conditional on the value of its own prediction. Calibration on its own is a weak condition, because +it only asks for f to be unbiased on average over all points x such that fpxq “ v. For example, the con- +stant predictor that predicts fpxq “ Epx,yq„Drys is calibrated. Thus calibration does not imply accuracy—a +calibrated predictor need not make predictions with lower squared error than the best constant predictor. +Calibration also does not imply that f is equally representative of the label distribution on different subsets +of the feature space X. For example, given a subset of the feature space G Ď X, even if f is calibrated, it +may be that f is not calibrated on the conditional distribution conditional on x P G—it might be e.g. that +Ery|fpxq “ v, x P Gs " v, and Ery|fpxq “ v, x R Gs ! v. To correct this last deficiency, Hébert-Johnson et al. +[2018] defined multi-calibration, which is a condition parameterized by a subset of groups G Ď X each defined +by an indicator function h : X Ñ t0, 1u in some class H. It asks (informally) that for each such h P H, and for +each v in the range of f, that Erhpxqpy ´vq|fpxq “ vs “ 0. Since h is a binary indicator function for some set +G, this is equivalent to asking for calibration not just marginally over D, but simultaneously for calibration +over D conditional on x P G. Kim et al. [2019] and Gopalan et al. [2022] generalize multicalibration beyond +group indicator functions to arbitrary real valued functions h : X Ñ R. Intuitively, as H becomes a richer +and richer set of functions, multicalibration becomes an increasingly stringent condition. But if H consists +of the indicator functions for e.g. even a very large number of randomly selected subsets G Ď X, then the +constant predictor fpxq “ Epx,yq„Drys will still be approximately multicalibrated with respect to H. What +property of H ensures that multicalibration with respect to H implies that f is a Bayes optimal regression +function? This question was recently asked by Burhanpurkar et al. [2021] — and we provide a necessary and +sufficient condition. +Boosting for Regression Boosting refers broadly to a collection of learning techniques that reduce the +problem of “strong learning” (informally, finding an error optimal model) to a series of “weak learning” tasks +(informally, finding a model that has only a small improvement over a trivial model)—See Schapire and Fre- +und [2013] for a textbook treatment. The vast majority of theoretical work on boosting studies the problem +of binary classification, in which a weak learner is a learner that obtains classification error bounded below +1{2. Several recent papers Kim et al. [2019], Gopalan et al. [2022] have made connections between algorithms +for guaranteeing multicalibration and boosting algorithms for binary classification. +In this paper, we show a direct connection between multicalibration and the much less well-studied +problem of boosting for squared error regression [Friedman, 2001, Duffy and Helmbold, 2002]. There is not +a single established notion for what constitutes a weak learner in the regression setting (Duffy and Helmbold +[2002] introduce several different notions), and unlike boosting algorithms for classification problems which +often work by calling a weak learner on a reweighting of the data distribution, existing algorithms for +boosting for regression typically resort to calling a learning algorithm on relabelled examples. We give a +boosting algorithm for regression that only requires calling a squared error regression learning algorithm +on subsets of examples from the original distribution (without relabelling), which lets us formulate a weak +learning condition that is sufficient to converge to the Bayes optimal predictor, without making the kinds of +realizability assumptions implicit in the analysis of boosting algorithms that assume one can drive error to +zero. +1.1 +Our Results +We focus on classes of real valued functions H that are closed under affine transformations — i.e. classes +such that if fpxq P H, then for any pair of constants a, b P R, pafpxq ` bq P H as well. Many natural classes +of models satisfy this condition already (e.g. linear and polynomial functions and regression trees), and any +neural network architecture that does not already satisfy this condition can be made to satisfy it by adding +two additional parameters (a and b) while maintaining differentiability. Thus we view closure under affine +transformations to be a weak assumption that is enforceable if necessary. +First in Section 3 we prove the following characterization for multicalibration over H, for any class H that +is closed under affine transformations. Informally, we show that a model f is multicalibrated with respect +2 + +to H if and only if, for every v in the range of f: +E +px,yq„Drpfpxq ´ yq2|fpxq “ vs ď min +hPH +E +px,yq„Drphpxq ´ yq2|fpxq “ vs +(See Theorem 3.2 for the formal statement). This is a “swap regret”-like condition (as in Foster and Vohra +[1999] and Blum and Mansour [2005]), that states that f must have lower squared error than any model +h P H, even conditional on its own prediction. Using this characterization, in Section 4 we give an exceedingly +simple algorithm for learning a multicalibrated predictor over H given a squared error regression oracle +for H. The algorithm simply repeats the following over t rounds until convergence, maintaining a model +f : X Ñ t0, 1{m, 2{m, . . . , 1u with a discrete range with support over multiples of 1{m for some discretization +factor m: +1. For each level set v P t0, 1{m, 2{m, . . . , 1u, run a regression algorithm to find the ht +v P H that minimizes +squared error on the distribution D|pft´1pxq “ vq, the distribution conditional on ft´1pxq “ v. +2. Replace each level set v of ft´1pxq with ht +vpxq to produce a new model ft, and round its output to the +discrete range t0, 1{m, 2{m, . . . , 1u +Each iteration decreases the squared error of ft, ensuring convergence, and our characterization of multi- +calibration ensures that we are multicalibrated with respect to H at convergence. Compared to existing +multicalibration algorithms (e.g. the split and merge algorithm of Gopalan et al. [2022]), our algorithm is +exceptionally simple and makes use of a standard squared-error regression oracle on subsets of the original +distribution, rather than using a classification oracle or requiring example relabelling. +We can also view the same algorithm as a boosting algorithm for squared error regression. Suppose H (or +equivalently our weak learning algorithm) satisfies the following weak learning assumption: informally, that +on any restriction of D on which the Bayes optimal predictor is non-constant, there should be some h P H +that obtains squared error better than that of the best constant predictor. Then our algorithm converges +to the Bayes optimal predictor. In Section A we give uniform convergence bounds which guarantee that the +algorithm’s accuracy and multicalibration guarantees generalize out of sample, with sample sizes that are +linear in the pseudodimension of H. +We then show in Section 5 that in a strong sense this is the “right” weak learning assumption: Multical- +ibration with respect to H implies Bayes optimality if and only if H satisfies this weak learning condition. +This gives a complete answer to the question of when multicalibration implies Bayes optimality. +In Section 6, we generalize our weak learning condition to a weak learning condition relative to a con- +strained class of functions C (rather than relative to the Bayes optimal predictor), and show that if H satisfies +the weak learning condition relative to C, then multicalibration with respect to H implies multicalibration +with respect to C, and hence error that is competitive with the best model in C. +We give a fast, parallelizable implementation of our algorithm and in Section 7 demonstrate its con- +vergence to Bayes optimality on two-dimensional datasets useful for visualization, as well as evaluate the +accuracy and calibration guarantees of our algorithm on real Census derived data using the Folktables pack- +age Ding et al. [2021]. +1.2 +Additional Related Work +Calibration as a statistical objective dates back at least to Dawid [1982]. Foster and Vohra [1999] showed +a tight connection between marginal calibration and internal (equivalently swap) regret. We extend this +characterization to multicalibration. Multicalibration was introduced by Hébert-Johnson et al. [2018], and +variants of the original definition have been studied by a number of works [Kim et al., 2019, Jung et al., +2021, Gopalan et al., 2022, Kim et al., 2022, Roth, 2022]. We use the ℓ2 variant of multicalibration studied +in Roth [2022]—but this definition implies all of the other variants of multicalibration up to a change in +parameters. Burhanpurkar et al. [2021] first asked the question “when does multicalibration with respect to +H imply accuracy”, and gave a sufficient condition: when H contains (refinements of) the levelsets of the +3 + +Bayes optimal regression function, together with techniques for attempting to find these. This can be viewed +as a “strong learning” assumption, in contrast to our weak learning assumption on H. +Boosting for binary classification was introduced by Schapire [1990] and has since become a major topic +of both theoretical and empirical study — see Schapire and Freund [2013] for a textbook overview. Both Kim +et al. [2019] and Gopalan et al. [2022] have drawn connections between algorithms for multicalibration and +boosting for binary classification. In particular, Gopalan et al. [2022] draw direct connections between their +split-and-merge multicalibration algorithm and agnostic boosting algorithms of Kalai [2004], Kanade and +Kalai [2009], Kalai et al. [2008]. Boosting for squared error regression is much less well studied. Freund and +Schapire [1997] give a variant of Adaboost (Adaboost.R) that reduces regression examples to infinite sets of +classification examples, and requires a base regressor that optimizes a non-standard loss function. Friedman +[2001] introduced the popular gradient boosting method, which for squared error regression corresponds to +iteratively fitting the residuals of the current model and then applying an additive update, but did not give +a theoretical analysis. Duffy and Helmbold [2002] give a theoretical analysis of several different boosting +algorithms for squared error regression under several different weak learning assumptions. Their algorithms +require base regression algorithms that can be called (and guaranteed to succeed) on arbitrarily relabelled +examples from the training distribution, and given their weak learning assumption, their analysis shows how +to drive the error of the final model arbitrarily close to 0. Weak learning assumptions in this style implicitly +make very strong realizabilty assumptions (that the Bayes error is close to 0), but because the weak learner +is called on relabelled samples, it is difficult to enunciate a weak learning condition that is consistent with +obtaining Bayes optimal error, but not better. The boosting algorithm we introduce only requires calling +a standard regression algorithm on subsets of the examples from the training distribution, which makes it +easy for us to define a weak learning condition that lets us drive error to the Bayes optimal rate without +realizability assumptions — thus our results can be viewed as giving an agnostic boosting algorithm for +regression. +2 +Preliminaries +We study prediction tasks over a domain Z “ X ˆY. Here X represents the feature domain and Y represents +the label domain. We focus on the bounded regression setting where Y “ r0, 1s (the scaling to r0, 1s is +arbitrary). We write D P ∆Z to denote a distribution over labelled examples, DX to denote the induced +marginal distribution over features, and write D „ Dn to denote a dataset consisting of n labelled examples +sampled i.i.d. from D. We will be interested in the squared error of a model f with respect to distribution +D, Epx,yq„Drpy ´ fpxqq2s. We abuse notation and identify datasets D “ tpx1, y1q, . . . , pxn, ynqu with the +empirical distribution over the examples they contain, and so we can write the empirical squared error over +D: as Epx,yq„Drpy ´ fpxqq2s “ 1 +n +řn +i“1pyi ´ fpxiqq2. When taking expectations over a distribution that is +clear from context, we will frequently suppress notation indicating the relevant distribution for readability. +We write Rpfq to denote the range of a function f, and when Rpfq is finite, use m to denote the cardinality +of its range: m “ |Rpfq|. We are interested in finding models that are multicalibrated with respect to a class +of real valued functions H. We use an ℓ2 notion of multicalibration as used in Roth [2022]: +Definition 2.1 (Multicalibration). Fix a distribution D P ∆Z and a model f : X Ñ r0, 1s that maps onto +a countable subset of its range. Let H be an arbitrary collection of real valued functions h : X Ñ R. We say +that f is α-approximately multicalibrated with respect to D and H if for every h P H: +K2pf, h, Dq “ +ÿ +vPRpfq +Pr +px,yq„Drfpxq “ vs +ˆ +E +px,yq„Drhpxqpy ´ vq|fpxq “ vs +˙2 +ď α. +We say that f is α-approximately calibrated if: +K2pf, Dq “ +ÿ +vPRpfq +Pr +px,yq„Drfpxq “ vs +ˆ +E +px,yq„Drpy ´ vq|fpxq “ vs +˙2 +ď α. +4 + +If α “ 0, then we simply say that a model is multicalibrated or calibrated. We will sometimes refer to +K2pf, Dq as the mean squared calibration error of a model f. +Remark 2.2. When the functions hpxq have binary range, we can view them as indicator functions for some +subset of the data domain S Ď X, in which case multicalibration corresponds to asking for calibration condi- +tional on membership in these subsets S. Allowing the functions h to have real valued range is only a more +general condition. Our notion of approximate multicalibration takes a weighted average over the level sets v of +the predictor f, weighted by the probability that fpxq “ v. This is necessary for any kind of out of sample gen- +eralization statement — otherwise we could not even necessarily measure calibration error from a finite sam- +ple. Other work on multicalibration use related measures of multicalibration that we think of as ℓ1 or ℓ8 vari- +ants, that we can write as K1pf, h, Dq “ ř +vPRpfq Prpx,yq„Drfpxq “ vs +ˇˇEpx,yq„Drhpxqpy ´ vq|fpxq “ vs +ˇˇ and +K8pf, h, Dq “ maxvPRpfq Prpx,yq„Drfpxq “ vs +` +Epx,yq„Drhpxqpy ´ vq|fpxq “ vs +˘ +. These notions are related +to each other: K2pf, h, Dq ď K1pf, h, Dq ď +a +K2pf, h, Dq and K8pf, h, Dq ď K1pf, h, Dq ď mK8pf, h, Dq +[Roth, 2022]. +We will characterize the relationship between multicalibration and Bayes optimality. +Definition 2.3 (Bayes Optimal Predictor). Let f ˚ : X Ñ r0, 1s. We say that f ˚ is the Bayes optimal +predictor for D if: +E +px,yq„Drpy ´ f ˚pxqq2s ď +min +f:XÑr0,1srpy ´ fpxqq2s +The Bayes Optimal predictor satisfies: f ˚pxq “ Epx1,yq„D ry|x1 “ xs . We say that a function f : X Ñ r0, 1s +is γ-approximately Bayes optimal if +E +px,yq„Drpy ´ fpxqq2s ď +E +px,yq„Drpy ´ f ˚pxqq2s ` γ. +Throughout this paper, we will denote the Bayes optimal predictor as f ˚. +3 +A Characterization of Multicalibration +In this section we give a simple “swap-regret” like characterization of multicalibration for any class of functions +H that is closed under affine transformations: +Definition 3.1. A class of functions H is closed under affine transformations if for every a, b P R, if +hpxq P H then h1pxq :“ ahpxq ` b P H. +As already discussed, closure under affine transformation is a mild assumption: it is already satisfied by +many classes of functions H like linear and polynomial functions and decision trees, and can be enforced for +neural network architectures when it is not already satisfied by adding two additional parameters a and b +without affecting our ability to optimize over the class. +The first direction of our characterization states that if f fails the multicalibration condition for some +h P H, then there is some other h1 P H that improves over f in terms of squared error, when restricted to a +level set of f. The second direction states the opposite: if f is calibrated (but not necessarily multicalibrated), +and if there is some level set of f on which h improves over f in terms of squared error, then in fact f must +fail the multicalibration condition for h. +Theorem 3.2. Suppose H is closed under affine transformation. Fix a model f : X Ñ R and a levelset +v P Rpfq of f. Then: +1. If there exists an h P H such that: +Erhpxqpy ´ vq|fpxq “ vs ě α, +for α ą 0, then there exists an h1 P H such that: +Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě +α2 +Erhpxq2|fpxq “ vs, +5 + +2. If f is calibrated and there exists an h P H such that +Erpfpxq ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, +then: +Erhpxqpy ´ vq|fpxq “ vs ě α +2 . +Proof. We prove each direction in turn. +Lemma 3.3. Fix a model f : X Ñ R. Suppose for some v P Rpfq there is an h P H such that: +Erhpxqpy ´ vq|fpxq “ vs ě α +Let h1 “ v ` ηhpxq for η “ +α +Erhpxq2|fpxq“vs. Then: +Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě +α2 +Erhpxq2|fpxq “ vs +Proof. We calculate: +Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs +“ +Erpv ´ yq2 ´ pv ` ηhpxq ´ yq2|fpxq “ vs +“ +Erv2 ´ 2vy ` y2 ´ pv ` ηhpxqq2 ` 2ypv ` ηhpxqq ´ y2|fpxq “ vs +“ +Er2yηhpxq ´ 2vηhpxq ´ η2hpxq2|fpxq “ vs +“ +Er2ηhpxqpy ´ vq ´ η2hpxq2|fpxq “ vs +ě +2ηα ´ η2 Erhpxq2|fpxq “ vs +“ +α2 +Erhpxq2|fpxq “ vs +Where the last line follows from the definition of η. +The first direction of Theorem 3.2 follows from Lemma 3.3, and the observation that since H is closed +under affine transformations, the function h1 defined in the statement of Lemma 3.3 is in H. Now for the +second direction. +Lemma 3.4. Fix a model f : X Ñ R. Suppose for some v P Rpfq there is an h P H such that: +Erp¯yv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, +where ¯yv “ Ery | fpxq “ vs. Then it must be that: +Erhpxqpy ´ ¯yvq|fpxq “ vs ě α +2 +Proof. We calculate: +6 + +E +px,yq„Drhpxqpy ´ ¯yvq|fpxq “ vs +“ +E +px,yq„Drhpxqy|fpxq “ vs ´ ¯yv +E +px,yq„Drhpxq|fpxq “ vs +“ +1 +2 +ˆ +2 +E +px,yq„Drhpxqy|fpxq “ vs ´ 2¯yv +E +px,yq„Drhpxq|fpxq “ vs +˙ +ě +1 +2 +ˆ +2 +E +px,yq„Drhpxqy|fpxq “ vs ´ 2¯yv +E +px,yq„Drhpxq|fpxq “ vs ´ +E +px,yq„Drphpxq ´ ¯yvq2|fpxq “ vs +˙ +“ +1 +2 +ˆ +E +px,yq„Dr2hpxqy ´ hpxq2 ´ ¯y2 +v|fpxq “ vs +˙ +“ +1 +2 +ˆ +E +px,yq„Dr2hpxqy ´ hpxq2 ´ 2¯yvy ` ¯y2 +v|fpxq “ vs +˙ +“ +1 +2 +ˆ +E +px,yq„Drp¯yv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs +˙ +ě +α +2 +where the 3rd to last line follows from adding and subtracting ¯y2 +v. +For any calibrated f it follows that v “ Ery | fpxq “ vs “ ¯yv, and so for calibrated f we have that if +Erpv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, +then: +Erhpxqpy ´ vq|fpxq “ vs ě α +2 . +4 +An Algorithm (For Multicalibration And Regression Boosting) +We now give a single algorithm, and then show how to analyze it both as an algorithm for obtaining a +multicalibrated predictor f, and as a boosting algorithm for squared error regression. +Let m P N` be a discretization term, and let r1{ms :“ t0, 1 +m, . . . , m´1 +m , 1u denote the set of points in r0, 1s +that are multiples of 1{m. We will learn a model f whose range is r1{ms, which we will enforce by rounding +its outputs to this range as necessary using the following operation: +Definition 4.1 (Roundpf; mq). Let F be the family of all functions f : X Ñ R. Let Round : F ˆ N` Ñ F +be a function such that Roundpf; mq outputs ˜hpxq “ minvPr1{ms |hpxq ´ v|. +Unlike other algorithms for multicalibration which make use of agnostic learning oracles for binary +classification, our algorithm makes use of an algorithm for solving squared-error regression problems over H: +Definition 4.2. AH is a squared error regression oracle for a class of real valued functions H if for every +D P ∆Z, AHpDq outputs a function h P H such that +h P arg min +h1PH +E +px,yq„Drph1pxq ´ yq2s. +7 + +For example, if H is the set of all linear functions, then AH simply solves a linear regression problem +(which has a closed form solution). Algorithm 1 (LSBoost2)repeats the following operation until it no longer +decreases overall squared error: it runs squared error regression on each of the level-sets of ft, and then +replaces those levelsets with the solutions to the regression problems, and rounds the output to r1{ms. +We will now analyze the algorithm first as a multicalibration algorithm, and then as a boosting algorithm. +For simplicity, in this section we will analyze the algorithm as if it is given direct access to the distribution +D. In practice, the algorithm will be run on the empirical distribution over a dataset D „ Dn, and the +multicalibration guarantees proven in this section will hold for this empirical distribution. In Section A +we prove generalization theorems, which allow us to translate our in-sample error and multicalibration +guarantees over D to out-of-sample guarantees over D. +Algorithm 1: LSBoost(f, α, AH, D, B) +Let m “ 2B +α . +Let f0 “ Roundpf; mq, err0 “ Epx,yq„Drpf0pxq ´ yq2s, +err´1 “ 8 and t “ 0. +while perrt´1 ´ errtq ě +α +2B do +for each v P r1{ms do +Let Dt`1 +v +“ D|pftpxq “ vq. +Let ht`1 +v +“ AHpDt`1 +v +q. +Let: +˜ft`1pxq “ +ÿ +vPr1{ms +1rftpxq “ vs ¨ ht`1 +v +pxq +ft`1 “ Roundp ˜ft`1, mq +Let errt`1 “ Epx,yq„Drpft`1pxq ´ yq2s and t “ t ` 1. +Output ft´1. +4.1 +Analysis as a Multicalibration Algorithm +Theorem 4.3. Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, any α ă 1, any class of real valued +functions H that is closed under affine transformations, and a squared error regression oracle AH for H. +For any bound B ą 0 let: +HB “ th P H : max +xPX hpxq2 ď Bu +be the set of functions in h with squared magnitude bounded by B. Then LSBoostpf, α, AH, D, Bq (Algorithm +1) halts after at most T ď 2B +α many iterations and outputs a model fT ´1 such that fT ´1 is α-approximately +multicalibrated with respect to D and HB. +Remark 4.4. Note the form of this theorem — we do not promise multicalibration at approximation param- +eter α for all of H, but only for HB — i.e. those functions in H satisfying a bound on their squared value. +This is necessary, since H is closed under affine transformations. To see this, note that if Erhpxqpy´vqs ě α, +then it must be that Erc ¨ hpxqpy ´ vqs ě c ¨ α. Since h1pxq “ chpxq is also in H by assumption, approximate +multicalibration bounds must always also be paired with a bound on the norm of the functions for which we +promise those bounds. +Proof. Since f0 takes values in r0, 1s and y P r0, 1s, we have err0 ď 1, and by definition errT ě 0 for all T. +By construction, if the algorithm has not halted at round t it must be that errt ď errt´1 ´ +α +2B , and so the +algorithm must halt after at most T ď 2B +α many iterations to avoid a contradiction. +It remains to show that when the algorithm halts at round T, the model fT ´1 that it outputs is α- +approximately multi-calibrated with respect to D and HB. We will show that if this is not the case, then +errT ´1 ´ errT ą +α +2B , which will be a contradiction to the halting criterion of the algorithm. +2LSBoost can be taken to stand for either “Level Set Boost" or “Least Squares Boost”, at the reader’s discretion. +8 + +Suppose that fT ´1 is not α-approximately multicalibrated with respect to D and HB. This means there +must be some h P HB such that: +ÿ +vPr1{ms +Pr +px,yq„DrfT ´1pxq “ vs +ˆ +E +px,yq„Drhpxqpy ´ vq|fT ´1pxq “ vs +˙2 +ą α +For each v P r1{ms define +αv “ +Pr +px,yq„DrfT ´1pxq “ vs +ˆ +E +px,yq„Drhpxqpy ´ vq|fT ´1pxq “ vs +˙2 +So we have ř +vPr1{ms αv ą α. +Applying the 1st part of Theorem 3.2 we learn that for each v, there must be some hv P H such that: +ErpfT ´1pxq ´ yq2 ´ phvpxq ´ yq2|fT ´1pxq “ vs +ą +1 +Erhpxq2|fT ´1pxq “ vs ¨ +αv +Prpx,yq„DrfT ´1pxq “ vs +ě +1 +B +αv +Prpx,yq„DrfT ´1pxq “ vs +where the last inequality follows from the fact that h P HB Now we can compute: +E +px,yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2s +“ +ÿ +vPr1{ms +Pr +px,yq„DrfT ´1pxq “ vs +E +px,yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2|fT ´1pxq “ vs +“ +ÿ +vPr1{ms +Pr +px,yq„DrfT ´1pxq “ vs +E +px,yq„DrpfT ´1pxq ´ yq2 ´ phT +v pxq ´ yq2|fT ´1pxq “ vs +ě +ÿ +vPr1{ms +Pr +px,yq„DrfT ´1pxq “ vs +E +px,yq„DrpfT ´1pxq ´ yq2 ´ phvpxq ´ yq2|fT ´1pxq “ vs +ě +ÿ +vPr1{ms +αv +B +ą +α +B +Here the third line follows from the definition of ˜fT and the fourth line follows from the fact hv P H and +that hT +v minimizes squared error on DT +v amongst all h P H. +Finally we calculate: +errT ´1 ´ errT +“ +E +px,yq„DrpfT ´1pxq ´ yq2 ´ pfT pxq ´ yq2s +“ +E +px,yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2s ` +E +px,yq„Drp ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2s +ą +α +B ` +E +px,yq„Drp ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2s +ą +α +B ´ 1 +m +ě +α +2B +where the last equality follows from the fact that m ě 2B +α . +9 + +The 2nd inequality follows from the fact that for every pair px, yq: +p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 ě ´ 1 +m +To see this we consider two cases. Since y P r0, 1s, if ˜fT pxq ą 1 or ˜fT pxq ă 0 then the Round operation +decreases squared error and we have p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 ě 0. +In the remaining case we have +fT pxq P r0, 1s and ∆ “ ˜fT pxq ´ fT pxq is such that |∆| ď +1 +2m. In this case we can compute: +p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 +“ +pfT pxq ` ∆ ´ yq2 ´ pfT pxq ´ yq2 +“ +2∆pfpxq ´ yq ` ∆2 +ě +´2|∆| ` ∆2 +ě +´ 1 +m +4.2 +Analysis as a Boosting Algorithm +We now analyze the same algorithm (Algorithm 1) as a boosting algorithm designed to boost a “weak +learning” algorithm AH to a strong learning algorithm. Often in the boosting literature, a “strong learning” +algorithm is one that can obtain accuracy arbitrarily close to perfect, which is only possible under strong +realizability assumptions. In this paper, by “strong learning”, we mean that Algorithm 1 should output +a model that is close to Bayes optimal, which is a goal we can enunciate for any distribution D without +needing to make realizability assumptions. (Observe that if the Bayes optimal predictor has zero error, then +our meaning of strong learning corresponds to the standard meaning, so our analysis is only more general). +We now turn to our definition of weak learning. Intuitively, a weak learning algorithm should return a +hypothesis that makes predictions that are slightly better than trivial whenever doing so is possible. We +take “trivial” predictions to be those of the best constant predictor as measured by squared error — i.e. +the squared error obtained by simply returning the label mean. A “weak learning” algorithm for us can be +run on any restriction of the data distribution D to a subset S Ď X, and must return a hypothesis with +squared error slightly better than the squared error of the best constant prediction, whenever the Bayes +optimal predictor f ˚ has squared error slightly better than a constant predictor; on restrictions for which +the Bayes optimal predictor also does not improve over constant prediction, our weak learning algorithm is +not required to do better either. +Traditionally, “weak learning” assumptions do not distinguish between the optimization ability of the +algorithm and the representation ability of the hypothesis class it optimizes over. Since we have defined a +squared error regression oracle AH as exactly optimizing the squared error over some class H, we will state +our weak learning assumption as an assumption on the representation ability of H—but this is not important +for our analysis here. To prove Theorem 4.6 we could equally well assume that AH returns a hypothesis +h that improves over a constant predictor whenever one exists, without assuming that h optimizes squared +error over all of H. +Definition 4.5 (Weak Learning Assumption). Fix a distribution D P ∆Z and a class of functions H. Let +f ˚pxq “ Ey„Dpxqrys denote the true conditional label expectation conditional on x. We say that H satisfies +the γ-weak learning condition relative to D if for every S Ď X with Prx„DX rx P Ss ą 0, if: +Erpf ˚pxq ´ yq2|x P Ss ă min +cPR Erpc ´ yq2|x P Ss ´ γ +then there exists an h P H such that: +Erphpxq ´ yq2|x P Ss ă min +cPR Erpc ´ yq2|x P Ss ´ γ +When γ “ 0 we simply say that H satisfies the weak learning condition relative to D. +10 + +Observe why our weak learning assumption is “weak”: the Bayes optimal predictor f ˚ may improve +arbitrarily over the best constant predictor on some set S in terms of squared error, but in this case we only +require of H that it include a hypothesis that improves by some γ which might be very small. +Since the γ-weak learning condition does not make any requirements on H on sets for which f ˚pxq +improves over a constant predictor by less than γ, the best we can hope to prove under this assumption is +γ-approximate Bayes optimality, which is what we do next. +Theorem 4.6. Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, any γ ą 0, any class of real valued +functions H that satisfies the γ-weak learning condition relative to D, and a squared error regression oracle +AH for H. Let α “ γ and B “ 1{γ (or any pair such that α{B “ γ2). Then LSBoostpf, α, AH, D, Bq halts +after at most T ď +2 +γ2 many iterations and outputs a model fT ´1 such that fT ´1 is 2γ-approximately Bayes +optimal over D: +E +px,yq„DrpfT ´1pxq ´ yq2s ď +E +px,yq„Drpf ˚pxq ´ yq2s ` 2γ +where f ˚pxq “ Epx,yq„Drys is the function that minimizes squared error over D. +Proof. At each round t before the algorithm halts, we have by construction that errt ď errt´1 ´ +α +2B , and +since the squared error of f0 is at most 1, and squared error is non-negative, we must have T ď 2B +α “ +2 +γ2 . +Now suppose the algorithm halts at round T and outputs fT ´1. It must be that errT ą errT ´1 ´ γ2 +2 . +Suppose also that fT ´1 is not 2γ-approximately Bayes optimal: +E +px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2s ą 2γ +We can write this condition as: +ÿ +vPr1{ms +PrrfT ´1pxq “ vs ¨ +E +px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ą 2γ +Define the set: +S “ tv P r1{ms : +E +px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ě γu +to denote the set of values v in the range of fT ´1 such that conditional on fT ´1pxq “ v, fT ´1 is at least +γ-sub-optimal. Since we have both y P r0, 1s and fT ´1pxq P r0, 1s, for every v we must have that ErpfT ´1pxq´ +yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ď 1. Therefore we can bound: +2γ +ă +ÿ +vPr1{ms +PrrfT ´1pxq “ vs ¨ +E +px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs +ď +Pr +px,yq„Drx P Ss ` p1 ´ +Pr +px,yq„Drx P Ssqγ +Solving we learn that: +Pr +px,yq„Drx P Ss ě 2γ ´ γ +p1 ´ γq ě 2γ ´ γ “ γ +Now observe that by the fact that H is assumed to satisfy the γ-weak learning assumption with respect +to D, at the final round T of the algorithm, for every v P S we have that hT +v satisfies: +E +px,yq„DrpfT ´1pxq ´ yq2 ´ phT +v pxq ´ yq2|fT ´1pxq “ vs ě γ +Let ˜ +errT “ Epx,yq„Drp ˜fT pxq ´ yq2s Therefore we have: +errT ´1 ´ ˜ +errT +“ +ÿ +vPr1{ms +Pr +px,yq„DrfT ´1pxq “ vs +E +px,yq„DrpfT ´1pxq ´ yq2 ´ phT +v pxq ´ yq2|fT ´1pxq “ vs +ě +Pr +px,yq„DrfT ´1pxq P Ssγ +ě +γ2 +11 + +We recall that | ˜ +errT ´ errT | ď 1{m “ γ2 +2 and so we can conclude that +errT ´1 ´ errT ě γ2 +2 +which contradicts the fact that the algorithm halted at round T, completing the proof. +5 +When Multicalibration Implies Accuracy +We analyzed the same algorithm (Algorithm 1) as both an algorithm for obtaining multicalibration with +respect to H, and, when H satisfied the weak learning condition given in Definition 4.5, as a boosting +algorithm that converges to the Bayes optimal model. In this section we show that this is no coincidence: +multicalibration with respect to H implies Bayes optimality if and only if H satisfies the weak learning +condition from Definition 4.5, +First we define what we mean when we say that multicalibration with respect to H implies Bayes opti- +mality. Note that the Bayes optimal model f ˚pxq is multicalibrated with respect to any set of functions, so +it is not enough to require that there exist Bayes optimal functions f that are multicalibrated with respect +to H. Instead, we have to require that every function that is multicalibrated with respect to H is Bayes +optimal: +Definition 5.1. Fix a distribution D P ∆Z. We say that multicalibration with respect to H implies Bayes +optimality over D if for every f : X Ñ R that is multicalibrated with respect to D and H, we have: +E +px,yq„Drpfpxq ´ yq2s “ +E +px,yq„Drpf ˚pxq ´ yq2s +Where f ˚pxq “ Ey„Dpxqrys is the function that has minimum squared error over the set of all functions. +Recall that when the weak learning parameter γ in Definition 4.5 is set to 0, we simply call it the “weak +learning condition” relative to D. We first state and prove our characterization for the exact case when +γ “ 0, because it leads to an exceptionally simple statement. We subsequently extend this characterization +to relate approximate Bayes optimality and approximate multicalibration under quantitative weakenings of +the weak learning condition. +Theorem 5.2. Fix a distribution D P ∆Z. +Let H be a class of functions that is closed under affine +transformation. Multicalibration with respect to H implies Bayes optimality over D if and only if H satisfies +the weak learning condition relative to D. +Proof. To avoid measurability issues we assume that models f have a countable range (which is true in +particular whenever X is countable). +First we show that if H satisfies the weak learning condition relative to D, then multicalibration with +respect to H implies Bayes optimality over D. Suppose not. Then there exists a function f that is multical- +ibrated with respect to D and H, but is such that: +E +px,yq„Drpfpxq ´ yq2s ą +E +px,yq„Drpf ˚pxq ´ yq2s +By linearity of expectation we have: +ÿ +vPRpfq +Prrfpxq “ vs ¨ +E +px,yq„Drpfpxq ´ yq2 ´ pf ˚pxq ´ yq2|fpxq “ vs ą 0 +In particular there must be some v P Rpfq with Prx„DX rfpxq “ vs ą 0 such that: +E +px,yq„Drpfpxq ´ yq2|fpxq “ vs ą +E +px,yq„Drpf ˚pxq ´ yq2|fpxq “ vs +12 + +Let S “ tx : fpxq “ vu. Observe that if H is closed under affine transformation, the constant function +hpxq “ 1 is in H, and hence multicalibration with respect to H implies calibration. Since f is calibrated, we +know that: +E +px,yq„Drpv ´ yq2|x P Ss “ min +cPR +E +px,yq„Drpc ´ yq2|x P Ss +Thus by the weak learning assumption there must exist some h P H such that: +Erpv ´ yq2 ´ phpxq ´ yq2|x P Ss “ Erpfpxq ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ą 0 +By Theorem 3.2, there must therefore exist some h1 P H such that: +E +px,yq„Drh1pxqpy ´ vq|fpxq “ vs ą 0 +implying that f is not multicalibrated with respect to D and H, a contradiction. +In the reverse direction, we show that for any H that does not satisfy the weak learning condition with +respect to D, then multicalibration with respect to H and D does not imply Bayes optimality over D. In +particular, we exhibit a function f such that f is multicalibrated with respect to H and D, but such that: +E +px,yq„Drpfpxq ´ yq2s ą +E +px,yq„Drpf ˚pxq ´ yq2s +Since H does not satisfy the weak learning assumption over D, there must exist some set S Ď X with +Prrx P Ss ą 0 such that +E +px,yq„Drpf ˚pxq ´ yq2|x P Ss ă min +cPR +E +px,yq„Drpc ´ yq2|x P Ss +but for every h P H: +E +px,yq„Drphpxq ´ yq2|x P Ss ě min +cPR +E +px,yq„Drpc ´ yq2|x P Ss +. +Let cpSq “ Epx,yq„Dry|x P Ss. We define fpxq as follows: +fpxq “ +# +f ˚pxq +x R S +cpSq +x P S +We can calculate that: +E +px,yq„Drpfpxq ´ yq2s +“ +Pr +px,yq„Drx P Ss +E +px,yq„DrpcpSq ´ yq2|x P Ss ` +Pr +px,yq„Drx R Ss +E +px,yq„Drpf ˚pxq ´ yq2|x R Ss +ą +Pr +px,yq„Drx P Ss +E +px,yq„Drpf ˚pxq ´ yq2|x P Ss ` +Pr +px,yq„Drx R Ss +E +px,yq„Drpf ˚pxq ´ yq2|x R Ss +“ +E +px,yq„Drpf ˚pxq ´ yq2s +In other words, f is not Bayes optimal. So if we can demonstrate that f is multicalibrated with respect to +H and D we are done. Suppose otherwise. Then there exists some h P H and some v P Rpfq such that +E +px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 +By Theorem 3.2, there exists some h1 P H such that: +E +px,yq„Drph1pxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpfpxq ´ yq2|fpxq “ vs +13 + +We first observe that it must be that v “ cpSq. If this were not the case, by definition of f we would +have that: +E +px,yq„Drph1pxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpf ˚pxq ´ yq2|fpxq “ vs +which would contradict the Bayes optimality of f ˚. Having established that v “ cpSq we can calculate: +E +px,yq„Drph1pxq ´ yq2|fpxq “ cpSqs +“ +Pr +px,yq„Drx P Ss +E +px,yq„Drph1pxq ´ yq2|x P Ss ` +Pr +px,yq„Drx R S, fpxq “ cpSqs +E +px,yq„Drph1pxq ´ yq2|x R S, fpxq “ cpSqs +ě +Pr +px,yq„Drx P Ss +E +px,yq„Drph1pxq ´ yq2|x P Ss ` +Pr +px,yq„Drx R S, fpxq “ cpSqs +E +px,yq„Drpfpxq ´ yq2|x R S, fpxq “ cpSqs +where in the last inequality we have used the fact that by definition, fpxq “ f ˚pxq for all x R S, and so is +pointwise Bayes optimal for all x R S. +Hence the only way we can have Epx,yq„Drph1pxq ´ yq2|fpxq “ cpSqs ă Epx,yq„Drpfpxq ´ yq2|fpxq “ cpSqs +is if: +E +px,yq„Drph1pxq ´ yq2|x P Ss ă +E +px,yq„DrpcpSq ´ yq2|x P Ss +But this contradicts our assumption that H violates the weak learning condition on S, which completes the +proof. +We now turn our attention to deriving a relationship between approximate multicalibration and approx- +imate Bayes optimality. To do so, we’ll introduce an even weaker weak learning condition that has one +additional parameter ρ, lower bounding the mass of sets S that we can condition on while still requiring +the weak learning condition to hold. We remark that Algorithm 1 can be analyzed as a boosting algorithm +under this weaker weak learning assumption as well, with only minor modifications in the analysis. +Definition 5.3 ( pγ, ρq-weak learning condition). Fix a distribution D P ∆Z and let H be a class of arbitrary +real-valued functions. We say that H satisfies the pγ, ρq-weak learning condition for D if the following holds. +For every set S Ď X such that Prx„DX rx P Ss ą ρ, if +E +px,yq„Drpf ˚ ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, +where ¯yS “ Epx,yq„Dry | x P Ss, then there exists h P H such that +E +px,yq„Drphpxq ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ. +We may now prove our theorem showing that approximate multicalibration with respect to a class H +implies approximate Bayes optimality if and only if H satisfies the pγ, ρq-weak learning condition. We recall +Remark 4.4, which notes that we must restrict approximate multicalibration to a bounded subset of H, as +we will assume that H is closed under affine transformation. +Theorem 5.4. Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, and any class of real valued functions +H that is closed under affine transformation. Let: +H1 “ th P H : max +xPX hpxq2 ď 1u +14 + +be the set of functions in H upper-bounded by 1 on X. Let m “ |Rpfq|, γ ą 0, and α ď +γ3 +16m. Then if H +satisfies the pγ, γ{mq-weak learning condition and f is α-approximately multicalibrated with respect to H1 on +D, then f has squared error +E +px,yq„Drpfpxq ´ yq2s ď +E +px,yq„Drpf ˚ ´ yq2s ` 3γ. +Conversely, if H does not satisfy the pγ, γ{mq-weak learning condition, there exists a model f : X Ñ r0, 1s +that is α-approximately multicalibrated with respect to H1 on D, for α “ γ, and is perfectly calibrated on D, +but f has squared error +E +px,yq„Drpfpxq ´ yq2s ě +E +px,yq„Drpf ˚ ´ yq2s ` γ2{m. +Proof. We begin by arguing that α-approximate multicalibration with respect to H1 on D implies approxi- +mate Bayes optimality when H satisfies the pγ, γ{mq-weak learning condition. Suppose not, and there exists +a function f that is α-multicalibrated with respect to H1, but +E +px,yq„Drpf ˚ ´ yq2s ă +E +px,yq„Drpfpxq ´ yq2s ´ 3γ. +Then there must exist some v P Rpfq such that Prpx,yq„Drfpxq “ vs ą γ{m and +E +px,yq„Drpf ˚ ´ yq2 | fpxq “ vs ă +E +px,yq„Drpfpxq ´ yq2 | fpxq “ vs ´ 2γ. +We observe that since H is closed under affine transformation, the constant function hpxq “ 1 is in H, and +so α-approximate multicalibration with respect to H1 implies α-approximate calibration as well. Thus by +definition, +Prrfpxq “ vs ¨ +ˆ +E +px,yq„Drv ´ y | fpxq “ vs +˙2 +ď α. +Letting ¯yv “ Ery | fpxq “ vs, our lower-bound that Prrfpxq “ vs ą γ{m gives us that pv ´ ¯yvq2 ă αm{γ ď +` γ +4 +˘2. We now use this upper-bound on calibration error in conjuction with our lower-bound on distance +from Bayes optimality to show that the squared error of the constant predictor ¯yv must also be far from +Bayes optimal. +E +px,yq„Drpf ˚pxq ´ yq2 | fpxq “ vs ă +E +px,yq„Drpfpxq ´ yq2 | fpxq “ vs ´ 2γ +“ +E +px,yq„Drpv ´ ¯yv ` ¯yv ´ yq2 | fpxq “ vs ´ 2γ +“ +E +px,yq„Drp¯yv ´ yq2 | fpxq “ vs ` pv ´ ¯yvq2 ´ 2γ +ă +E +px,yq„Drp¯yv ´ yq2 | fpxq “ vs ´ γ. +The pγ, γ{mq-weak learning condition then guarantees that there exists some h P H such that +E +px,yq„Drph ´ yq2 | fpxq “ vs ă +E +px,yq„Drp¯yv ´ yq2 | fpxq “ vs ´ γ. +By Lemma 3.4, the fact that h improves on the squared loss of ¯yv by an additive factor γ, on the set of x +such that fpxq “ v, implies that Erhpxqpy ´ ¯yvq | fpxq “ vs ą γ{2. Because f is α-approximately calibrated +on D, we can use the existence of such an h to witness a failure of multicalibration: +Erhpy ´ vq | fpxq “ vs +“ Erhpxqpy ´ ¯yv ` ¯yv ´ vq | fpxq “ vs +“ Erhpxqpy ´ ¯yvq | fpxq “ vs ` Erhpxqp¯yv ´ vq | fpxq “ vs +ą γ{2 ´ |¯yv ´ v| +ą γ{4. +15 + +Then +Prrfpxq “ vs ¨ +ˆ +E +px,yq„Drhpxqpy ´ vq | fpxq “ vs +˙2 +ą +γ3 +16m, +contradicting our assumption that f is α-approximately multicalibrated with respect to H1 for α ă +γ3 +16m. +Therefore approximate multicalibration with respect to H1 must imply that f is approximately Bayes opti- +mal. +It remains to show the other direction, that α-approximate multicalibration with respect to a class H1 +implies approximate Bayes optimality only if H satisfies the pγ, γ{mq-weak learning condition. If this claim +were not true for the stated parameters, then there must exist a class H such that every predictor f that: +• is α-approximately multicalibrated with respect to H1 +• is perfectly calibrated on D +• has range with cardinality |Rpfq| “ m +also has squared error within γ2{m of Bayes optimal, but H does not satisfy the weak learning condition. +We will show that no such class exists by defining, for any class H not satisfying the weak learning condition, +a predictor f that is α-approximately multicalibrated with respect to that class, but has squared error that +is not within γ2{m of Bayes optimal. +Recall that if a class H does not satisfy the pγ, γ{mq-weak learning condition, then there must be some +set SH such that Prrx P SHs ą γ{m, there does not exist an h P H such that +E +px,yq„Drph ´ yq2 | x P SHs ă +E +px,yq„Drp¯ySH ´ yq2 | x P SHs ´ γ, +but for the Bayes optimal predictor, it holds that its squared loss satisfies +E +px,yq„Drpf ˚ ´ yq2 | x P SHs ă +E +px,yq„Drp¯ySH ´ yq2 | x P SHs ´ γ, +where ¯ySH “ Ery | x P SHs. For some hypothesis class H not satisfying the weak learning condition, and +associated set SH, let fH be defined as follows: +fHpxq “ +# +f ˚pxq, +x R SH +¯ySH, +x P SH. . +Note that, because fH is constant on SH, there must be some v P Rpfq such that the level set Sv “ tx P +X : fpxq “ vu contains SH. To see that fH is α-approximately multicalibrated with respect to H1, we first +consider the contribution to multicalibration error from the level sets not containing SH. For all h P H and +v P Rpfq such that v ‰ ¯ySH, +E +px,yq„Drhpxqpy ´ fHpxqq | fHpxq “ vs “ +E +px,yq„Drhpxqpy ´ f ˚pxqq | fHpxq “ vs +“ +E +x„Dx +E +y„Dypxqrhpxqy | fHpxq “ vs ´ +E +x„Dx +rhpxqf ˚pxq | fHpxq “ vs +“ +E +x„Dx +E +y„Dypxqrhpxqy | fHpxq “ vs ´ +E +x„Dx +E +y„Dypxqrhpxqy | fHpxq “ vs +“ 0. +For the level set Sv for which SH Ď Sv, we know from the argument above that the elements x P SvzSH +contribute nothing to the multicalibration error, as fpxq “ f ˚pxq on these elements. So, +E +px,yq„Drhpxqpy ´ fHpxqq | fpxq “ vs “ +Pr +x„DXrx P SHs ¨ +E +px,yq„Drhpxqpy ´ ¯ySHq | x P SHs +` +Pr +x„DXrx R SHs ¨ +E +px,yq„Drhpxqpy ´ f ˚pxqq | x P SvzSHs +“ +Pr +x„DXrx P SHs ¨ +E +px,yq„Drhpxqpy ´ ¯ySHq | x P SHs +16 + +Therefore if fH is not α-approximately multicalibrated with respect to H1 on D, it must be the case that +there exists some h P H1 such that Erhpxqpy ´ ¯ySHq | x P SHs ą ?α. Then by Theorem 3.2, there must exist +a h1 P H such that +E +px,yq„Drp¯ySH ´ yq2 ´ ph1pxq ´ yq2 | x P SHs ą α “ γ. +But SH was defined to be a subset of X for which no such h1 exists and for which Prrx P SHs ą γ{m. This +would contradict our assumption that H does not satisfy the pγ, γ{mq-weak learning condition on D, and +therefore fH is α-approximately multicalibrated with respect to H1 on D. +It remains to prove that fH is far from Bayes optimal. +E +px,yq„DrpfHpxq ´ yq2s “ +Pr +x„DXrx P SHs +E +px,yq„Drp¯ySH ´ yq2 | x P SHs ` Prrx R SHs +E +px,yq„Drpf ˚pxq ´ yq2 | x R SHs +ě +Pr +x„DXrx P SHs +ˆ +E +px,yqDrpf ˚ ´ yq2 | x P SHs ` γ +˙ +` Prrx R SHs +E +px,yq„Drpf ˚pxq ´ yq2 | x R SHs +“ +E +px,yq„Drpf ˚ ´ yq2s ` γ +Pr +x„DXrx P SHs +ě +E +px,yq„Drpf ˚ ´ yq2s ` γ2{m. +6 +Weak Learners With Respect to Constrained Classes +Thus far we have studied function classes H that satisfy a weak learning condition with respect to the Bayes +optimal predictor f ˚. But we can also study function classes H that satisfy a weak learning condition defined +with respect to another constrained class of real valued functions. +Definition 6.1 (Weak Learning Assumption Relative to C). Fix a distribution D P ∆Z and two classes of +functions H and C. We say that H satisfies the γ-weak learner condition relative to C and D if for every +S Ď X with Prx„DX rx P Ss ą 0, if: +min +cPC +E +px,yq„Drpcpxq ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, +where ¯yS “ Epx,yq„Dry | x P Ss, then there exists h P H such that +E +px,yq„Drphpxq ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ. +When γ “ 0 we simply say that H satisfies the weak learning condition relative to C and D. +We will show that if a predictor f is multicalibrated with respect to H, and H satisfies the weak learning +assumption with respect to C, then in fact: +1. f is multicalibrated with respect to C, and +2. f has squared error at most that of the minimum error predictor in C. +In fact, Gopalan et al. [2022] show that if f is multicalibrated with respect to C, then it is an omnipredictor +for C, which implies that f has loss no more than the best function cpxq P C, where loss can be measured +with respect to any Lipschitz convex loss function (not just squared error). Thus our results imply that to +obtain an omnipredictor for C, it is sufficient to be multicalibrated with respect to a class H that satisfies +our weak learning assumption with respect to C. +Theorem 6.2. Fix a distribution D P ∆Z and two classes of functions H and C that are closed under affine +transformations. Then if f : X Ñ r0, 1s is multicalibrated with respect to D and H, and if H satisfies the +weak learning condition relative to C and D, then in fact f is multicalibrated with respect to D and C as well. +17 + +Proof. We assume for simplicity that f has a countable range (which is without loss of generality e.g. +whenever X is countable). Suppose for contradiction that f is not multicalibrated with respect to C and D. +In this case there must be some c P C such that: +ÿ +vPRpfq +Prrfpxq “ vs +ˆ +E +px,yq„Drcpxqpy ´ vq|fpxq “ vs +˙2 +ą 0 +Since C is closed under affine transformations (and so both c and ´c are in C), there must be some c1 P C +and some v P Rpfq with Prrfpxq “ vs ą 0 such that: +E +px,yq„Drc1pxqpy ´ vq|fpxq “ vs ą 0 +Therefore, by the first part of Theorem 3.2, there must be some c2 P C such that: +E +px,yq„Drpc2pxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpv ´ yq2|fpxq “ vs +Since H is closed under affine transformations, the function hpxq “ 1 is in H and so multicalibration with +respect to H implies calibration. Thus v “ ¯ySv for Sv “ tx : fpxq “ vu. Therefore, the fact that H satisfies +the weak learning condition relative to C and D implies that there must be some h P H such that: +E +px,yq„Drphpxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpv ´ yq2|fpxq “ vs +Finally, the second part of Theorem 3.2 implies that: +E +px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 +which is a violation of our assumption that f is multicalibrated with respect to H and D, a contradiction. +Theorem 6.3. Fix a distribution D P ∆Z and two classes of functions H and C. Then if f : X Ñ r0, 1s is +calibrated and multicalibrated with respect to D and H, and if H satisfies the weak learning condition relative +to C and D, then: +E +px,yq„Drpfpxq ´ yq2s ď min +cPC +E +px,yq„Drpcpxq ´ yq2s +Proof. We assume for simplicity that f has a countable range (which is without loss of generality e.g. +whenever X is countable). Suppose for contradiction that there is some c P C such that: +E +px,yq„Drpcpxq ´ yq2s ă +E +px,yq„Drpfpxq ´ yq2s +Then there must be some v P Rpfq with Prrfpxq “ vs ą 0 and: +E +px,yq„Drpcpxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpv ´ yq2|fpxq “ vs +Since f is calibrated, v “ ¯ySv for Sv “ tx : fpxq “ vu. Therefore, the fact that H satisfies the weak learning +condition relative to C and D implies that there must be some h P H such that: +E +px,yq„Drphpxq ´ yq2|fpxq “ vs ă +E +px,yq„Drpv ´ yq2|fpxq “ vs +Finally, the second part of Theorem 3.2 implies that: +E +px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 +which is a violation of our assumption that f is multicalibrated with respect to H and D, a contradiction. +18 + +We now turn to approximate versions of these statements. To do so, we need a refined version of one +direction of Theorem 3.2 that shows us that if f witnesses a failure of multicalibration with respect to some +h P H, then there is another function h1 P H that can be used to improve on f’s squared error, while +controlling the norm of h1. +Lemma 6.4. Suppose H is closed under affine transformation. Fix a model f : X Ñ r0, 1s, a levelset +v P Rpfq, and a bound B ą 0. Then if there exists an h P H such that maxxPX hpxq2 ď B and +Erhpxqpy ´ vq|fpxq “ vs ě α, +for α ě 0, then there exists an h1 P H such that maxxPX h1pxq2 ď p1 ` +? +B +α q2 and: +Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 +B . +Proof. Let h1pxq “ v ` ηhpxq where η “ +α +Erhpxq2|fpxq“vs, as in Theorem 3.2. Because hpxq2 is uniformly +bounded by B on X, it follows that Erhpxq2s ď B, and we have already shown in the proof of Theorem 3.2 +that this implies +Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 +B . +It only remains to bound maxxPX h1pxq2. We begin by lower-bounding Erhpxq2 | fpxq “ vs in terms of α. +Erhpxq2 | fpxq “ vs ě Erhpxq | fpxq “ vs2 +ě Erhpxqpy ´ vq | fpxq “ vs2 +ě α2. +It follows that η ď 1{α, and so +max +xPX h1pxq2 “ max +xPX pv ` ηhpxqq2 +ď p1 ` η +? +Bq2 +ď +˜ +1 ` +? +B +α +¸2 +. +We will also need a parameterized version of our weak learning condition. Recalling Remark 4.4, for +approximate multicalibration to be meaningful with respect to a class that is closed under affine transfor- +mation, we must specify a bounded subset of that class with respect to which a predictor is approximately +multicalibrated. Then to show that approximate multicalibration with respect to one potentially unbounded +class implies approximate multicalibration with respect to another, we will need to specify the subsets of +each class with respect to which a predictor is claimed to be approximately multicalibrated. This motivates +a parameterization of our previous weak learning condition relative to a class C. We will need to assume that +whenever there is a B-bounded function in C that improves over the best constant predictor on a restriction +of D, there also exists a B-bounded function in H that improves on the restriction as well. +Definition 6.5 (B-Bounded Weak Learning Assumption Relative to C). Fix a distribution D P ∆Z and +two classes of functions H and C. Fix a bound B ą 0 and let HB and CB denote the sets +HB “ th P H : max +xPX hpxq2 ď Bu +and +CB “ tc P C : max +xPX cpxq2 ď Bu +19 + +respectively. We say that H satisfies the B-bounded γ-bounded weak learning condition relative to C and D +if for every S Ď X with Prx„DX rx P Ss ą 0, if: +min +cPCB +E +px,yq„Drpcpxq ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, +where ¯yS “ Ery | x P Ss, then there exists h P HB such that +E +px,yq„Drphpxq ´ yq2 | x P Ss ă +E +px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ. +Theorem 6.6. Fix a distribution D P ∆Z and two classes of functions H and C that are closed under affine +transformations. Fix αC, B ą 0. Let B1 “ p1 ` +b +2B +αC q2 and γ “ αC +4B . Fix a function f : X Ñ r0, 1s that maps +into a countable subset of its range, and let m “ |Rpfq|, αH ă +α3 +C +29mB12 , and α ă +αCγ2 +32mB12 . Then if +• H satisfies the B1-bounded γ-weak learning condition relative to C and D +• f is αH-approximately multicalibrated with respect to D and HB1 +• f is α-approximately calibrated on D, +then f is αC-approximately multicalibrated with respect to D and CB. +Proof. Suppose not and there exists some c P CB such that +ÿ +vPRpfq +Pr +x„Dxrfpxq “ vs ¨ +ˆ +E +px,yq„Drcpxqpy ´ vq | fpxq “ vs +˙2 +ą αC. +Then there must exist some v P Rpfq such that Prrfpxq “ vs ą αC +2m and +E +px,yq„Drcpxqpy ´ vq | fpxq “ vs2 ą αC{2. +Because C is closed under affine transformations, CB is closed under negation, so there must also exist some +c1 P CB such that +E +px,yq„Drc1pxqpy ´ vq | fpxq “ vs ą +a +αC{2. +Then Lemma 3.3 shows that there is a c2 P Cp1` +b +2B +αC q2 “ CB1 such that +E +px,yq„Drpy ´ fpxqq2 ´ py ´ c2pxqq2 | fpxq “ vs ě αC +2B “ 2γ. +Because f is α-calibrated on D, by definition we have +Pr +x„Dxrfpxq “ vs ¨ +ˆ +E +px,yq„Drv ´ y | fpxq “ vs +˙2 +ă α. +Letting ¯yv “ Ery | fpxq “ vs, our lower-bound that Prrfpxq “ vs ą αC +2m gives us that pv ´ ¯yvq2 ă 2αm +αC +ď +γ2 +16B12 ă γ. So, because v is close to ¯yv, we can show the squared error of f must be close to the squared +error of ¯yv on this level set. +E +px,yq„Drpy ´ fpxqq2 | fpxq “ vs “ +E +px,yq„Drpy ´ ¯yv ` ¯yv ´ fpxqq2 | fpxq “ vs +“ +E +px,yq„Drpy ´ ¯yvq2 ` 2py ´ ¯yvqp¯yv ´ vq | fpxq “ vs ` p¯yv ´ vq2 +“ +E +px,yq„Drpy ´ ¯yvq2 | fpxq “ vs ` p¯yv ´ vq2 +ă +E +px,yq„Drpy ´ ¯yvq2 | fpxq “ vs ` γ. +20 + +Then, because the squared error of c2 on this level set is much less than the squared error of f, we find that +c2 must also have squared error less than that of ¯yv: +E +px,yq„Drpy ´ ¯yvq2 ´ py ´ c2pxqq2 | fpxq “ vs ą +E +px,yq„Drpy ´ fpxqq2 ´ γ ´ py ´ c2pxqq2 | fpxq “ vs +ě 2γ ´ γ +“ γ +We assumed H satisfies the B1-bounded γ-weak learning condition relative to C, so this gives us a function +h P HB1 such that +E +px,yq„Drpy ´ ¯yvq2 ´ py ´ hpxqq2 | fpxq “ vs ą γ. +Then Lemma 3.3 shows that +Erhpxqpy ´ ¯yvq | fpxq “ vs ą γ{2. +So h witnesses a failure of multicalibration of f, since it follows that +Erhpxqpy ´ vq | fpxq “ vs “ Erhpxqpy ´ ¯yvq | fpxq “ vs ` Erhpxqp¯yv ´ vq | fpxq “ vs +ą γ{2 ´ B1 |¯yv ´ v| +ě γ{2 ´ B1γ +4B1 +“ γ{4 +and so +Pr +x„Dxrfpxq “ vs +ˆ +E +px,yq„Drhpxqpy ´ vq | fpxq “ vs +˙2 +ą αCγ2 +32m ą αH, +contradicting αH-approximate multicalibration of f on HB1 and D. +In Gopalan et al. [2022], Gopalan, Kalai, Reingold, Sharan, and Wieder show that any predictor that is +approximately multicalibrated for a class H and distribution D can be efficiently post-processed to approxi- +mately minimize any convex, Lipschitz loss function relative to the class H. The theorem we have just proved +can now be used to extend their result to approximate loss minimization over any other class C, so long as +H satisfies the B-bounded γ-weak learning assumption relative to C. Intuitively, this follows from the fact +that if f is approximately multicalibrated with respect to H on D, it is also approximately multicalibrated +with respect to C. However, the notion of approximate multicalibration adopted in Gopalan et al. [2022] +differs from the one in this work. So, to formalize our intuition above, we will first state the covariance-based +definition of approximate multicalibration appearing in Gopalan et al. [2022] and prove a lemma relating it +to our own. We note that, going forward, we will restrict ourselves to distributions D over X ˆ t0, 1u, as in +this case the two definitions of approximate multicalibration are straightforwardly connected. +Definition 6.7 (Approximate Covariance Multicalibration Gopalan et al. [2022]). Fix a distribution D over +X ˆ t0, 1u and a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq. Let +H be an arbitrary collection of real valued functions h : X Ñ R. Then f is α-approximately covariance +multicalibrated with respect to H on D if +ÿ +vPRpfq +Pr +x„DXrfpxq “ vs ¨ +ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs +ˇˇ ď α, +where ¯hv “ Erhpxq | fpxq “ vs and ¯yv “ Ery | fpxq “ vs. +Lemma 6.8. Fix a distribution D over X ˆ t0, 1u and a class of functions on X, H. Let HB denote the +subset +HB “ th P H : max +xPX hpxq2 ď Bu. +21 + +Fix a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq. Then if f is +α-approximately multicalibrated with respect to HB on D, then f is p?αp1 ` +? +Bqq-approximately covariance +multicalibrated. That is, for all h P HB, f satisfies +ÿ +vPRpfq +Prrfpxq “ vs ¨ +ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs +ˇˇ ď ?αp1 ` +? +Bq. +Proof. +ÿ +vPRpfq +Prrfpxq “ vs¨ +ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs +ˇˇ +“ +ÿ +vPRpfq +Prrfpxq “ vs ¨ +ˇˇErhpxqy | fpxq “ vs ´ ¯yv¯hv +ˇˇ +“ +ÿ +vPRpfq +Prrfpxq “ vs ¨ +ˇˇErhpxqy | fpxq “ vs ´ v¯hv ` v¯hv ´ ¯yv¯hv +ˇˇ +“ +ÿ +vPRpfq +Prrfpxq “ vs ¨ +ˇˇErhpxqpy ´ vq | fpxq “ vs ` ¯hvpv ´ ¯yvq +ˇˇ +ď +ÿ +vPRpfq +Prrfpxq “ vs ¨ +` +|Erhpxqpy ´ vq | fpxq “ vs| ` +ˇˇ¯hvpv ´ ¯yvq +ˇˇ˘ +ď ?α ` +? +B +ÿ +vPRpfq +Prrfpxq “ vs ¨ |v ´ ¯yv| +ď ?αp1 ` +? +Bq. +where the second inequality follows from the fact that Erxs ď +a +Erx2s and the bound maxxPX hpxq2 ď B. +We now recall a theorem of Gopalan et al. [2022], showing that approximate covariance multicalibration +with respect to a class H implies approximate loss minimization relative to H, for convex, Lipschitz losses. +Theorem 6.9. Fix a distribution D over X ˆ t0, 1u and a class of real-valued functions on X, H. Fix a +function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq. Let L be a class of +functions on t0, 1u ˆ R that are convex and L-Lipschitz in their second argument. If f is α-approximately +covariance multicalibrated with respect to HB on D, then for every ℓ P L there exists an efficient post- +processing function kℓ such that +E +px,yq„Drℓpy, kℓpfpxqqqs ď min +hPHB +E +px,yq„Drℓpy, hpxqqs ` 2αL. +Corollary 6.10. Fix a distribution D over X ˆ t0, 1u and two classes of real-valued functions on X that +are closed under affine transformation, H and C. Fix a function f : X Ñ r0, 1s that maps onto a countable +subset of its range, denoted Rpfq. Let L be a class of functions on t0, 1uˆR that are convex and L-Lipschitz +in their second argument. Fix αC, B ą 0. Let B1 “ p1 ` +b +2B +αC q2 and γ “ +αC +4B . Let αH ă +α3 +C +29mB12 , and +α ă +αCγ2 +32mB12 . Then if +• H satisfies the B1-bounded γ-weak learning condition relative to C and D +• f is αH-approximately multicalibrated with respect to D and HB1 +• f is α-approximately calibrated on D, +then for every ℓ P L there exists an efficient post-processing function kℓ such that +E +px,yq„Drℓpy, kℓpfpxqqqs ď min +cPCB +E +px,yq„Drℓpy, cpxqqs ` 2L?αCp1 ` +? +Bq. +22 + +Figure 1: The update process at round t with m level sets during training. +Proof. We have from Theorem 6.6 that given the assumed conditions, f will be αC-approximately multicali- +brated with respect to CB on D. It follows from Lemma 6.8 that f is ?αCp1` +? +Bq-approximately covariance +multicalibrated with respect to CB on D. The result of Gopalan et al. [2022] then gives us that for all ℓ P L, +there exists an efficient post-processing function kℓ such that +E +px,yq„Drℓpy, kℓpfpxqqqs ď min +cPCB +E +px,yq„Drℓpy, cpxqqs ` 2L?αCp1 ` +? +Bq. +7 +Empirical Evaluation +In this section, we study Algorithm 1 empirically via an efficient, open-source Python implementation of our +algorithm on both synthetic and real regression problems. Our code is available here: https://github.com/ +Declancharrison/Level-Set-Boosting. An important feature of Algorithm 1 which distinguishes it from +traditional boosting algorithms is the ability to parallelize not only during inference, but also during training. +Let ft be the model maintained by Algorithm 1 at round t with m level sets. Given a data set X, ft creates +a partition of X defined by Xt`1 +i +“ tx|ftpxq “ viu. Since the Xi are disjoint, each call ht`1 +i +“ AHpXt`1 +i +q +can be made on a separate worker followed by a combine and round operation to obtain ˜ft`1 and ft`1 +respectively, as shown in Figure 1. A parallel inference pass at round t works nearly identically, but uses the +historical weak learners ht`1 +i +obtained from training and applies them to each set Xt`1 +i +. +7.1 +Prediction on Synthetic Data +From Theorem 5.2, we know that multicalibration with respect to a hypothesis class H satisfying our weak +learning condition implies Bayes optimality. To visualize the fast convergence of our algorithm to Bayes +optimality, we create two synthetic datasets; each dataset contains one million samples with two features. +We label these points using two functions, C0 and C1, defined below and pictured in Figure 2). We attempt +to learn the underlying function with Algorithm 1. +C0pxq “ +$ +’ +’ +’ +& +’ +’ +’ +% +px ` 1q2 ` py ´ 1q2, +if x ď 0, y ě 0 +px ´ 1q2 ` py ´ 1q2, +if x ą 0, y ě 0 +px ` 1q2 ` py ` 1q2, +if x ď 0, y ă 0 +px ´ 1q2 ` py ` 1q2, +if x ą 0, y ă 0 +(C0) +23 + += A(Xi) +21) +二 +α(ft(α) +αl(ft(α) = vi) +Round(ft) +Round( +ft +ht+1 +AH(Xi) +Jt+1 +1(α) +hm += AH(Xm)C1pxq “ +$ +’ +’ +’ +’ +’ +’ +& +’ +’ +’ +’ +’ +’ +% +x ` 20xy2 cosp´8xq sinp8yq +´ +p1.5x`4qpx`1q2 +y`3 +` py ´ 1q2¯ +, +if x ď 0, y ě 0 +x ` 20xy2 cosp8xq sinp8yq +´ +p1.5x`4qpx´1q2 +y`3 +` py ´ 1q2¯ +, +if x ą 0, y ě 0 +x ` 20xy2 cosp´8xq sinp8yq +´ +p1.5x`4qpx`1q2 +y`3 +` py ` 1q2¯ +, +if x ď 0, y ă 0 +x ` 20xy2 cosp8xq sinp8yq +´ +p1.5x`4qpx´1q2 +y`3 +` py ` 1q2¯ +, +if x ą 0, y ă 0 +(C1) +In Figure 3, we show an example of Algorithm 1 learning C0 using a discretization of five-hundred level +sets and a weak learner hypothesis class of depth one decision trees. Each image in figure 3 corresponds to +the map produced by Algorithm 1 at the round listed in the top of the image. As the round count increases, +the number of non-empty level sets increases until each level set is filled, at which point the updates become +more granular. The termination round titled ‘final round’ occurs at T “ 199 and paints an approximate +map of C0. The image titled ‘out of sample’ is the map produced on a set of one million points randomly +drawn outside of the training sample, and shows that Algorithm 1 is in fact an approximation of the Bayes +Optimal C0. +Figure 2: C0 maps x1, x2 P r´2, 2s to four cylindrical cones symmetric about the origin. C1 maps x1, x2 P +r´1, 1s to a hilly terrain from a more complex function. +Figure 4 plots the same kind of progression as Figure 3, but with a more complicated underlying function +C1 using a variety of weak learner classes. We are able to learn this more complex surface out of sample +with all base classes except for linear regression, which results in a noisy out-of-sample plot. +7.2 +Prediction on Census Data +We evaluate the empirical performance of Algorithm 1 on US Census data compiled using the Python +folktables package Ding et al. [2021]. In this dataset, the feature space consists of demographic information +about individuals (see Table 1), and the labels correspond to the individual’s annual income. +24 + +Co(X1, X2) +¥1.0 +0.8 +0.6 +y +0.4 +0.2 +0.0 +2.0 +1.5 +1.0 +0.5 +-2.0 +0.0 +1.5 +-1.0 +-0.5 +-0.5 +0.0 +-1.0 +0.5 ++1 +1.0 +-1.5 +1.5 +-2.0 +2.0C1(X1, X2) +1.0 +0.8 +0.6 +y +0.4 +0.2 +0.0 +1.00 +0.75 +0.50 +0.25 +-1.00 +0.00 +0.75 +-0.50 +-0.25 +-0.25 +0.00 +0.50 +0.25 ++1 +-0.75 +0.50 +0.75 +1.00 -1.00Figure 3: Evolution of Algorithm 1 learning C0. +feature +description +feature +description +AGEP +age +POBP +place of birth +COW +class of worker +RELP +relationship +SCHL +education level +WKHP +work hours per week +MAR +marital status +SEX +binary sex +OCCP +occupation +RAC1P +race +Table 1: Features included in income prediction task. +We cap income at $100,000 and then rescale all labels into r0, 1s. On an 80/20% train-test split with +500,000 total samples, we compare the performance of Algorithm 1 with Gradient Boosting with two perfor- +mance metrics: mean squared error (MSE), and mean squared calibration error (MSCE). For less expressive +weak learner classes (such as DT(1), see Figure 5), Algorithm 1 has superior MSE out of sample compared to +Gradient Boosting through one hundred rounds while maintaining significantly lower MSCE, and converges +quicker. However, as the weak learning class becomes more expressive (e.g. increasing decision tree depths), +Algorithm 1 is more prone to overfitting than gradient boosting (see Figure 6). +25 + +Figure 4: Stages of Algorithm 1 learning C1 with linear regression (LR) and varying depth d decision trees +(DT(d)). In the out of sample plot for linear regression, points are not mapped to their proper position, +implying C1 cannot be learned by boosting linear functions. All other hypothesis classes eventually converge +to C1. +26 + +H +T=0 +T=1 +T=2 +Final Round +Out of Sample +LR +DT(1) +DT(2) +DT(3) +DT(4)Figure 5: Comparison of Algorithm 1 (LS) and Gradient Boosting (GB), both using depth 1 regression trees. +* indicates termination round of Algorithm 1. +In Table 2, we compare the time taken to train n weak learners with Algorithm 1 and with scikit-learn’s +version of Gradient Boosting on our census data. Recall that our algorithm trains multiple weak learners +per round of boosting, and so comparing the two algorithms for a fixed number of calls to the weak learner +is distinct from comparing them for a fixed number of rounds. Because models output by Algorithm 1 may +be more complex than those produced by Gradient Boosting run for the same number of rounds, we use +number of weak learners trained as a proxy for model complexity, and compare the two algorithms holding +this measure fixed. We see the trend for Gradient Boosting is linear with respect to number of weak learners, +whereas Algorithm 1 does not follow the same linear pattern upfront. This is due to not being able to fully +27 + +MSE with Weak Learner Depth 1 Decision Trees +train error (LS) +0.055 +test error (LS) +train error (GB) +test error (GB) +0.050 +0.040 +0.035 +0 +20 +40 +60 +80 +100 +Number of roundsMSCE with Weak Learner Depth 1 Decision Trees +train error (LS) +0.025 +test error (LS) +train error (GB) +--- test error (GB) +0.020 +0.015 +MSCE +0.010 +0.005 +0.000 +0 +20 +40 +60 +80 +100 +Number of roundsleverage parallelization of training weak learners in early stages of boosting. At each round, Algorithm 1 +calls the weak learner on every large enough level set of the current model, and it is these independent calls +that can be easily parallelized. However, in the early rounds of boosting the model may be relatively simple, +and so many level sets may be sparsely populated. As the model becomes more expressive over subsequent +rounds, the weak learner will be invoked on more sets per round, allowing us to fully utilize parallelizability. +# Weak Learners +DT(1) +DT(2) +DT(3) +LS +GB +Faster? +LS +GB +Faster? +LS +GB +Faster? +50 level sets +100 +9.11 +11.97 +✓ +5.86 +23.01 +✓ +6.88 +32.92 +✓ +300 +18.70 +35.81 +✓ +14.90 +69.17 +✓ +15.64 +102.14 +✓ +500 +27.00 +58.19 +✓ +21.74 +115.65 +✓ +24.77 +169.90 +✓ +1000 +46.73 +116.49 +✓ +42.92 +231.74 +✓ +46.38 +336.89 +✓ +100 level sets +100 +7.18 +11.97 +✓ +5.29 +23.01 +✓ +5.06 +32.92 +✓ +300 +13.08 +35.81 +✓ +13.55 +69.17 +✓ +14.72 +102.14 +✓ +500 +21.20 +58.19 +✓ +19.57 +115.65 +✓ +21.79 +169.90 +✓ +1000 +41.99 +116.49 +✓ +36.26 +231.74 +✓ +40.92 +336.89 +✓ +300 level sets +100 +5.87 +11.97 +✓ +9.18 +23.01 +✓ +6.54 +32.92 +✓ +300 +13.21 +35.81 +✓ +17.46 +69.17 +✓ +11.13 +102.14 +✓ +500 +19.05 +58.19 +✓ +22.20 +115.65 +✓ +19.64 +169.90 +✓ +1000 +32.80 +116.49 +✓ +36.61 +231.74 +✓ +27.12 +336.89 +✓ +Table 2: Time (in seconds) comparison of Algorithm 1 (LS) with fifty level sets and Gradient Boosting to +train certain numbers of estimators for various weak learner classes. +In Figure 6, we measure MSE and MSCE for Algorithm 1 and Gradient Boosting over rounds of training +on our census data. Again, we note that one round of Algorithm 1 is not equivalent to one round of Gradient +Boosting, but intend to demonstrate error comparisons and rates of convergence. For the linear regression +plots, Gradient Boosting does not reduce either error since combinations of linear models are also linear. As +the complexity of the underlying model class increases, Gradient Boosting surpasses Algorithm 1 in terms +of MSE, though it does not minimize calibration error. +We notice that Algorithm 1, like most machine learning algorithms, is prone to overfitting when allowed. +Future performance hueristics we intend to investigate include validating updates, complexity penalties, and +weighted mixtures of updates. +28 + +Figure 6: MSE and MSCE comparison of Algorithm 1 (LS) and Gradient Boosting (GB) on linear regression +and decision trees of varying depths. * indicates termination round of LS and occurs, from top to bottom, +at T “ 41, 23, 39, 20. +29 + +MSE with Weak Learner Linear Regression +0.043 +train error (LS) +test error (LS) +train error (GB) +0.042 +test error (GB) +0.041 +0.039 +0.038 +0.037 +0 +20 +40 +60 +80 +100 +Number of roundsMSCE with Weak Learner Linear Regression +train error (LS) +0.04 +test error (LS) +train error (GB) +test error (GB) +0.03 +0.01 +0.00 +0 +20 +40 +60 +80 +100 +Number of roundsMSE with Weak Learner Depth 1 Decision Trees +train error (LS) +0.055 +test error (LS) +train error (GB) +test error (GB) +0.050 +0.040 +0.035 +0 +20 +40 +60 +80 +100 +Number of roundsMSE with Weak Learner Depth 2 Decision Trees +train error (LS) +test error (LS) +train error (GB) +0.0475 +test error (GB) +0.0450 +0.0425 +0.0375 +0.0350 +0.0325 +0 +20 +40 +60 +80 +100 +Number of roundsMSCE with Weak Learner Depth 2 Decision Trees +train error (LS) +test error (LS) +train error (GB) +0.030 +--- test error (GB) +0.025 +0.020 +0.010 +0.005 +0.000 +0 +20 +40 +60 +80 +100 +Number of roundsMSE with Weak Learner Depth 3 Decision Trees +train error (LS) +test error (LS) +0.044 +train error (GB) +test error (GB) +0.042 +0.040 +M +0.036 +0.034 +0.032 +0 +20 +40 +60 +80 +100 +Number of roundsMSCE with Weak Learner Depth 3 Decision Trees +train error (LS) +test error (LS) +train error (GB) +0.030 +--- test error (GB) +0.025 +0.020 +0.010 +0.005 +0.000 +0 +20 +40 +60 +80 +100 +Number of roundsReferences +Avrim Blum and Yishay Mansour. 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Springer Science & Business Media, 2012. +30 + +Aaron +Roth. +Uncertain: +Modern +topics +in +uncertainty +estimation. +https://www.cis.upenn.edu/ aaroth/uncertainty-notes.pdf, 2022. +Robert E Schapire. The strength of weak learnability. Machine learning, 5(2):197–227, 1990. +Robert E Schapire and Yoav Freund. Boosting: Foundations and algorithms. Kybernetes, 2013. +Eliran Shabat, Lee Cohen, and Yishay Mansour. Sample complexity of uniform convergence for multicali- +bration. Advances in Neural Information Processing Systems, 33:13331–13340, 2020. +Shai Shalev-Shwartz and Shai Ben-David. Understanding machine learning: From theory to algorithms. +Cambridge university press, 2014. +V.N. Vapnik and A. YA. Chervonenkis. On the uniform convergence of relative frequencies of events to their +probabilities, 1971. +A +Generalization Bounds +Our analysis of Algorithm 1 assumed direct access to the data distribution D. In practice, we will run the +algorithm on the empirical distribution over a sample of n points D „ Dn. In this section, we show that +when we do this, so long as n is sufficiently large, both our squared error and our multicalibration guarantees +carry over from the empirical distribution over D to the distribution D from which D was sampled. Most +generalization bounds for multicalibration algorithms (e.g. Hébert-Johnson et al. [2018], Jung et al. [2021, +2022], Shabat et al. [2020]) are either stated and proven for finite classes H, or are proven for algorithms that +do not operate as empirical risk minimization algorithms, but instead gain access to a fresh sample of data +from the distribution at each iteration, or are proven for hypotheses classes that are fixed independently of +the algorithm. We have a different challenge: Like Hébert-Johnson et al. [2018], Jung et al. [2021] we study +an iterative algorithm whose final hypothesis class is not fixed up front, but implicitly defined as a function +of H. But we wish to study the algorithms as they are used—as empirical risk minimization algorithms—so +we do not want our analysis to depend on using a fresh sample of data at each iteration. And unlike the +analysis in Jung et al. [2022], for us H is continuously large (since it is closed under affine transformations), +so we cannot rely on bounds that depend on log |H|. Instead we give a uniform convergence analysis that +depends on the pseudo-dimension of our class of weak learners H: +Definition A.1. Pseudodimension[Pollard [2012]] Let H be a class of functions from X to R. We say that +a set S “ px1, . . . , xm, y1, . . . , ymq P X m ˆRm is pseudo-shattered by H if for any pb1, . . . , bmq P t0, 1um there +exists h P H such that @i, hpxiq ą y ðñ bi “ 1 The pseudodimension of H, denoted PdimpHq is the largest +integer m for which H pseudo-shatters some set S of cardinality m. +Although hypotheses in H are continuously valued, Algorithm 1 outputs functions that have finite range +r1{ms, and so we can view them as multi-class classification functions. Our analysis will proceed by study- +ing the generalization properties of these multiclass functions, which we will characterize using Natarajan +dimension: +Definition A.2 (Shattering for multiclass functions). Natarajan [1989], Shalev-Shwartz and Ben-David +[2014] A set C Ď X is shattered by H if there exists two functions f0, f1 : C Ñ rks such that +1. For every x P C, f0pxq ‰ f1pxq. +2. For every B Ď C there exists a function h P H such that +@x P B, hpxq “ f0pxq and @x P C B, hpxq “ f1pxq. +Definition A.3 (Natarajan dimension). Natarajan [1989], Shalev-Shwartz and Ben-David [2014] The Natara- +jan dimension of H, denoted NdimpHq, is the maximal size of a shattered set C Ď X. +31 + +We can then rely the following standard uniform convergence bound for multiclass classification. This +statement is slightly modified from the result in Shalev-Schwartz and Ben-David to account for our use of +squared error. The result still holds on account of the fact that the Cherhoff bound only relies on the loss +function being bounded, and ours is indeed bounded between 0 and 1. +Theorem A.4 (Multiclass uniform convergence). Shalev-Shwartz and Ben-David [2014] Let ϵ, δ ą 0 and let +H be a class of functions h : X Ñ r1{ks such that the Natarajan dimension of H is d. Let D P ∆pX ˆ r0, 1sq +be an arbitrary distribution and let D “ tpx1, y1q, . . . , pxn, ynqupxi,yiq„D be a sample of n points from D. +Then for +n “ O +ˆd logpkq ` logp1{δq +ε2 +˙ +, +Pr +„ +max +hPH +ˇˇˇˇ +E +px,yq„Drpy ´ hpxqq2s ´ +E +px,yq„Drpy ´ hpxqq2s +ˇˇˇˇ ě ϵs +ȷ +ď δ. +Our strategy will be to bound the Natarajan dimension of the class of models that can be output by +Algorithm 1 in terms of the pseudodimension of the underlying weak learner, then apply the above uniform +convergence result. To do so, we will first use the following lemma, which bounds the Natarajan dimension +of functions that can be described as post-processings of binary valued-functions from a class of bounded +VC-dimension. +Lemma A.5. Shalev-Shwartz and Ben-David [2014] Suppose we have ℓ binary classifiers from binary class +Hbin and a rule r : t0, 1uℓ Ñ rks that determines a multiclass label according to the predictions of the ℓ binary +classifiers. Define the hypothesis class corresponding to this rule as +H “ trph1p¨q, . . . , hℓp¨qq : ph1, . . . , hℓq P pHbinqℓu. +Then, if d “ VCdimpHbinq, +NdimpHq ď 3ℓd logpℓdq. +Recall that the VC-dimension of a binary classifier is defined as follows: +Definition A.6 (VC-dimension). Vapnik and Chervonenkis [1971] Let H be a class of binary classifiers +h : X Ñ t0, 1u. Let S “ tx1, . . . , xmu and let ΠHpSq “ tphpx1q, . . . , hpxmqq : h P Hu Ď t0, 1um. We say +that S is shattered by H if ΠHpSq “ t0, 1um. The Vapnik-Chervonenkis (VC) dimension of H, denoted +VCdimpHq, is the cardinality of the largest set S shattered by H. +Lemma A.7. Let Hboost be the class of models output by RegressionMulticalibratepf, α, AH, ¨, Bq (Algorithm +1) for any input distribution D and let d be the pseudodimension of its input weak learner class H. +Ndim pHboostq ď 24pB{αq3d log +` +p2B{αq3d +˘ +. +Proof. Let m be defined (as in RegressionMulticalibratepf, α, AH, D, Bq) to be 2B{α. Because our models +are always rounded to the nearest value in r1{ms, we can think of the model ft generated in every round +of the algorithm multiclass classification problems over m classes. We will show that our final model can +be written as a decision rule that maps the outputs of some ℓ Boolean classifiers to r1{ms, and that these +Boolean classifiers have VC dimension that is bounded by the pseudodimension of the weak learner class. +Then, we will apply Lemma A.5 to get an upper bound on the Natarajan dimension of the class of models +in terms of α, B, and the pseudodimension of the input weak learner class H. +Consider the initial round of the algorithm. We can convert our (rounded) initial regressor f0 to a series +of m Boolean thresholdings gv which return 1 when f0pxq ě v: +g0 +v “ +# +1 +if f0pxq ě v, +0 +otherwise. +. +32 + +These m Boolean thresholdings can then be mapped back to the original prediction over r1{ms using a +decision rule r : t0, 1um Ñ r1{ms which picks the largest of the thresholds that evaluates to 1, and assigns +that index to the prediction: +r0ptg0 +vuvPr1{msqpxq “ arg max +iPr1{ms i1rgvpxq “ 1s. +Note that since our initial predictor f0 was already rounded to take values in r1{ms, the largest v +such that f0pxq ě v will be exactly f0pxq, so r0 is exactly equivalent to f0. Similarly, at round t ` 1 of +RegressionMulticalibratepf, α, AH, D, Bq, we will show that the model ft`1 can be written as a decision rule +rt`1 over m ` pt ` 1qm2 binary classifiers g, where +gt +v,i “ +# +1 +if ht +vpxq ě i ´ 1{p2mq, +0 +otherwise. +, +Here, the thresholds measure halfway between each level set, as ht +vpxq has yet to be rounded to the +nearest level set. We can write a decision rule that maps these thresholds to classifications over r1{ms: +rt`1 +` +rt, tgt`1 +v,i ui,vPr1{ms +˘ +pxq “ +ÿ +vPr1{ms +1rrtpxq “ vs arg max +iPr1{ms +` +i ¨ 1rgt`1 +v,i pxq “ 1s +˘ +, +Now, we need to show that this decision rule evaluated at round t is equivalent to ft. +We proceed +inductively. For our base case, we have already argued that our initial decision rule r0 is equivalent to the +classifier f0. Now, say that we have decision rule rt over binary classifiers g that is equivalent to model ft. +Then, we can write +rt`1 +` +rt, tgt`1 +v,i ui,vPr1{ms +˘ +pxq “ +ÿ +vPr1{ms +1rrtpxq “ vs arg max +iPr1{ms +` +i ¨ 1rgt`1 +v,i pxq “ 1s +˘ +, +“ +ÿ +vPr1{ms +1rftpxq “ vs arg max +iPr1{ms +` +i ¨ 1rgt`1 +v,i pxq “ 1s +˘ +“ +ÿ +vPr1{ms +1rftpxq “ vs arg max +iPr1{ms +` +i ¨ 1rht`1 +v +pxq ě i ´ 1{p2mqs +˘ +“ +ÿ +vPr1{ms +1rftpxq “ vsRoundpht`1 +v +pxqq +“ ft`1pxq, +where the second line comes from the inductive hypothesis and the second to last line’s equality comes from +the fact that the largest i such that ht`1 +v +pxq ´ 1{p2mq ě i will be the exact rounded prediction of ht`1 +v +pxq. +Now, we need to show that at round t ` 1, the decision rule is a decision rule over m ` pt ` 1qm2 binary +classifiers. Note that our initial decision rule r0 has m “ m`0¨m2 binary classifiers. Say that at round t we +have a decision rule rt over m ` tm2 classifiers. In the following round, we build m2 new Boolean classifiers +gv, i for v, i P r1{ms. So, at round t ` 1 we have m ` tm2 ` m2 “ m ` pt ` 1qm2 classifiers total. +From Theorem 4.3, we know that Algorithm 1 halts after at most T ď 2B{α rounds, at which point it +outputs model fT ´1. So, we can rewrite fT ´1 as a decision rule rT ´1 composed of at most m ` pT ´ 1qm2 ă +Tm2 Boolean models. Plugging in our bound for T and definition of m, this gives us a decision rule rT ´1 +composed of at most +` 2B +α +˘3 Boolean classifiers. +Let G be the class of Boolean threshold functions over H, i.e. functions g : X Ñ t0, 1u such that +gpxq “ +# +1 +hpxq ě i +0 +hpxq ă i, +33 + +for some h P H and i P R. Say that the VC-dimension of G is d1. Then, applying lemma A.5, it follows that +NdimpHboostq ď 3 +ˆ2B +α +˙3 +d1 log +˜ˆ2B +α +˙3 +d1 +¸ +, +“ 24 +ˆB +α +˙ +d1 log +˜ˆ2B +α +˙3 +d1 +¸ +. +Now, it remains to show that we can bound the VC-dimension of these thresholding functions by the +pseudodimension of the weak learner class H. Note that G as we have defined it above is a richer hypothesis +class than the actual class of thresholding functions used in the above analysis, because it can threshold +at any value in R rather than being restricted to r1{ms. Thus, its VC dimension can only be greater than +the VC dimension of the class of threshold functions over H restricted to r1{ms, and hence an upper bound +on the VC dimension of G in terms of the pseudodimension of H will also be an upper bound on the VC +dimension of the restricted class of threshold functions. +Let d be the pseudodimension of H, and say that d ă d1. By the definition of VC-dimension, t0, 1ud`1 +must be shattered by G. I.e., for any set of d ` 1 points x1, . . . , xd`1 P X with arbitrary labels b1, . . . , bd`1, +there is some hypothesis g P G that realizes those labels on px1, . . . , xd`1q. Consider the function g that, +given the d ` 1 points in X, realizes the labels b1, . . . , bd`1. By the construction of G, g is a thresholding of +some function h P H at some point i. So, there is be some i P R such that hpxiq ą i ñ bi “ 1 and such that +bi “ 1 ñ hpxiq ą i. But this means that t0, 1ud`1 is pseudo-shattered by H, and thus the pseudodimension +of H is not d. Thus, it cannot be the case that d ă d1, and hence d1 ď d, i.e. the VC dimension of G is +bounded above by the pseudodimension of H. Plugging this bound into the above bound on the Natarajan +dimension gives us that +NdimpHboostq ď 24 +ˆB +α +˙ +d1 log +˜ˆ2B +α +˙3 +d1 +¸ +, +ď 24 +ˆB +α +˙ +d log +˜ˆ2B +α +˙3 +d +¸ +. +Now, we can state the following uniform convergence theorem for our final model. +Theorem A.8 (Squared Error Generalization for Algorithm 1.). Let ϵ, δ, α, B ą 0. Let Hboost be the class of +models that can be output by RegressionMulticalibratepf, α, AH, ¨, Bq (Algorithm 1) for any input distribution +D and let d be the pseudodimension of its input weak learner class H. Let D “ tpx1, y1q, . . . , pxn, ynqupxi,yiq„D +be a sample of n points drawn i.i.d. from D. Then if +n “ O +ˆdB3 log2pdB{αq +α3ϵ2 +` logp1{δq +ϵ2 +˙ +Pr +„ +max +hPHboost +ˇˇˇˇ +E +px,yq„Drpy ´ hpxqq2s ´ +E +px,yq„Drpy ´ hpxqq2s +ˇˇˇˇ ě ϵs +ȷ +ď δ. +Proof. This follows directly from Theorem A.4 and the bound on the Natarajan dimension in Lemma A.7. +We also would like to know that our multicalibration guarantees are generalizable. Rather than doing a +bespoke analysis here, we can rely on the connection that we have established between failure of multicali- +bration and ability to improve squared error and argue that if the final hypothesis output by the algorithm +was not multicalibrated with high probability then it would be possible to improve its squared error out-of- +sample. Thus, by our previous generalization result for squared error, it would be possible to improve the +squared error in-sample as well, giving us a contradiction. +34 + +Theorem A.9 (Multicalibration generalization guarantee). Let ϵ, δ, α, B ą 0 and consider the model fT ´1 +output by RegressionMulticalibratepf, α, AH, D, Bq for some sample D of n points drawn i.i.d. from distri- +bution D such that +n “ O +ˆdB3 log2pdB{αq +α3ϵ2 +` logp1{δq +ϵ2 +˙ +Then if ϵ ď +α +4B , with probability greater than or equal to 1 ´ 2δ it follows that fT ´1 is 2α-approximately +multicalibrated with respect to the distribution D. +Proof. Let D “ tpx1, y1q, . . . , pxn, ynqupxi,yiq„D. Consider the model fT ´1 output by +RegressionMulticalibratepf, α, AH, D, Bq, and recall that within the run of the algorithm there was also a +model fT defined in the final round. Say that model fT ´1 is not 2α-approximately multicalibrated with +respect to HB and the true distribution D. +Since the algorithm running on the sample halted, it must have been that the model in the final round +improved in squared error by less than α{p2Bq when measured with respect to the sample D: +E +px,yq„DrpfT ´1 ´ yq2s ´ +E +px,yq„DrpfT ´ yq2s ď pα{2Bq. +Consider what happens if we run the algorithm again, but with fT ´1 as its initial model and now +with the underlying distribution as input rather than the sample of n points. Let f 1 +T be the model found +in the first round of running this process RegressionMulticalibratepfT ´1, α, AH, D, Bq. Since fT ´1 is not +2α´approximately multicalibrated with respect to D and HB, then by an identical argument as in the proof +of Theorem 4.3, it it must be that a single round of the algorithm improves the squared error on D by at +least α{B. Thus, Epx,yq„DrpfT ´1 ´ yq2s ´ Epx,yq„Drpf 1 +T ´ yq2s ą α{B. +We know from our previous convergence bound, Theorem A.8, that with probability 1´δ, | Epx,yq„Drpf 1 +T ´ +yq2s ´ Epx,yq„DrpfT ´ yq2s| ă ϵ. So, f 1 +T must with high probability also improve on the sample D: +α +B ă +E +px,yq„DrpfT ´1 ´ yq2s ´ +E +px,yq„Drpf 1 +T ´ yq2s +ă +E +px,yq„DrpfT ´1 ´ yq2s ´ +E +px,yq„Drpf 1 +T ´ yq2s ` ϵ +(with probability ě 1 ´ δ) +ă +E +px,yq„DrpfT ´1 ´ yq2s ´ +E +px,yq„Drpf 1 +T ´ yq2s ` 2ϵ +(with probability ě 1 ´ 2δ) +ă α +2B ` 2ϵ, +where the last line comes from the fact that the error of f 1 +T on D cannot be less than the error of fT on +D, or else the regression oracle would have found it. Now we have a contradiction: since we have set ϵ ď +α +4B , +α +B ă α +2B ` 2 α +4B +“ α +B . +So, it must follow that fT ´1 is, with probability 1 ´ 2δ, 2α´approximately multicalibrated. +35 + diff --git a/x9FST4oBgHgl3EQfSzhz/content/tmp_files/load_file.txt b/x9FST4oBgHgl3EQfSzhz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3554bc7c23dc56d189f3560fcd114c025a5b2d1 --- /dev/null +++ b/x9FST4oBgHgl3EQfSzhz/content/tmp_files/load_file.txt @@ -0,0 +1,1088 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf,len=1087 +page_content='Multicalibration as Boosting for Regression Ira Globus-Harris Declan Harrison Michael Kearns Aaron Roth Jessica Sorrell February 1, 2023 Abstract We study the connection between multicalibration and boosting for squared error regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' First we prove a useful characterization of multicalibration in terms of a “swap regret” like condition on squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Using this characterization, we give an exceedingly simple algorithm that can be analyzed both as a boosting algorithm for regression and as a multicalibration algorithm for a class H that makes use only of a standard squared error regression oracle for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We give a weak learning assumption on H that ensures convergence to Bayes optimality without the need to make any realizability assumptions — giving us an agnostic boosting algorithm for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We then show that our weak learning assumption on H is both necessary and sufficient for multicalibration with respect to H to imply Bayes optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We also show that if H satisfies our weak learning condition relative to another class C then multicalibration with respect to H implies multicalibration with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Finally we investigate the empirical performance of our algorithm experimentally using an open source implementation that we make available on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 1 Introduction We revisit the problem of boosting for regression, and develop a new agnostic regression boosting algorithm via a connection to multicalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In doing so, we shed additional light on multicalibration, a recent learning objective that has emerged from the algorithmic fairness literature [Hébert-Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In particular, we characterize multicalibration in terms of a “swap-regret” like condition, and use it to answer the question “what property must a collection of functions H have so that multicalibration with respect to H implies Bayes optimality?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', giving a complete answer to problem asked by Burhanpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Using our swap-regret characterization, we derive an especially simple algorithm for learning a multicalibrated predictor for a class of functions H by reduction to a standard squared-error regression algorithm for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The same algorithm can also be analyzed as a boosting algorithm for squared error regression that makes calls to a weak learner for squared error regression on subsets of the original data distribution without the need to relabel examples (in contrast to Gradient Boosting as well as existing multicalibration algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This lets us specify a weak learning condition that is sufficient for convergence to the Bayes optimal predictor (even if the Bayes optimal predictor does not have zero error), avoiding the kinds of realizability assumptions that are implicit in analyses of boosting algorithms that converge to zero error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We conclude that ensuring multicalibration with respect to H corresponds to boosting for squared error regression in which H forms the set of weak learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Finally we define a weak learning condition for H relative to a constrained class of functions C (rather than with respect to the Bayes optimal predictor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We show that multicalibration with respect to H implies multicalibration with respect to C if H satisfies the weak learning condition with respect to C, which in turn implies accuracy at least that of the best function in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Multicalibration Consider a distribution D P ∆Z defined over a domain Z “ X ˆ R of feature vectors x P X paired with real valued labels y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Informally, a regression function f : X Ñ R is calibrated if for every v in the range of f, Epx,yq„Dry|fpxq “ vs “ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In other words, fpxq must be an unbiased estimator of y, 1Our code repository can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='com/Declancharrison/Level-Set-Boosting 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='13767v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='LG] 31 Jan 2023 even conditional on the value of its own prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Calibration on its own is a weak condition, because it only asks for f to be unbiased on average over all points x such that fpxq “ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For example, the con- stant predictor that predicts fpxq “ Epx,yq„Drys is calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus calibration does not imply accuracy—a calibrated predictor need not make predictions with lower squared error than the best constant predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Calibration also does not imply that f is equally representative of the label distribution on different subsets of the feature space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For example, given a subset of the feature space G Ď X, even if f is calibrated, it may be that f is not calibrated on the conditional distribution conditional on x P G—it might be e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' that Ery|fpxq “ v, x P Gs " v, and Ery|fpxq “ v, x R Gs !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To correct this last deficiency, Hébert-Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2018] defined multi-calibration, which is a condition parameterized by a subset of groups G Ď X each defined by an indicator function h : X Ñ t0, 1u in some class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It asks (informally) that for each such h P H, and for each v in the range of f, that Erhpxqpy ´vq|fpxq “ vs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since h is a binary indicator function for some set G, this is equivalent to asking for calibration not just marginally over D, but simultaneously for calibration over D conditional on x P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2019] and Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] generalize multicalibration beyond group indicator functions to arbitrary real valued functions h : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Intuitively, as H becomes a richer and richer set of functions, multicalibration becomes an increasingly stringent condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' But if H consists of the indicator functions for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' even a very large number of randomly selected subsets G Ď X, then the constant predictor fpxq “ Epx,yq„Drys will still be approximately multicalibrated with respect to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' What property of H ensures that multicalibration with respect to H implies that f is a Bayes optimal regression function?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This question was recently asked by Burhanpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021] — and we provide a necessary and sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Boosting for Regression Boosting refers broadly to a collection of learning techniques that reduce the problem of “strong learning” (informally, finding an error optimal model) to a series of “weak learning” tasks (informally, finding a model that has only a small improvement over a trivial model)—See Schapire and Fre- und [2013] for a textbook treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The vast majority of theoretical work on boosting studies the problem of binary classification, in which a weak learner is a learner that obtains classification error bounded below 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Several recent papers Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2019], Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] have made connections between algorithms for guaranteeing multicalibration and boosting algorithms for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this paper, we show a direct connection between multicalibration and the much less well-studied problem of boosting for squared error regression [Friedman, 2001, Duffy and Helmbold, 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' There is not a single established notion for what constitutes a weak learner in the regression setting (Duffy and Helmbold [2002] introduce several different notions), and unlike boosting algorithms for classification problems which often work by calling a weak learner on a reweighting of the data distribution, existing algorithms for boosting for regression typically resort to calling a learning algorithm on relabelled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We give a boosting algorithm for regression that only requires calling a squared error regression learning algorithm on subsets of examples from the original distribution (without relabelling), which lets us formulate a weak learning condition that is sufficient to converge to the Bayes optimal predictor, without making the kinds of realizability assumptions implicit in the analysis of boosting algorithms that assume one can drive error to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 Our Results We focus on classes of real valued functions H that are closed under affine transformations — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' classes such that if fpxq P H, then for any pair of constants a, b P R, pafpxq ` bq P H as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Many natural classes of models satisfy this condition already (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' linear and polynomial functions and regression trees), and any neural network architecture that does not already satisfy this condition can be made to satisfy it by adding two additional parameters (a and b) while maintaining differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus we view closure under affine transformations to be a weak assumption that is enforceable if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' First in Section 3 we prove the following characterization for multicalibration over H, for any class H that is closed under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Informally, we show that a model f is multicalibrated with respect 2 to H if and only if, for every v in the range of f: E px,yq„Drpfpxq ´ yq2|fpxq “ vs ď min hPH E px,yq„Drphpxq ´ yq2|fpxq “ vs (See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 for the formal statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This is a “swap regret”-like condition (as in Foster and Vohra [1999] and Blum and Mansour [2005]), that states that f must have lower squared error than any model h P H, even conditional on its own prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Using this characterization, in Section 4 we give an exceedingly simple algorithm for learning a multicalibrated predictor over H given a squared error regression oracle for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The algorithm simply repeats the following over t rounds until convergence, maintaining a model f : X Ñ t0, 1{m, 2{m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , 1u with a discrete range with support over multiples of 1{m for some discretization factor m: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For each level set v P t0, 1{m, 2{m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , 1u, run a regression algorithm to find the ht v P H that minimizes squared error on the distribution D|pft´1pxq “ vq, the distribution conditional on ft´1pxq “ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Replace each level set v of ft´1pxq with ht vpxq to produce a new model ft, and round its output to the discrete range t0, 1{m, 2{m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , 1u Each iteration decreases the squared error of ft, ensuring convergence, and our characterization of multi- calibration ensures that we are multicalibrated with respect to H at convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Compared to existing multicalibration algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' the split and merge algorithm of Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022]), our algorithm is exceptionally simple and makes use of a standard squared-error regression oracle on subsets of the original distribution, rather than using a classification oracle or requiring example relabelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We can also view the same algorithm as a boosting algorithm for squared error regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose H (or equivalently our weak learning algorithm) satisfies the following weak learning assumption: informally, that on any restriction of D on which the Bayes optimal predictor is non-constant, there should be some h P H that obtains squared error better than that of the best constant predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then our algorithm converges to the Bayes optimal predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Section A we give uniform convergence bounds which guarantee that the algorithm’s accuracy and multicalibration guarantees generalize out of sample, with sample sizes that are linear in the pseudodimension of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We then show in Section 5 that in a strong sense this is the “right” weak learning assumption: Multical- ibration with respect to H implies Bayes optimality if and only if H satisfies this weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This gives a complete answer to the question of when multicalibration implies Bayes optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Section 6, we generalize our weak learning condition to a weak learning condition relative to a con- strained class of functions C (rather than relative to the Bayes optimal predictor), and show that if H satisfies the weak learning condition relative to C, then multicalibration with respect to H implies multicalibration with respect to C, and hence error that is competitive with the best model in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We give a fast, parallelizable implementation of our algorithm and in Section 7 demonstrate its con- vergence to Bayes optimality on two-dimensional datasets useful for visualization, as well as evaluate the accuracy and calibration guarantees of our algorithm on real Census derived data using the Folktables pack- age Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 Additional Related Work Calibration as a statistical objective dates back at least to Dawid [1982].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Foster and Vohra [1999] showed a tight connection between marginal calibration and internal (equivalently swap) regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We extend this characterization to multicalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Multicalibration was introduced by Hébert-Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2018], and variants of the original definition have been studied by a number of works [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', 2019, Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', 2021, Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', 2022, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', 2022, Roth, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We use the ℓ2 variant of multicalibration studied in Roth [2022]—but this definition implies all of the other variants of multicalibration up to a change in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Burhanpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021] first asked the question “when does multicalibration with respect to H imply accuracy”, and gave a sufficient condition: when H contains (refinements of) the levelsets of the 3 Bayes optimal regression function, together with techniques for attempting to find these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This can be viewed as a “strong learning” assumption, in contrast to our weak learning assumption on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Boosting for binary classification was introduced by Schapire [1990] and has since become a major topic of both theoretical and empirical study — see Schapire and Freund [2013] for a textbook overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Both Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2019] and Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] have drawn connections between algorithms for multicalibration and boosting for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In particular, Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] draw direct connections between their split-and-merge multicalibration algorithm and agnostic boosting algorithms of Kalai [2004], Kanade and Kalai [2009], Kalai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Boosting for squared error regression is much less well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Freund and Schapire [1997] give a variant of Adaboost (Adaboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='R) that reduces regression examples to infinite sets of classification examples, and requires a base regressor that optimizes a non-standard loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Friedman [2001] introduced the popular gradient boosting method, which for squared error regression corresponds to iteratively fitting the residuals of the current model and then applying an additive update, but did not give a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Duffy and Helmbold [2002] give a theoretical analysis of several different boosting algorithms for squared error regression under several different weak learning assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Their algorithms require base regression algorithms that can be called (and guaranteed to succeed) on arbitrarily relabelled examples from the training distribution, and given their weak learning assumption, their analysis shows how to drive the error of the final model arbitrarily close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Weak learning assumptions in this style implicitly make very strong realizabilty assumptions (that the Bayes error is close to 0), but because the weak learner is called on relabelled samples, it is difficult to enunciate a weak learning condition that is consistent with obtaining Bayes optimal error, but not better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The boosting algorithm we introduce only requires calling a standard regression algorithm on subsets of the examples from the training distribution, which makes it easy for us to define a weak learning condition that lets us drive error to the Bayes optimal rate without realizability assumptions — thus our results can be viewed as giving an agnostic boosting algorithm for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 2 Preliminaries We study prediction tasks over a domain Z “ X ˆY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Here X represents the feature domain and Y represents the label domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We focus on the bounded regression setting where Y “ r0, 1s (the scaling to r0, 1s is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We write D P ∆Z to denote a distribution over labelled examples, DX to denote the induced marginal distribution over features, and write D „ Dn to denote a dataset consisting of n labelled examples sampled i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will be interested in the squared error of a model f with respect to distribution D, Epx,yq„Drpy ´ fpxqq2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We abuse notation and identify datasets D “ tpx1, y1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , pxn, ynqu with the empirical distribution over the examples they contain, and so we can write the empirical squared error over D: as Epx,yq„Drpy ´ fpxqq2s “ 1 n řn i“1pyi ´ fpxiqq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' When taking expectations over a distribution that is clear from context, we will frequently suppress notation indicating the relevant distribution for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We write Rpfq to denote the range of a function f, and when Rpfq is finite, use m to denote the cardinality of its range: m “ |Rpfq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We are interested in finding models that are multicalibrated with respect to a class of real valued functions H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We use an ℓ2 notion of multicalibration as used in Roth [2022]: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 (Multicalibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and a model f : X Ñ r0, 1s that maps onto a countable subset of its range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let H be an arbitrary collection of real valued functions h : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that f is α-approximately multicalibrated with respect to D and H if for every h P H: K2pf, h, Dq “ ÿ vPRpfq Pr px,yq„Drfpxq “ vs ˆ E px,yq„Drhpxqpy ´ vq|fpxq “ vs ˙2 ď α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that f is α-approximately calibrated if: K2pf, Dq “ ÿ vPRpfq Pr px,yq„Drfpxq “ vs ˆ E px,yq„Drpy ´ vq|fpxq “ vs ˙2 ď α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 4 If α “ 0, then we simply say that a model is multicalibrated or calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will sometimes refer to K2pf, Dq as the mean squared calibration error of a model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' When the functions hpxq have binary range, we can view them as indicator functions for some subset of the data domain S Ď X, in which case multicalibration corresponds to asking for calibration condi- tional on membership in these subsets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Allowing the functions h to have real valued range is only a more general condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Our notion of approximate multicalibration takes a weighted average over the level sets v of the predictor f, weighted by the probability that fpxq “ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This is necessary for any kind of out of sample gen- eralization statement — otherwise we could not even necessarily measure calibration error from a finite sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Other work on multicalibration use related measures of multicalibration that we think of as ℓ1 or ℓ8 vari- ants, that we can write as K1pf, h, Dq “ ř vPRpfq Prpx,yq„Drfpxq “ vs ˇˇEpx,yq„Drhpxqpy ´ vq|fpxq “ vs ˇˇ and K8pf, h, Dq “ maxvPRpfq Prpx,yq„Drfpxq “ vs ` Epx,yq„Drhpxqpy ´ vq|fpxq “ vs ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' These notions are related to each other: K2pf, h, Dq ď K1pf, h, Dq ď a K2pf, h, Dq and K8pf, h, Dq ď K1pf, h, Dq ď mK8pf, h, Dq [Roth, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will characterize the relationship between multicalibration and Bayes optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 (Bayes Optimal Predictor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let f ˚ : X Ñ r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that f ˚ is the Bayes optimal predictor for D if: E px,yq„Drpy ´ f ˚pxqq2s ď min f:XÑr0,1srpy ´ fpxqq2s The Bayes Optimal predictor satisfies: f ˚pxq “ Epx1,yq„D ry|x1 “ xs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that a function f : X Ñ r0, 1s is γ-approximately Bayes optimal if E px,yq„Drpy ´ fpxqq2s ď E px,yq„Drpy ´ f ˚pxqq2s ` γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Throughout this paper, we will denote the Bayes optimal predictor as f ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 3 A Characterization of Multicalibration In this section we give a simple “swap-regret” like characterization of multicalibration for any class of functions H that is closed under affine transformations: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A class of functions H is closed under affine transformations if for every a, b P R, if hpxq P H then h1pxq :“ ahpxq ` b P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' As already discussed, closure under affine transformation is a mild assumption: it is already satisfied by many classes of functions H like linear and polynomial functions and decision trees, and can be enforced for neural network architectures when it is not already satisfied by adding two additional parameters a and b without affecting our ability to optimize over the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The first direction of our characterization states that if f fails the multicalibration condition for some h P H, then there is some other h1 P H that improves over f in terms of squared error, when restricted to a level set of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The second direction states the opposite: if f is calibrated (but not necessarily multicalibrated), and if there is some level set of f on which h improves over f in terms of squared error, then in fact f must fail the multicalibration condition for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose H is closed under affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a model f : X Ñ R and a levelset v P Rpfq of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' If there exists an h P H such that: Erhpxqpy ´ vq|fpxq “ vs ě α, for α ą 0, then there exists an h1 P H such that: Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 Erhpxq2|fpxq “ vs, 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' If f is calibrated and there exists an h P H such that Erpfpxq ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, then: Erhpxqpy ´ vq|fpxq “ vs ě α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We prove each direction in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a model f : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose for some v P Rpfq there is an h P H such that: Erhpxqpy ´ vq|fpxq “ vs ě α Let h1 “ v ` ηhpxq for η “ α Erhpxq2|fpxq“vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then: Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 Erhpxq2|fpxq “ vs Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We calculate: Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs “ Erpv ´ yq2 ´ pv ` ηhpxq ´ yq2|fpxq “ vs “ Erv2 ´ 2vy ` y2 ´ pv ` ηhpxqq2 ` 2ypv ` ηhpxqq ´ y2|fpxq “ vs “ Er2yηhpxq ´ 2vηhpxq ´ η2hpxq2|fpxq “ vs “ Er2ηhpxqpy ´ vq ´ η2hpxq2|fpxq “ vs ě 2ηα ´ η2 Erhpxq2|fpxq “ vs “ α2 Erhpxq2|fpxq “ vs Where the last line follows from the definition of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The first direction of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3, and the observation that since H is closed under affine transformations, the function h1 defined in the statement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 is in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now for the second direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a model f : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose for some v P Rpfq there is an h P H such that: Erp¯yv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, where ¯yv “ Ery | fpxq “ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then it must be that: Erhpxqpy ´ ¯yvq|fpxq “ vs ě α 2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We calculate: 6 E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxqpy ´ ¯yvq|fpxq “ vs “ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxqy|fpxq “ vs ´ ¯yv E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxq|fpxq “ vs “ 1 2 ˆ 2 E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxqy|fpxq “ vs ´ 2¯yv E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxq|fpxq “ vs ˙ ě 1 2 ˆ 2 E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxqy|fpxq “ vs ´ 2¯yv E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drhpxq|fpxq “ vs ´ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drphpxq ´ ¯yvq2|fpxq “ vs ˙ “ 1 2 ˆ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Dr2hpxqy ´ hpxq2 ´ ¯y2 v|fpxq “ vs ˙ “ 1 2 ˆ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Dr2hpxqy ´ hpxq2 ´ 2¯yvy ` ¯y2 v|fpxq “ vs ˙ “ 1 2 ˆ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drp¯yv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ˙ ě α 2 where the 3rd to last line follows from adding and subtracting ¯y2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For any calibrated f it follows that v “ Ery | fpxq “ vs “ ¯yv, and so for calibrated f we have that if Erpv ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ě α, then: Erhpxqpy ´ vq|fpxq “ vs ě α 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 4 An Algorithm (For Multicalibration And Regression Boosting) We now give a single algorithm, and then show how to analyze it both as an algorithm for obtaining a multicalibrated predictor f, and as a boosting algorithm for squared error regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let m P N` be a discretization term, and let r1{ms :“ t0, 1 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , m´1 m , 1u denote the set of points in r0, 1s that are multiples of 1{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will learn a model f whose range is r1{ms, which we will enforce by rounding its outputs to this range as necessary using the following operation: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 (Roundpf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' mq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let F be the family of all functions f : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let Round : F ˆ N` Ñ F be a function such that Roundpf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' mq outputs ˜hpxq “ minvPr1{ms |hpxq ´ v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Unlike other algorithms for multicalibration which make use of agnostic learning oracles for binary classification, our algorithm makes use of an algorithm for solving squared-error regression problems over H: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' AH is a squared error regression oracle for a class of real valued functions H if for every D P ∆Z, AHpDq outputs a function h P H such that h P arg min h1PH E px,yq„Drph1pxq ´ yq2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 7 For example, if H is the set of all linear functions, then AH simply solves a linear regression problem (which has a closed form solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Algorithm 1 (LSBoost2)repeats the following operation until it no longer decreases overall squared error: it runs squared error regression on each of the level-sets of ft, and then replaces those levelsets with the solutions to the regression problems, and rounds the output to r1{ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will now analyze the algorithm first as a multicalibration algorithm, and then as a boosting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For simplicity, in this section we will analyze the algorithm as if it is given direct access to the distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In practice, the algorithm will be run on the empirical distribution over a dataset D „ Dn, and the multicalibration guarantees proven in this section will hold for this empirical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Section A we prove generalization theorems, which allow us to translate our in-sample error and multicalibration guarantees over D to out-of-sample guarantees over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Algorithm 1: LSBoost(f, α, AH, D, B) Let m “ 2B α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let f0 “ Roundpf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' mq, err0 “ Epx,yq„Drpf0pxq ´ yq2s, err´1 “ 8 and t “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' while perrt´1 ´ errtq ě α 2B do for each v P r1{ms do Let Dt`1 v “ D|pftpxq “ vq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let ht`1 v “ AHpDt`1 v q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let: ˜ft`1pxq “ ÿ vPr1{ms 1rftpxq “ vs ¨ ht`1 v pxq ft`1 “ Roundp ˜ft`1, mq Let errt`1 “ Epx,yq„Drpft`1pxq ´ yq2s and t “ t ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Output ft´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 Analysis as a Multicalibration Algorithm Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, any α ă 1, any class of real valued functions H that is closed under affine transformations, and a squared error regression oracle AH for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For any bound B ą 0 let: HB “ th P H : max xPX hpxq2 ď Bu be the set of functions in h with squared magnitude bounded by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then LSBoostpf, α, AH, D, Bq (Algorithm 1) halts after at most T ď 2B α many iterations and outputs a model fT ´1 such that fT ´1 is α-approximately multicalibrated with respect to D and HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note the form of this theorem — we do not promise multicalibration at approximation param- eter α for all of H, but only for HB — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' those functions in H satisfying a bound on their squared value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This is necessary, since H is closed under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To see this, note that if Erhpxqpy´vqs ě α, then it must be that Erc ¨ hpxqpy ´ vqs ě c ¨ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since h1pxq “ chpxq is also in H by assumption, approximate multicalibration bounds must always also be paired with a bound on the norm of the functions for which we promise those bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since f0 takes values in r0, 1s and y P r0, 1s, we have err0 ď 1, and by definition errT ě 0 for all T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' By construction, if the algorithm has not halted at round t it must be that errt ď errt´1 ´ α 2B , and so the algorithm must halt after at most T ď 2B α many iterations to avoid a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It remains to show that when the algorithm halts at round T, the model fT ´1 that it outputs is α- approximately multi-calibrated with respect to D and HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will show that if this is not the case, then errT ´1 ´ errT ą α 2B , which will be a contradiction to the halting criterion of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 2LSBoost can be taken to stand for either “Level Set Boost" or “Least Squares Boost”, at the reader’s discretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 8 Suppose that fT ´1 is not α-approximately multicalibrated with respect to D and HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This means there must be some h P HB such that: ÿ vPr1{ms Pr px,yq„DrfT ´1pxq “ vs ˆ E px,yq„Drhpxqpy ´ vq|fT ´1pxq “ vs ˙2 ą α For each v P r1{ms define αv “ Pr px,yq„DrfT ´1pxq “ vs ˆ E px,yq„Drhpxqpy ´ vq|fT ´1pxq “ vs ˙2 So we have ř vPr1{ms αv ą α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Applying the 1st part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 we learn that for each v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' there must be some hv P H such that: ErpfT ´1pxq ´ yq2 ´ phvpxq ´ yq2|fT ´1pxq “ vs ą 1 Erhpxq2|fT ´1pxq “ vs ¨ αv Prpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs ě 1 B αv Prpx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs where the last inequality follows from the fact that h P HB Now we can compute: E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2s “ ÿ vPr1{ms Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2|fT ´1pxq “ vs “ ÿ vPr1{ms Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ phT v pxq ´ yq2|fT ´1pxq “ vs ě ÿ vPr1{ms Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ phvpxq ´ yq2|fT ´1pxq “ vs ě ÿ vPr1{ms αv B ą α B Here the third line follows from the definition of ˜fT and the fourth line follows from the fact hv P H and that hT v minimizes squared error on DT v amongst all h P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Finally we calculate: errT ´1 ´ errT “ E px,yq„DrpfT ´1pxq ´ yq2 ´ pfT pxq ´ yq2s “ E px,yq„DrpfT ´1pxq ´ yq2 ´ p ˜fT pxq ´ yq2s ` E px,yq„Drp ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2s ą α B ` E px,yq„Drp ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2s ą α B ´ 1 m ě α 2B where the last equality follows from the fact that m ě 2B α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 9 The 2nd inequality follows from the fact that for every pair px, yq: p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 ě ´ 1 m To see this we consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since y P r0, 1s, if ˜fT pxq ą 1 or ˜fT pxq ă 0 then the Round operation decreases squared error and we have p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In the remaining case we have fT pxq P r0, 1s and ∆ “ ˜fT pxq ´ fT pxq is such that |∆| ď 1 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this case we can compute: p ˜fT pxq ´ yq2 ´ pfT pxq ´ yq2 “ pfT pxq ` ∆ ´ yq2 ´ pfT pxq ´ yq2 “ 2∆pfpxq ´ yq ` ∆2 ě ´2|∆| ` ∆2 ě ´ 1 m 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 Analysis as a Boosting Algorithm We now analyze the same algorithm (Algorithm 1) as a boosting algorithm designed to boost a “weak learning” algorithm AH to a strong learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Often in the boosting literature, a “strong learning” algorithm is one that can obtain accuracy arbitrarily close to perfect, which is only possible under strong realizability assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this paper, by “strong learning”, we mean that Algorithm 1 should output a model that is close to Bayes optimal, which is a goal we can enunciate for any distribution D without needing to make realizability assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' (Observe that if the Bayes optimal predictor has zero error, then our meaning of strong learning corresponds to the standard meaning, so our analysis is only more general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We now turn to our definition of weak learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Intuitively, a weak learning algorithm should return a hypothesis that makes predictions that are slightly better than trivial whenever doing so is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We take “trivial” predictions to be those of the best constant predictor as measured by squared error — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' the squared error obtained by simply returning the label mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A “weak learning” algorithm for us can be run on any restriction of the data distribution D to a subset S Ď X, and must return a hypothesis with squared error slightly better than the squared error of the best constant prediction, whenever the Bayes optimal predictor f ˚ has squared error slightly better than a constant predictor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' on restrictions for which the Bayes optimal predictor also does not improve over constant prediction, our weak learning algorithm is not required to do better either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Traditionally, “weak learning” assumptions do not distinguish between the optimization ability of the algorithm and the representation ability of the hypothesis class it optimizes over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since we have defined a squared error regression oracle AH as exactly optimizing the squared error over some class H, we will state our weak learning assumption as an assumption on the representation ability of H—but this is not important for our analysis here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6 we could equally well assume that AH returns a hypothesis h that improves over a constant predictor whenever one exists, without assuming that h optimizes squared error over all of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5 (Weak Learning Assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and a class of functions H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let f ˚pxq “ Ey„Dpxqrys denote the true conditional label expectation conditional on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that H satisfies the γ-weak learning condition relative to D if for every S Ď X with Prx„DX rx P Ss ą 0, if: Erpf ˚pxq ´ yq2|x P Ss ă min cPR Erpc ´ yq2|x P Ss ´ γ then there exists an h P H such that: Erphpxq ´ yq2|x P Ss ă min cPR Erpc ´ yq2|x P Ss ´ γ When γ “ 0 we simply say that H satisfies the weak learning condition relative to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 10 Observe why our weak learning assumption is “weak”: the Bayes optimal predictor f ˚ may improve arbitrarily over the best constant predictor on some set S in terms of squared error, but in this case we only require of H that it include a hypothesis that improves by some γ which might be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since the γ-weak learning condition does not make any requirements on H on sets for which f ˚pxq improves over a constant predictor by less than γ, the best we can hope to prove under this assumption is γ-approximate Bayes optimality, which is what we do next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, any γ ą 0, any class of real valued functions H that satisfies the γ-weak learning condition relative to D, and a squared error regression oracle AH for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let α “ γ and B “ 1{γ (or any pair such that α{B “ γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then LSBoostpf, α, AH, D, Bq halts after at most T ď 2 γ2 many iterations and outputs a model fT ´1 such that fT ´1 is 2γ-approximately Bayes optimal over D: E px,yq„DrpfT ´1pxq ´ yq2s ď E px,yq„Drpf ˚pxq ´ yq2s ` 2γ where f ˚pxq “ Epx,yq„Drys is the function that minimizes squared error over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' At each round t before the algorithm halts, we have by construction that errt ď errt´1 ´ α 2B , and since the squared error of f0 is at most 1, and squared error is non-negative, we must have T ď 2B α “ 2 γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now suppose the algorithm halts at round T and outputs fT ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It must be that errT ą errT ´1 ´ γ2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose also that fT ´1 is not 2γ-approximately Bayes optimal: E px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2s ą 2γ We can write this condition as: ÿ vPr1{ms PrrfT ´1pxq “ vs ¨ E px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ą 2γ Define the set: S “ tv P r1{ms : E px,yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ě γu to denote the set of values v in the range of fT ´1 such that conditional on fT ´1pxq “ v, fT ´1 is at least γ-sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since we have both y P r0, 1s and fT ´1pxq P r0, 1s, for every v we must have that ErpfT ´1pxq´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Therefore we can bound: 2γ ă ÿ vPr1{ms PrrfT ´1pxq “ vs ¨ E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ pf ˚pxq ´ yq2|fT ´1pxq “ vs ď Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx P Ss ` p1 ´ Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx P Ssqγ Solving we learn that: Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx P Ss ě 2γ ´ γ p1 ´ γq ě 2γ ´ γ “ γ Now observe that by the fact that H is assumed to satisfy the γ-weak learning assumption with respect to D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' at the final round T of the algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' for every v P S we have that hT v satisfies: E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ phT v pxq ´ yq2|fT ´1pxq “ vs ě γ Let ˜ errT “ Epx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drp ˜fT pxq ´ yq2s Therefore we have: errT ´1 ´ ˜ errT “ ÿ vPr1{ms Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq “ vs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrpfT ´1pxq ´ yq2 ´ phT v pxq ´ yq2|fT ´1pxq “ vs ě Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„DrfT ´1pxq P Ssγ ě γ2 11 We recall that | ˜ errT ´ errT | ď 1{m “ γ2 2 and so we can conclude that errT ´1 ´ errT ě γ2 2 which contradicts the fact that the algorithm halted at round T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 5 When Multicalibration Implies Accuracy We analyzed the same algorithm (Algorithm 1) as both an algorithm for obtaining multicalibration with respect to H, and, when H satisfied the weak learning condition given in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5, as a boosting algorithm that converges to the Bayes optimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this section we show that this is no coincidence: multicalibration with respect to H implies Bayes optimality if and only if H satisfies the weak learning condition from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5, First we define what we mean when we say that multicalibration with respect to H implies Bayes opti- mality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note that the Bayes optimal model f ˚pxq is multicalibrated with respect to any set of functions, so it is not enough to require that there exist Bayes optimal functions f that are multicalibrated with respect to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Instead, we have to require that every function that is multicalibrated with respect to H is Bayes optimal: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that multicalibration with respect to H implies Bayes optimality over D if for every f : X Ñ R that is multicalibrated with respect to D and H, we have: E px,yq„Drpfpxq ´ yq2s “ E px,yq„Drpf ˚pxq ´ yq2s Where f ˚pxq “ Ey„Dpxqrys is the function that has minimum squared error over the set of all functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Recall that when the weak learning parameter γ in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5 is set to 0, we simply call it the “weak learning condition” relative to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We first state and prove our characterization for the exact case when γ “ 0, because it leads to an exceptionally simple statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We subsequently extend this characterization to relate approximate Bayes optimality and approximate multicalibration under quantitative weakenings of the weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let H be a class of functions that is closed under affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Multicalibration with respect to H implies Bayes optimality over D if and only if H satisfies the weak learning condition relative to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To avoid measurability issues we assume that models f have a countable range (which is true in particular whenever X is countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' First we show that if H satisfies the weak learning condition relative to D, then multicalibration with respect to H implies Bayes optimality over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then there exists a function f that is multical- ibrated with respect to D and H, but is such that: E px,yq„Drpfpxq ´ yq2s ą E px,yq„Drpf ˚pxq ´ yq2s By linearity of expectation we have: ÿ vPRpfq Prrfpxq “ vs ¨ E px,yq„Drpfpxq ´ yq2 ´ pf ˚pxq ´ yq2|fpxq “ vs ą 0 In particular there must be some v P Rpfq with Prx„DX rfpxq “ vs ą 0 such that: E px,yq„Drpfpxq ´ yq2|fpxq “ vs ą E px,yq„Drpf ˚pxq ´ yq2|fpxq “ vs 12 Let S “ tx : fpxq “ vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Observe that if H is closed under affine transformation, the constant function hpxq “ 1 is in H, and hence multicalibration with respect to H implies calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since f is calibrated, we know that: E px,yq„Drpv ´ yq2|x P Ss “ min cPR E px,yq„Drpc ´ yq2|x P Ss Thus by the weak learning assumption there must exist some h P H such that: Erpv ´ yq2 ´ phpxq ´ yq2|x P Ss “ Erpfpxq ´ yq2 ´ phpxq ´ yq2|fpxq “ vs ą 0 By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2, there must therefore exist some h1 P H such that: E px,yq„Drh1pxqpy ´ vq|fpxq “ vs ą 0 implying that f is not multicalibrated with respect to D and H, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In the reverse direction, we show that for any H that does not satisfy the weak learning condition with respect to D, then multicalibration with respect to H and D does not imply Bayes optimality over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In particular, we exhibit a function f such that f is multicalibrated with respect to H and D, but such that: E px,yq„Drpfpxq ´ yq2s ą E px,yq„Drpf ˚pxq ´ yq2s Since H does not satisfy the weak learning assumption over D, there must exist some set S Ď X with Prrx P Ss ą 0 such that E px,yq„Drpf ˚pxq ´ yq2|x P Ss ă min cPR E px,yq„Drpc ´ yq2|x P Ss but for every h P H: E px,yq„Drphpxq ´ yq2|x P Ss ě min cPR E px,yq„Drpc ´ yq2|x P Ss .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let cpSq “ Epx,yq„Dry|x P Ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We define fpxq as follows: fpxq “ # f ˚pxq x R S cpSq x P S We can calculate that: E px,yq„Drpfpxq ´ yq2s “ Pr px,yq„Drx P Ss E px,yq„DrpcpSq ´ yq2|x P Ss ` Pr px,yq„Drx R Ss E px,yq„Drpf ˚pxq ´ yq2|x R Ss ą Pr px,yq„Drx P Ss E px,yq„Drpf ˚pxq ´ yq2|x P Ss ` Pr px,yq„Drx R Ss E px,yq„Drpf ˚pxq ´ yq2|x R Ss “ E px,yq„Drpf ˚pxq ´ yq2s In other words, f is not Bayes optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So if we can demonstrate that f is multicalibrated with respect to H and D we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then there exists some h P H and some v P Rpfq such that E px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2, there exists some h1 P H such that: E px,yq„Drph1pxq ´ yq2|fpxq “ vs ă E px,yq„Drpfpxq ´ yq2|fpxq “ vs 13 We first observe that it must be that v “ cpSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' If this were not the case, by definition of f we would have that: E px,yq„Drph1pxq ´ yq2|fpxq “ vs ă E px,yq„Drpf ˚pxq ´ yq2|fpxq “ vs which would contradict the Bayes optimality of f ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Having established that v “ cpSq we can calculate: E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drph1pxq ´ yq2|fpxq “ cpSqs “ Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx P Ss E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drph1pxq ´ yq2|x P Ss ` Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx R S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' fpxq “ cpSqs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drph1pxq ´ yq2|x R S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' fpxq “ cpSqs ě Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx P Ss E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drph1pxq ´ yq2|x P Ss ` Pr px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drx R S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' fpxq “ cpSqs E px,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='yq„Drpfpxq ´ yq2|x R S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' fpxq “ cpSqs where in the last inequality we have used the fact that by definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' fpxq “ f ˚pxq for all x R S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' and so is pointwise Bayes optimal for all x R S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Hence the only way we can have Epx,yq„Drph1pxq ´ yq2|fpxq “ cpSqs ă Epx,yq„Drpfpxq ´ yq2|fpxq “ cpSqs is if: E px,yq„Drph1pxq ´ yq2|x P Ss ă E px,yq„DrpcpSq ´ yq2|x P Ss But this contradicts our assumption that H violates the weak learning condition on S, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We now turn our attention to deriving a relationship between approximate multicalibration and approx- imate Bayes optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To do so, we’ll introduce an even weaker weak learning condition that has one additional parameter ρ, lower bounding the mass of sets S that we can condition on while still requiring the weak learning condition to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We remark that Algorithm 1 can be analyzed as a boosting algorithm under this weaker weak learning assumption as well, with only minor modifications in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 ( pγ, ρq-weak learning condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and let H be a class of arbitrary real-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that H satisfies the pγ, ρq-weak learning condition for D if the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For every set S Ď X such that Prx„DX rx P Ss ą ρ, if E px,yq„Drpf ˚ ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, where ¯yS “ Epx,yq„Dry | x P Ss, then there exists h P H such that E px,yq„Drphpxq ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We may now prove our theorem showing that approximate multicalibration with respect to a class H implies approximate Bayes optimality if and only if H satisfies the pγ, ρq-weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We recall Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4, which notes that we must restrict approximate multicalibration to a bounded subset of H, as we will assume that H is closed under affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix any distribution D P ∆Z, any model f : X Ñ r0, 1s, and any class of real valued functions H that is closed under affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let: H1 “ th P H : max xPX hpxq2 ď 1u 14 be the set of functions in H upper-bounded by 1 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let m “ |Rpfq|, γ ą 0, and α ď γ3 16m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if H satisfies the pγ, γ{mq-weak learning condition and f is α-approximately multicalibrated with respect to H1 on D, then f has squared error E px,yq„Drpfpxq ´ yq2s ď E px,yq„Drpf ˚ ´ yq2s ` 3γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Conversely, if H does not satisfy the pγ, γ{mq-weak learning condition, there exists a model f : X Ñ r0, 1s that is α-approximately multicalibrated with respect to H1 on D, for α “ γ, and is perfectly calibrated on D, but f has squared error E px,yq„Drpfpxq ´ yq2s ě E px,yq„Drpf ˚ ´ yq2s ` γ2{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We begin by arguing that α-approximate multicalibration with respect to H1 on D implies approxi- mate Bayes optimality when H satisfies the pγ, γ{mq-weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose not, and there exists a function f that is α-multicalibrated with respect to H1, but E px,yq„Drpf ˚ ´ yq2s ă E px,yq„Drpfpxq ´ yq2s ´ 3γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then there must exist some v P Rpfq such that Prpx,yq„Drfpxq “ vs ą γ{m and E px,yq„Drpf ˚ ´ yq2 | fpxq “ vs ă E px,yq„Drpfpxq ´ yq2 | fpxq “ vs ´ 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We observe that since H is closed under affine transformation, the constant function hpxq “ 1 is in H, and so α-approximate multicalibration with respect to H1 implies α-approximate calibration as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus by definition, Prrfpxq “ vs ¨ ˆ E px,yq„Drv ´ y | fpxq “ vs ˙2 ď α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Letting ¯yv “ Ery | fpxq “ vs, our lower-bound that Prrfpxq “ vs ą γ{m gives us that pv ´ ¯yvq2 ă αm{γ ď ` γ 4 ˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We now use this upper-bound on calibration error in conjuction with our lower-bound on distance from Bayes optimality to show that the squared error of the constant predictor ¯yv must also be far from Bayes optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' E px,yq„Drpf ˚pxq ´ yq2 | fpxq “ vs ă E px,yq„Drpfpxq ´ yq2 | fpxq “ vs ´ 2γ “ E px,yq„Drpv ´ ¯yv ` ¯yv ´ yq2 | fpxq “ vs ´ 2γ “ E px,yq„Drp¯yv ´ yq2 | fpxq “ vs ` pv ´ ¯yvq2 ´ 2γ ă E px,yq„Drp¯yv ´ yq2 | fpxq “ vs ´ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The pγ, γ{mq-weak learning condition then guarantees that there exists some h P H such that E px,yq„Drph ´ yq2 | fpxq “ vs ă E px,yq„Drp¯yv ´ yq2 | fpxq “ vs ´ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4, the fact that h improves on the squared loss of ¯yv by an additive factor γ, on the set of x such that fpxq “ v, implies that Erhpxqpy ´ ¯yvq | fpxq “ vs ą γ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because f is α-approximately calibrated on D, we can use the existence of such an h to witness a failure of multicalibration: Erhpy ´ vq | fpxq “ vs “ Erhpxqpy ´ ¯yv ` ¯yv ´ vq | fpxq “ vs “ Erhpxqpy ´ ¯yvq | fpxq “ vs ` Erhpxqp¯yv ´ vq | fpxq “ vs ą γ{2 ´ |¯yv ´ v| ą γ{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 15 Then Prrfpxq “ vs ¨ ˆ E px,yq„Drhpxqpy ´ vq | fpxq “ vs ˙2 ą γ3 16m, contradicting our assumption that f is α-approximately multicalibrated with respect to H1 for α ă γ3 16m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Therefore approximate multicalibration with respect to H1 must imply that f is approximately Bayes opti- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It remains to show the other direction, that α-approximate multicalibration with respect to a class H1 implies approximate Bayes optimality only if H satisfies the pγ, γ{mq-weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' If this claim were not true for the stated parameters, then there must exist a class H such that every predictor f that: is α-approximately multicalibrated with respect to H1 is perfectly calibrated on D has range with cardinality |Rpfq| “ m also has squared error within γ2{m of Bayes optimal, but H does not satisfy the weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will show that no such class exists by defining, for any class H not satisfying the weak learning condition, a predictor f that is α-approximately multicalibrated with respect to that class, but has squared error that is not within γ2{m of Bayes optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Recall that if a class H does not satisfy the pγ, γ{mq-weak learning condition, then there must be some set SH such that Prrx P SHs ą γ{m, there does not exist an h P H such that E px,yq„Drph ´ yq2 | x P SHs ă E px,yq„Drp¯ySH ´ yq2 | x P SHs ´ γ, but for the Bayes optimal predictor, it holds that its squared loss satisfies E px,yq„Drpf ˚ ´ yq2 | x P SHs ă E px,yq„Drp¯ySH ´ yq2 | x P SHs ´ γ, where ¯ySH “ Ery | x P SHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For some hypothesis class H not satisfying the weak learning condition, and associated set SH, let fH be defined as follows: fHpxq “ # f ˚pxq, x R SH ¯ySH, x P SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note that, because fH is constant on SH, there must be some v P Rpfq such that the level set Sv “ tx P X : fpxq “ vu contains SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To see that fH is α-approximately multicalibrated with respect to H1, we first consider the contribution to multicalibration error from the level sets not containing SH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For all h P H and v P Rpfq such that v ‰ ¯ySH, E px,yq„Drhpxqpy ´ fHpxqq | fHpxq “ vs “ E px,yq„Drhpxqpy ´ f ˚pxqq | fHpxq “ vs “ E x„Dx E y„Dypxqrhpxqy | fHpxq “ vs ´ E x„Dx rhpxqf ˚pxq | fHpxq “ vs “ E x„Dx E y„Dypxqrhpxqy | fHpxq “ vs ´ E x„Dx E y„Dypxqrhpxqy | fHpxq “ vs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For the level set Sv for which SH Ď Sv, we know from the argument above that the elements x P SvzSH contribute nothing to the multicalibration error, as fpxq “ f ˚pxq on these elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, E px,yq„Drhpxqpy ´ fHpxqq | fpxq “ vs “ Pr x„DXrx P SHs ¨ E px,yq„Drhpxqpy ´ ¯ySHq | x P SHs ` Pr x„DXrx R SHs ¨ E px,yq„Drhpxqpy ´ f ˚pxqq | x P SvzSHs “ Pr x„DXrx P SHs ¨ E px,yq„Drhpxqpy ´ ¯ySHq | x P SHs 16 Therefore if fH is not α-approximately multicalibrated with respect to H1 on D, it must be the case that there exists some h P H1 such that Erhpxqpy ´ ¯ySHq | x P SHs ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2, there must exist a h1 P H such that E px,yq„Drp¯ySH ´ yq2 ´ ph1pxq ´ yq2 | x P SHs ą α “ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' But SH was defined to be a subset of X for which no such h1 exists and for which Prrx P SHs ą γ{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This would contradict our assumption that H does not satisfy the pγ, γ{mq-weak learning condition on D, and therefore fH is α-approximately multicalibrated with respect to H1 on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It remains to prove that fH is far from Bayes optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' E px,yq„DrpfHpxq ´ yq2s “ Pr x„DXrx P SHs E px,yq„Drp¯ySH ´ yq2 | x P SHs ` Prrx R SHs E px,yq„Drpf ˚pxq ´ yq2 | x R SHs ě Pr x„DXrx P SHs ˆ E px,yqDrpf ˚ ´ yq2 | x P SHs ` γ ˙ ` Prrx R SHs E px,yq„Drpf ˚pxq ´ yq2 | x R SHs “ E px,yq„Drpf ˚ ´ yq2s ` γ Pr x„DXrx P SHs ě E px,yq„Drpf ˚ ´ yq2s ` γ2{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 6 Weak Learners With Respect to Constrained Classes Thus far we have studied function classes H that satisfy a weak learning condition with respect to the Bayes optimal predictor f ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' But we can also study function classes H that satisfy a weak learning condition defined with respect to another constrained class of real valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 (Weak Learning Assumption Relative to C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and two classes of functions H and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that H satisfies the γ-weak learner condition relative to C and D if for every S Ď X with Prx„DX rx P Ss ą 0, if: min cPC E px,yq„Drpcpxq ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, where ¯yS “ Epx,yq„Dry | x P Ss, then there exists h P H such that E px,yq„Drphpxq ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' When γ “ 0 we simply say that H satisfies the weak learning condition relative to C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will show that if a predictor f is multicalibrated with respect to H, and H satisfies the weak learning assumption with respect to C, then in fact: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' f is multicalibrated with respect to C, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' f has squared error at most that of the minimum error predictor in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In fact, Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] show that if f is multicalibrated with respect to C, then it is an omnipredictor for C, which implies that f has loss no more than the best function cpxq P C, where loss can be measured with respect to any Lipschitz convex loss function (not just squared error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus our results imply that to obtain an omnipredictor for C, it is sufficient to be multicalibrated with respect to a class H that satisfies our weak learning assumption with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and two classes of functions H and C that are closed under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if f : X Ñ r0, 1s is multicalibrated with respect to D and H, and if H satisfies the weak learning condition relative to C and D, then in fact f is multicalibrated with respect to D and C as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We assume for simplicity that f has a countable range (which is without loss of generality e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' whenever X is countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose for contradiction that f is not multicalibrated with respect to C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this case there must be some c P C such that: ÿ vPRpfq Prrfpxq “ vs ˆ E px,yq„Drcpxqpy ´ vq|fpxq “ vs ˙2 ą 0 Since C is closed under affine transformations (and so both c and ´c are in C), there must be some c1 P C and some v P Rpfq with Prrfpxq “ vs ą 0 such that: E px,yq„Drc1pxqpy ´ vq|fpxq “ vs ą 0 Therefore, by the first part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2, there must be some c2 P C such that: E px,yq„Drpc2pxq ´ yq2|fpxq “ vs ă E px,yq„Drpv ´ yq2|fpxq “ vs Since H is closed under affine transformations, the function hpxq “ 1 is in H and so multicalibration with respect to H implies calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus v “ ¯ySv for Sv “ tx : fpxq “ vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Therefore, the fact that H satisfies the weak learning condition relative to C and D implies that there must be some h P H such that: E px,yq„Drphpxq ´ yq2|fpxq “ vs ă E px,yq„Drpv ´ yq2|fpxq “ vs Finally, the second part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 implies that: E px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 which is a violation of our assumption that f is multicalibrated with respect to H and D, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and two classes of functions H and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if f : X Ñ r0, 1s is calibrated and multicalibrated with respect to D and H, and if H satisfies the weak learning condition relative to C and D, then: E px,yq„Drpfpxq ´ yq2s ď min cPC E px,yq„Drpcpxq ´ yq2s Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We assume for simplicity that f has a countable range (which is without loss of generality e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' whenever X is countable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose for contradiction that there is some c P C such that: E px,yq„Drpcpxq ´ yq2s ă E px,yq„Drpfpxq ´ yq2s Then there must be some v P Rpfq with Prrfpxq “ vs ą 0 and: E px,yq„Drpcpxq ´ yq2|fpxq “ vs ă E px,yq„Drpv ´ yq2|fpxq “ vs Since f is calibrated, v “ ¯ySv for Sv “ tx : fpxq “ vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Therefore, the fact that H satisfies the weak learning condition relative to C and D implies that there must be some h P H such that: E px,yq„Drphpxq ´ yq2|fpxq “ vs ă E px,yq„Drpv ´ yq2|fpxq “ vs Finally, the second part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 implies that: E px,yq„Drhpxqpy ´ vq|fpxq “ vs ą 0 which is a violation of our assumption that f is multicalibrated with respect to H and D, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 18 We now turn to approximate versions of these statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To do so, we need a refined version of one direction of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 that shows us that if f witnesses a failure of multicalibration with respect to some h P H, then there is another function h1 P H that can be used to improve on f’s squared error, while controlling the norm of h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose H is closed under affine transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a model f : X Ñ r0, 1s, a levelset v P Rpfq, and a bound B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if there exists an h P H such that maxxPX hpxq2 ď B and Erhpxqpy ´ vq|fpxq “ vs ě α, for α ě 0, then there exists an h1 P H such that maxxPX h1pxq2 ď p1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' B α q2 and: Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let h1pxq “ v ` ηhpxq where η “ α Erhpxq2|fpxq“vs, as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because hpxq2 is uniformly bounded by B on X, it follows that Erhpxq2s ď B, and we have already shown in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 that this implies Erpfpxq ´ yq2 ´ ph1pxq ´ yq2|fpxq “ vs ě α2 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It only remains to bound maxxPX h1pxq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We begin by lower-bounding Erhpxq2 | fpxq “ vs in terms of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Erhpxq2 | fpxq “ vs ě Erhpxq | fpxq “ vs2 ě Erhpxqpy ´ vq | fpxq “ vs2 ě α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It follows that η ď 1{α, and so max xPX h1pxq2 “ max xPX pv ` ηhpxqq2 ď p1 ` η ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq2 ď ˜ 1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' B α ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will also need a parameterized version of our weak learning condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Recalling Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4, for approximate multicalibration to be meaningful with respect to a class that is closed under affine transfor- mation, we must specify a bounded subset of that class with respect to which a predictor is approximately multicalibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then to show that approximate multicalibration with respect to one potentially unbounded class implies approximate multicalibration with respect to another, we will need to specify the subsets of each class with respect to which a predictor is claimed to be approximately multicalibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This motivates a parameterization of our previous weak learning condition relative to a class C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will need to assume that whenever there is a B-bounded function in C that improves over the best constant predictor on a restriction of D, there also exists a B-bounded function in H that improves on the restriction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5 (B-Bounded Weak Learning Assumption Relative to C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and two classes of functions H and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a bound B ą 0 and let HB and CB denote the sets HB “ th P H : max xPX hpxq2 ď Bu and CB “ tc P C : max xPX cpxq2 ď Bu 19 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that H satisfies the B-bounded γ-bounded weak learning condition relative to C and D if for every S Ď X with Prx„DX rx P Ss ą 0, if: min cPCB E px,yq„Drpcpxq ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ, where ¯yS “ Ery | x P Ss, then there exists h P HB such that E px,yq„Drphpxq ´ yq2 | x P Ss ă E px,yq„Drp¯yS ´ yq2 | x P Ss ´ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D P ∆Z and two classes of functions H and C that are closed under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix αC, B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let B1 “ p1 ` b 2B αC q2 and γ “ αC 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a function f : X Ñ r0, 1s that maps into a countable subset of its range, and let m “ |Rpfq|, αH ă α3 C 29mB12 , and α ă αCγ2 32mB12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if H satisfies the B1-bounded γ-weak learning condition relative to C and D f is αH-approximately multicalibrated with respect to D and HB1 f is α-approximately calibrated on D, then f is αC-approximately multicalibrated with respect to D and CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Suppose not and there exists some c P CB such that ÿ vPRpfq Pr x„Dxrfpxq “ vs ¨ ˆ E px,yq„Drcpxqpy ´ vq | fpxq “ vs ˙2 ą αC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then there must exist some v P Rpfq such that Prrfpxq “ vs ą αC 2m and E px,yq„Drcpxqpy ´ vq | fpxq “ vs2 ą αC{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because C is closed under affine transformations, CB is closed under negation, so there must also exist some c1 P CB such that E px,yq„Drc1pxqpy ´ vq | fpxq “ vs ą a αC{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 shows that there is a c2 P Cp1` b 2B αC q2 “ CB1 such that E px,yq„Drpy ´ fpxqq2 ´ py ´ c2pxqq2 | fpxq “ vs ě αC 2B “ 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because f is α-calibrated on D, by definition we have Pr x„Dxrfpxq “ vs ¨ ˆ E px,yq„Drv ´ y | fpxq “ vs ˙2 ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Letting ¯yv “ Ery | fpxq “ vs, our lower-bound that Prrfpxq “ vs ą αC 2m gives us that pv ´ ¯yvq2 ă 2αm αC ď γ2 16B12 ă γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, because v is close to ¯yv, we can show the squared error of f must be close to the squared error of ¯yv on this level set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' E px,yq„Drpy ´ fpxqq2 | fpxq “ vs “ E px,yq„Drpy ´ ¯yv ` ¯yv ´ fpxqq2 | fpxq “ vs “ E px,yq„Drpy ´ ¯yvq2 ` 2py ´ ¯yvqp¯yv ´ vq | fpxq “ vs ` p¯yv ´ vq2 “ E px,yq„Drpy ´ ¯yvq2 | fpxq “ vs ` p¯yv ´ vq2 ă E px,yq„Drpy ´ ¯yvq2 | fpxq “ vs ` γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 20 Then, because the squared error of c2 on this level set is much less than the squared error of f, we find that c2 must also have squared error less than that of ¯yv: E px,yq„Drpy ´ ¯yvq2 ´ py ´ c2pxqq2 | fpxq “ vs ą E px,yq„Drpy ´ fpxqq2 ´ γ ´ py ´ c2pxqq2 | fpxq “ vs ě 2γ ´ γ “ γ We assumed H satisfies the B1-bounded γ-weak learning condition relative to C, so this gives us a function h P HB1 such that E px,yq„Drpy ´ ¯yvq2 ´ py ´ hpxqq2 | fpxq “ vs ą γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 shows that Erhpxqpy ´ ¯yvq | fpxq “ vs ą γ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So h witnesses a failure of multicalibration of f, since it follows that Erhpxqpy ´ vq | fpxq “ vs “ Erhpxqpy ´ ¯yvq | fpxq “ vs ` Erhpxqp¯yv ´ vq | fpxq “ vs ą γ{2 ´ B1 |¯yv ´ v| ě γ{2 ´ B1γ 4B1 “ γ{4 and so Pr x„Dxrfpxq “ vs ˆ E px,yq„Drhpxqpy ´ vq | fpxq “ vs ˙2 ą αCγ2 32m ą αH, contradicting αH-approximate multicalibration of f on HB1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022], Gopalan, Kalai, Reingold, Sharan, and Wieder show that any predictor that is approximately multicalibrated for a class H and distribution D can be efficiently post-processed to approxi- mately minimize any convex, Lipschitz loss function relative to the class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The theorem we have just proved can now be used to extend their result to approximate loss minimization over any other class C, so long as H satisfies the B-bounded γ-weak learning assumption relative to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Intuitively, this follows from the fact that if f is approximately multicalibrated with respect to H on D, it is also approximately multicalibrated with respect to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' However, the notion of approximate multicalibration adopted in Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] differs from the one in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, to formalize our intuition above, we will first state the covariance-based definition of approximate multicalibration appearing in Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] and prove a lemma relating it to our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We note that, going forward, we will restrict ourselves to distributions D over X ˆ t0, 1u, as in this case the two definitions of approximate multicalibration are straightforwardly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='7 (Approximate Covariance Multicalibration Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D over X ˆ t0, 1u and a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let H be an arbitrary collection of real valued functions h : X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then f is α-approximately covariance multicalibrated with respect to H on D if ÿ vPRpfq Pr x„DXrfpxq “ vs ¨ ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs ˇˇ ď α, where ¯hv “ Erhpxq | fpxq “ vs and ¯yv “ Ery | fpxq “ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D over X ˆ t0, 1u and a class of functions on X, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let HB denote the subset HB “ th P H : max xPX hpxq2 ď Bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 21 Fix a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if f is α-approximately multicalibrated with respect to HB on D, then f is p?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αp1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bqq-approximately covariance multicalibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' That is, for all h P HB, f satisfies ÿ vPRpfq Prrfpxq “ vs ¨ ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs ˇˇ ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αp1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' ÿ vPRpfq Prrfpxq “ vs¨ ˇˇErphpxq ´ ¯hvqpy ´ ¯yvq | fpxq “ vs ˇˇ “ ÿ vPRpfq Prrfpxq “ vs ¨ ˇˇErhpxqy | fpxq “ vs ´ ¯yv¯hv ˇˇ “ ÿ vPRpfq Prrfpxq “ vs ¨ ˇˇErhpxqy | fpxq “ vs ´ v¯hv ` v¯hv ´ ¯yv¯hv ˇˇ “ ÿ vPRpfq Prrfpxq “ vs ¨ ˇˇErhpxqpy ´ vq | fpxq “ vs ` ¯hvpv ´ ¯yvq ˇˇ ď ÿ vPRpfq Prrfpxq “ vs ¨ ` |Erhpxqpy ´ vq | fpxq “ vs| ` ˇˇ¯hvpv ´ ¯yvq ˇˇ˘ ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='α ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' B ÿ vPRpfq Prrfpxq “ vs ¨ |v ´ ¯yv| ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αp1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' where the second inequality follows from the fact that Erxs ď a Erx2s and the bound maxxPX hpxq2 ď B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We now recall a theorem of Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022], showing that approximate covariance multicalibration with respect to a class H implies approximate loss minimization relative to H, for convex, Lipschitz losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D over X ˆ t0, 1u and a class of real-valued functions on X, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let L be a class of functions on t0, 1u ˆ R that are convex and L-Lipschitz in their second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' If f is α-approximately covariance multicalibrated with respect to HB on D, then for every ℓ P L there exists an efficient post- processing function kℓ such that E px,yq„Drℓpy, kℓpfpxqqqs ď min hPHB E px,yq„Drℓpy, hpxqqs ` 2αL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a distribution D over X ˆ t0, 1u and two classes of real-valued functions on X that are closed under affine transformation, H and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix a function f : X Ñ r0, 1s that maps onto a countable subset of its range, denoted Rpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let L be a class of functions on t0, 1uˆR that are convex and L-Lipschitz in their second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Fix αC, B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let B1 “ p1 ` b 2B αC q2 and γ “ αC 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let αH ă α3 C 29mB12 , and α ă αCγ2 32mB12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if H satisfies the B1-bounded γ-weak learning condition relative to C and D f is αH-approximately multicalibrated with respect to D and HB1 f is α-approximately calibrated on D, then for every ℓ P L there exists an efficient post-processing function kℓ such that E px,yq„Drℓpy, kℓpfpxqqqs ď min cPCB E px,yq„Drℓpy, cpxqqs ` 2L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αCp1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 22 Figure 1: The update process at round t with m level sets during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We have from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6 that given the assumed conditions, f will be αC-approximately multicali- brated with respect to CB on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='8 that f is ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αCp1` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq-approximately covariance multicalibrated with respect to CB on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The result of Gopalan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022] then gives us that for all ℓ P L, there exists an efficient post-processing function kℓ such that E px,yq„Drℓpy, kℓpfpxqqqs ď min cPCB E px,yq„Drℓpy, cpxqqs ` 2L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='αCp1 ` ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 7 Empirical Evaluation In this section, we study Algorithm 1 empirically via an efficient, open-source Python implementation of our algorithm on both synthetic and real regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Our code is available here: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='com/ Declancharrison/Level-Set-Boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' An important feature of Algorithm 1 which distinguishes it from traditional boosting algorithms is the ability to parallelize not only during inference, but also during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let ft be the model maintained by Algorithm 1 at round t with m level sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Given a data set X, ft creates a partition of X defined by Xt`1 i “ tx|ftpxq “ viu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since the Xi are disjoint, each call ht`1 i “ AHpXt`1 i q can be made on a separate worker followed by a combine and round operation to obtain ˜ft`1 and ft`1 respectively, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A parallel inference pass at round t works nearly identically, but uses the historical weak learners ht`1 i obtained from training and applies them to each set Xt`1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1 Prediction on Synthetic Data From Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2, we know that multicalibration with respect to a hypothesis class H satisfying our weak learning condition implies Bayes optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To visualize the fast convergence of our algorithm to Bayes optimality, we create two synthetic datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' each dataset contains one million samples with two features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We label these points using two functions, C0 and C1, defined below and pictured in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We attempt to learn the underlying function with Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' C0pxq “ $ ’ ’ ’ & ’ ’ ’ % px ` 1q2 ` py ´ 1q2, if x ď 0, y ě 0 px ´ 1q2 ` py ´ 1q2, if x ą 0, y ě 0 px ` 1q2 ` py ` 1q2, if x ď 0, y ă 0 px ´ 1q2 ` py ` 1q2, if x ą 0, y ă 0 (C0) 23 = A(Xi) 21) 二 α(ft(α) αl(ft(α) = vi) Round(ft) Round( ft ht+1 AH(Xi) Jt+1 1(α) hm = AH(Xm)C1pxq “ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % x ` 20xy2 cosp´8xq sinp8yq ´ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5x`4qpx`1q2 y`3 ` py ´ 1q2¯ , if x ď 0, y ě 0 x ` 20xy2 cosp8xq sinp8yq ´ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5x`4qpx´1q2 y`3 ` py ´ 1q2¯ , if x ą 0, y ě 0 x ` 20xy2 cosp´8xq sinp8yq ´ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5x`4qpx`1q2 y`3 ` py ` 1q2¯ , if x ď 0, y ă 0 x ` 20xy2 cosp8xq sinp8yq ´ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5x`4qpx´1q2 y`3 ` py ` 1q2¯ , if x ą 0, y ă 0 (C1) In Figure 3, we show an example of Algorithm 1 learning C0 using a discretization of five-hundred level sets and a weak learner hypothesis class of depth one decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Each image in figure 3 corresponds to the map produced by Algorithm 1 at the round listed in the top of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' As the round count increases, the number of non-empty level sets increases until each level set is filled, at which point the updates become more granular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The termination round titled ‘final round’ occurs at T “ 199 and paints an approximate map of C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The image titled ‘out of sample’ is the map produced on a set of one million points randomly drawn outside of the training sample, and shows that Algorithm 1 is in fact an approximation of the Bayes Optimal C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Figure 2: C0 maps x1, x2 P r´2, 2s to four cylindrical cones symmetric about the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' C1 maps x1, x2 P r´1, 1s to a hilly terrain from a more complex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Figure 4 plots the same kind of progression as Figure 3, but with a more complicated underlying function C1 using a variety of weak learner classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We are able to learn this more complex surface out of sample with all base classes except for linear regression, which results in a noisy out-of-sample plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 Prediction on Census Data We evaluate the empirical performance of Algorithm 1 on US Census data compiled using the Python folktables package Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this dataset, the feature space consists of demographic information about individuals (see Table 1), and the labels correspond to the individual’s annual income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 24 Co(X1, X2) ¥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6 y 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='00 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='00Figure 3: Evolution of Algorithm 1 learning C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' feature description feature description AGEP age POBP place of birth COW class of worker RELP relationship SCHL education level WKHP work hours per week MAR marital status SEX binary sex OCCP occupation RAC1P race Table 1: Features included in income prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We cap income at $100,000 and then rescale all labels into r0, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' On an 80/20% train-test split with 500,000 total samples, we compare the performance of Algorithm 1 with Gradient Boosting with two perfor- mance metrics: mean squared error (MSE), and mean squared calibration error (MSCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For less expressive weak learner classes (such as DT(1), see Figure 5), Algorithm 1 has superior MSE out of sample compared to Gradient Boosting through one hundred rounds while maintaining significantly lower MSCE, and converges quicker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' However, as the weak learning class becomes more expressive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' increasing decision tree depths), Algorithm 1 is more prone to overfitting than gradient boosting (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 25 Figure 4: Stages of Algorithm 1 learning C1 with linear regression (LR) and varying depth d decision trees (DT(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In the out of sample plot for linear regression, points are not mapped to their proper position, implying C1 cannot be learned by boosting linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' All other hypothesis classes eventually converge to C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 26 H T=0 T=1 T=2 Final Round Out of Sample LR DT(1) DT(2) DT(3) DT(4)Figure 5: Comparison of Algorithm 1 (LS) and Gradient Boosting (GB), both using depth 1 regression trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' indicates termination round of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Table 2, we compare the time taken to train n weak learners with Algorithm 1 and with scikit-learn’s version of Gradient Boosting on our census data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Recall that our algorithm trains multiple weak learners per round of boosting, and so comparing the two algorithms for a fixed number of calls to the weak learner is distinct from comparing them for a fixed number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because models output by Algorithm 1 may be more complex than those produced by Gradient Boosting run for the same number of rounds, we use number of weak learners trained as a proxy for model complexity, and compare the two algorithms holding this measure fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We see the trend for Gradient Boosting is linear with respect to number of weak learners, whereas Algorithm 1 does not follow the same linear pattern upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This is due to not being able to fully 27 MSE with Weak Learner Depth 1 Decision Trees train error (LS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='055 test error (LS) train error (GB) test error (GB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='035 0 20 40 60 80 100 Number of roundsMSCE with Weak Learner Depth 1 Decision Trees train error (LS) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='025 test error (LS) train error (GB) --- test error (GB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='015 MSCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='000 0 20 40 60 80 100 Number of roundsleverage parallelization of training weak learners in early stages of boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' At each round, Algorithm 1 calls the weak learner on every large enough level set of the current model, and it is these independent calls that can be easily parallelized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' However, in the early rounds of boosting the model may be relatively simple, and so many level sets may be sparsely populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' As the model becomes more expressive over subsequent rounds, the weak learner will be invoked on more sets per round, allowing us to fully utilize parallelizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' # Weak Learners DT(1) DT(2) DT(3) LS GB Faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' LS GB Faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' LS GB Faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 50 level sets 100 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='97 ✓ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='86 23.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='80 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='49 ✓ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='61 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='74 ✓ 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='12 336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='89 ✓ Table 2: Time (in seconds) comparison of Algorithm 1 (LS) with fifty level sets and Gradient Boosting to train certain numbers of estimators for various weak learner classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In Figure 6, we measure MSE and MSCE for Algorithm 1 and Gradient Boosting over rounds of training on our census data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Again, we note that one round of Algorithm 1 is not equivalent to one round of Gradient Boosting, but intend to demonstrate error comparisons and rates of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For the linear regression plots, Gradient Boosting does not reduce either error since combinations of linear models are also linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' As the complexity of the underlying model class increases, Gradient Boosting surpasses Algorithm 1 in terms of MSE, though it does not minimize calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We notice that Algorithm 1, like most machine learning algorithms, is prone to overfitting when allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Future performance hueristics we intend to investigate include validating updates, complexity penalties, and weighted mixtures of updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 28 Figure 6: MSE and MSCE comparison of Algorithm 1 (LS) and Gradient Boosting (GB) on linear regression and decision trees of varying depths.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In International Conference on Com- putational Learning Theory, pages 621–636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Springer, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Maya Burhanpurkar, Zhun Deng, Cynthia Dwork, and Linjun Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Scaffolding sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='03135, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A Philip Dawid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The well-calibrated bayesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Journal of the American Statistical Association, 77(379): 605–610, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Frances Ding, Moritz Hardt, John Miller, and Ludwig Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Retiring adult: New datasets for fair machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Nigel Duffy and David Helmbold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Boosting methods for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Machine Learning, 47(2):153–200, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Dean P Foster and Rakesh Vohra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Regret in the on-line decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Games and Economic Behavior, 29(1-2):7–35, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Yoav Freund and Robert E Schapire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A decision-theoretic generalization of on-line learning and an application to boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Journal of computer and system sciences, 55(1):119–139, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Jerome H Friedman.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Robert E Schapire and Yoav Freund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Boosting: Foundations and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Kybernetes, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Eliran Shabat, Lee Cohen, and Yishay Mansour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Sample complexity of uniform convergence for multicali- bration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:13331–13340, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Shai Shalev-Shwartz and Shai Ben-David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Understanding machine learning: From theory to algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Cambridge university press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Vapnik and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' YA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Chervonenkis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' On the uniform convergence of relative frequencies of events to their probabilities, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' A Generalization Bounds Our analysis of Algorithm 1 assumed direct access to the data distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In practice, we will run the algorithm on the empirical distribution over a sample of n points D „ Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In this section, we show that when we do this, so long as n is sufficiently large, both our squared error and our multicalibration guarantees carry over from the empirical distribution over D to the distribution D from which D was sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Most generalization bounds for multicalibration algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Hébert-Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2018], Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021, 2022], Shabat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2020]) are either stated and proven for finite classes H, or are proven for algorithms that do not operate as empirical risk minimization algorithms, but instead gain access to a fresh sample of data from the distribution at each iteration, or are proven for hypotheses classes that are fixed independently of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We have a different challenge: Like Hébert-Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2018], Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2021] we study an iterative algorithm whose final hypothesis class is not fixed up front, but implicitly defined as a function of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' But we wish to study the algorithms as they are used—as empirical risk minimization algorithms—so we do not want our analysis to depend on using a fresh sample of data at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' And unlike the analysis in Jung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' [2022], for us H is continuously large (since it is closed under affine transformations), so we cannot rely on bounds that depend on log |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Instead we give a uniform convergence analysis that depends on the pseudo-dimension of our class of weak learners H: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Pseudodimension[Pollard [2012]] Let H be a class of functions from X to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that a set S “ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , xm, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , ymq P X m ˆRm is pseudo-shattered by H if for any pb1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , bmq P t0, 1um there exists h P H such that @i, hpxiq ą y ðñ bi “ 1 The pseudodimension of H, denoted PdimpHq is the largest integer m for which H pseudo-shatters some set S of cardinality m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Although hypotheses in H are continuously valued, Algorithm 1 outputs functions that have finite range r1{ms, and so we can view them as multi-class classification functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Our analysis will proceed by study- ing the generalization properties of these multiclass functions, which we will characterize using Natarajan dimension: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='2 (Shattering for multiclass functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Natarajan [1989], Shalev-Shwartz and Ben-David [2014] A set C Ď X is shattered by H if there exists two functions f0, f1 : C Ñ rks such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For every x P C, f0pxq ‰ f1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For every B Ď C there exists a function h P H such that @x P B, hpxq “ f0pxq and @x P C B, hpxq “ f1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3 (Natarajan dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Natarajan [1989], Shalev-Shwartz and Ben-David [2014] The Natara- jan dimension of H, denoted NdimpHq, is the maximal size of a shattered set C Ď X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 31 We can then rely the following standard uniform convergence bound for multiclass classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This statement is slightly modified from the result in Shalev-Schwartz and Ben-David to account for our use of squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The result still holds on account of the fact that the Cherhoff bound only relies on the loss function being bounded, and ours is indeed bounded between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4 (Multiclass uniform convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Shalev-Shwartz and Ben-David [2014] Let ϵ, δ ą 0 and let H be a class of functions h : X Ñ r1{ks such that the Natarajan dimension of H is d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let D P ∆pX ˆ r0, 1sq be an arbitrary distribution and let D “ tpx1, y1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , pxn, ynqupxi,yiq„D be a sample of n points from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then for n “ O ˆd logpkq ` logp1{δq ε2 ˙ , Pr „ max hPH ˇˇˇˇ E px,yq„Drpy ´ hpxqq2s ´ E px,yq„Drpy ´ hpxqq2s ˇˇˇˇ ě ϵs ȷ ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Our strategy will be to bound the Natarajan dimension of the class of models that can be output by Algorithm 1 in terms of the pseudodimension of the underlying weak learner, then apply the above uniform convergence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' To do so, we will first use the following lemma, which bounds the Natarajan dimension of functions that can be described as post-processings of binary valued-functions from a class of bounded VC-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Shalev-Shwartz and Ben-David [2014] Suppose we have ℓ binary classifiers from binary class Hbin and a rule r : t0, 1uℓ Ñ rks that determines a multiclass label according to the predictions of the ℓ binary classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Define the hypothesis class corresponding to this rule as H “ trph1p¨q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , hℓp¨qq : ph1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , hℓq P pHbinqℓu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then, if d “ VCdimpHbinq, NdimpHq ď 3ℓd logpℓdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Recall that the VC-dimension of a binary classifier is defined as follows: Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='6 (VC-dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Vapnik and Chervonenkis [1971] Let H be a class of binary classifiers h : X Ñ t0, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let S “ tx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , xmu and let ΠHpSq “ tphpx1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , hpxmqq : h P Hu Ď t0, 1um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We say that S is shattered by H if ΠHpSq “ t0, 1um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' The Vapnik-Chervonenkis (VC) dimension of H, denoted VCdimpHq, is the cardinality of the largest set S shattered by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let Hboost be the class of models output by RegressionMulticalibratepf, α, AH, ¨, Bq (Algorithm 1) for any input distribution D and let d be the pseudodimension of its input weak learner class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Ndim pHboostq ď 24pB{αq3d log ` p2B{αq3d ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let m be defined (as in RegressionMulticalibratepf, α, AH, D, Bq) to be 2B{α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Because our models are always rounded to the nearest value in r1{ms, we can think of the model ft generated in every round of the algorithm multiclass classification problems over m classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We will show that our final model can be written as a decision rule that maps the outputs of some ℓ Boolean classifiers to r1{ms, and that these Boolean classifiers have VC dimension that is bounded by the pseudodimension of the weak learner class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then, we will apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5 to get an upper bound on the Natarajan dimension of the class of models in terms of α, B, and the pseudodimension of the input weak learner class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Consider the initial round of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We can convert our (rounded) initial regressor f0 to a series of m Boolean thresholdings gv which return 1 when f0pxq ě v: g0 v “ # 1 if f0pxq ě v, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 32 These m Boolean thresholdings can then be mapped back to the original prediction over r1{ms using a decision rule r : t0, 1um Ñ r1{ms which picks the largest of the thresholds that evaluates to 1, and assigns that index to the prediction: r0ptg0 vuvPr1{msqpxq “ arg max iPr1{ms i1rgvpxq “ 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note that since our initial predictor f0 was already rounded to take values in r1{ms, the largest v such that f0pxq ě v will be exactly f0pxq, so r0 is exactly equivalent to f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Similarly, at round t ` 1 of RegressionMulticalibratepf, α, AH, D, Bq, we will show that the model ft`1 can be written as a decision rule rt`1 over m ` pt ` 1qm2 binary classifiers g, where gt v,i “ # 1 if ht vpxq ě i ´ 1{p2mq, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , Here, the thresholds measure halfway between each level set, as ht vpxq has yet to be rounded to the nearest level set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We can write a decision rule that maps these thresholds to classifications over r1{ms: rt`1 ` rt, tgt`1 v,i ui,vPr1{ms ˘ pxq “ ÿ vPr1{ms 1rrtpxq “ vs arg max iPr1{ms ` i ¨ 1rgt`1 v,i pxq “ 1s ˘ , Now, we need to show that this decision rule evaluated at round t is equivalent to ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We proceed inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' For our base case, we have already argued that our initial decision rule r0 is equivalent to the classifier f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now, say that we have decision rule rt over binary classifiers g that is equivalent to model ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' we can write rt`1 ` rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' tgt`1 v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='vPr1{ms ˘ pxq “ ÿ vPr1{ms 1rrtpxq “ vs arg max iPr1{ms ` i ¨ 1rgt`1 v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i pxq “ 1s ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' “ ÿ vPr1{ms 1rftpxq “ vs arg max iPr1{ms ` i ¨ 1rgt`1 v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i pxq “ 1s ˘ “ ÿ vPr1{ms 1rftpxq “ vs arg max iPr1{ms ` i ¨ 1rht`1 v pxq ě i ´ 1{p2mqs ˘ “ ÿ vPr1{ms 1rftpxq “ vsRoundpht`1 v pxqq “ ft`1pxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' where the second line comes from the inductive hypothesis and the second to last line’s equality comes from the fact that the largest i such that ht`1 v pxq ´ 1{p2mq ě i will be the exact rounded prediction of ht`1 v pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now, we need to show that at round t ` 1, the decision rule is a decision rule over m ` pt ` 1qm2 binary classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note that our initial decision rule r0 has m “ m`0¨m2 binary classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Say that at round t we have a decision rule rt over m ` tm2 classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' In the following round, we build m2 new Boolean classifiers gv, i for v, i P r1{ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, at round t ` 1 we have m ` tm2 ` m2 “ m ` pt ` 1qm2 classifiers total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3, we know that Algorithm 1 halts after at most T ď 2B{α rounds, at which point it outputs model fT ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, we can rewrite fT ´1 as a decision rule rT ´1 composed of at most m ` pT ´ 1qm2 ă Tm2 Boolean models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Plugging in our bound for T and definition of m, this gives us a decision rule rT ´1 composed of at most ` 2B α ˘3 Boolean classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let G be the class of Boolean threshold functions over H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' functions g : X Ñ t0, 1u such that gpxq “ # 1 hpxq ě i 0 hpxq ă i, 33 for some h P H and i P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Say that the VC-dimension of G is d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then, applying lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='5, it follows that NdimpHboostq ď 3 ˆ2B α ˙3 d1 log ˜ˆ2B α ˙3 d1 ¸ , “ 24 ˆB α ˙ d1 log ˜ˆ2B α ˙3 d1 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now, it remains to show that we can bound the VC-dimension of these thresholding functions by the pseudodimension of the weak learner class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Note that G as we have defined it above is a richer hypothesis class than the actual class of thresholding functions used in the above analysis, because it can threshold at any value in R rather than being restricted to r1{ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus, its VC dimension can only be greater than the VC dimension of the class of threshold functions over H restricted to r1{ms, and hence an upper bound on the VC dimension of G in terms of the pseudodimension of H will also be an upper bound on the VC dimension of the restricted class of threshold functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let d be the pseudodimension of H, and say that d ă d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' By the definition of VC-dimension, t0, 1ud`1 must be shattered by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=', for any set of d ` 1 points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , xd`1 P X with arbitrary labels b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , bd`1, there is some hypothesis g P G that realizes those labels on px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , xd`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Consider the function g that, given the d ` 1 points in X, realizes the labels b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , bd`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' By the construction of G, g is a thresholding of some function h P H at some point i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, there is be some i P R such that hpxiq ą i ñ bi “ 1 and such that bi “ 1 ñ hpxiq ą i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' But this means that t0, 1ud`1 is pseudo-shattered by H, and thus the pseudodimension of H is not d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus, it cannot be the case that d ă d1, and hence d1 ď d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' the VC dimension of G is bounded above by the pseudodimension of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Plugging this bound into the above bound on the Natarajan dimension gives us that NdimpHboostq ď 24 ˆB α ˙ d1 log ˜ˆ2B α ˙3 d1 ¸ , ď 24 ˆB α ˙ d log ˜ˆ2B α ˙3 d ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now, we can state the following uniform convergence theorem for our final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='8 (Squared Error Generalization for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let ϵ, δ, α, B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let Hboost be the class of models that can be output by RegressionMulticalibratepf, α, AH, ¨, Bq (Algorithm 1) for any input distribution D and let d be the pseudodimension of its input weak learner class H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let D “ tpx1, y1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , pxn, ynqupxi,yiq„D be a sample of n points drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Then if n “ O ˆdB3 log2pdB{αq α3ϵ2 ` logp1{δq ϵ2 ˙ Pr „ max hPHboost ˇˇˇˇ E px,yq„Drpy ´ hpxqq2s ´ E px,yq„Drpy ´ hpxqq2s ˇˇˇˇ ě ϵs ȷ ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' This follows directly from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='4 and the bound on the Natarajan dimension in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We also would like to know that our multicalibration guarantees are generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Rather than doing a bespoke analysis here, we can rely on the connection that we have established between failure of multicali- bration and ability to improve squared error and argue that if the final hypothesis output by the algorithm was not multicalibrated with high probability then it would be possible to improve its squared error out-of- sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus, by our previous generalization result for squared error, it would be possible to improve the squared error in-sample as well, giving us a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 34 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='9 (Multicalibration generalization guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let ϵ, δ, α, B ą 0 and consider the model fT ´1 output by RegressionMulticalibratepf, α, AH, D, Bq for some sample D of n points drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' from distri- bution D such that n “ O ˆdB3 log2pdB{αq α3ϵ2 ` logp1{δq ϵ2 ˙ Then if ϵ ď α 4B , with probability greater than or equal to 1 ´ 2δ it follows that fT ´1 is 2α-approximately multicalibrated with respect to the distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let D “ tpx1, y1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' , pxn, ynqupxi,yiq„D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Consider the model fT ´1 output by RegressionMulticalibratepf, α, AH, D, Bq, and recall that within the run of the algorithm there was also a model fT defined in the final round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Say that model fT ´1 is not 2α-approximately multicalibrated with respect to HB and the true distribution D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since the algorithm running on the sample halted, it must have been that the model in the final round improved in squared error by less than α{p2Bq when measured with respect to the sample D: E px,yq„DrpfT ´1 ´ yq2s ´ E px,yq„DrpfT ´ yq2s ď pα{2Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Consider what happens if we run the algorithm again, but with fT ´1 as its initial model and now with the underlying distribution as input rather than the sample of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Let f 1 T be the model found in the first round of running this process RegressionMulticalibratepfT ´1, α, AH, D, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Since fT ´1 is not 2α´approximately multicalibrated with respect to D and HB, then by an identical argument as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='3, it it must be that a single round of the algorithm improves the squared error on D by at least α{B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Thus, Epx,yq„DrpfT ´1 ´ yq2s ´ Epx,yq„Drpf 1 T ´ yq2s ą α{B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' We know from our previous convergence bound, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content='8, that with probability 1´δ, | Epx,yq„Drpf 1 T ´ yq2s ´ Epx,yq„DrpfT ´ yq2s| ă ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, f 1 T must with high probability also improve on the sample D: α B ă E px,yq„DrpfT ´1 ´ yq2s ´ E px,yq„Drpf 1 T ´ yq2s ă E px,yq„DrpfT ´1 ´ yq2s ´ E px,yq„Drpf 1 T ´ yq2s ` ϵ (with probability ě 1 ´ δ) ă E px,yq„DrpfT ´1 ´ yq2s ´ E px,yq„Drpf 1 T ´ yq2s ` 2ϵ (with probability ě 1 ´ 2δ) ă α 2B ` 2ϵ, where the last line comes from the fact that the error of f 1 T on D cannot be less than the error of fT on D, or else the regression oracle would have found it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' Now we have a contradiction: since we have set ϵ ď α 4B , α B ă α 2B ` 2 α 4B “ α B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' So, it must follow that fT ´1 is, with probability 1 ´ 2δ, 2α´approximately multicalibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} +page_content=' 35' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9FST4oBgHgl3EQfSzhz/content/2301.13767v1.pdf'} diff --git a/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/2301.01430v1.pdf.txt b/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/2301.01430v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2488ea2e700d4bcddf1543576fd866d29fc192e6 --- /dev/null +++ b/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/2301.01430v1.pdf.txt @@ -0,0 +1,670 @@ +arXiv:2301.01430v1 [eess.SY] 4 Jan 2023 +1–14 +Multi-Task System Identification of Similar Linear Time-Invariant +Dynamical Systems +Yiting Chen +YITING.CHEN-1@COLORADO.EDU +Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder +Ana M. Ospina +ANA.OSPINA@COLORADO.EDU +Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder +Fabio Pasqualetti +FABIOPAS@ENGR.UCR.EDU +Department of Mechanical Engineering , University of California at Riverside +Emiliano Dall’Anese +EMILIANO.DALLANESE@COLORADO.EDU +Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder +Abstract +Existing works on identification of the dynamics of linear time-invariant (LTI) systems primarily +focus on the least squares (LS) method when the recorded trajectories are rich and satisfy conditions +such as the persistency of excitation. In this paper, we consider the case where the recorded states +and inputs are not sufficiently rich, and present a system identification framework – inspired by +multi-task learning – that estimates the matrices of a given number of LTI systems jointly, by +leveraging structural similarities across the LTI systems. By regularizing the LS fit for each system +with a function that enforces common structural properties, the proposed method alleviates the +ill-conditioning of the LS when the recorded trajectories are not sufficiently rich. We consider +priors where, for example, the LTI systems are similar in the sense that the system matrices share a +common sparsity pattern, some matrices are linear combinations of others, or their norm difference +is small. We outline a proximal-gradient method to solve the multi-task identification problem, and +we propose a decentralized algorithm in the spirit of existing federated learning architectures. We +provide empirical evidence of the effectiveness of the proposed method by considering a synthetic +dataset, and by applying our method to the problem of estimating the dynamics of brain networks. +For the latter, the proposed method requires a significantly smaller number of fMRI readings to +achieve similar error levels of the LS when estimating the brain dynamics across subjects. +Keywords: System identification, multi-task learning, regularized regression, LTI systems. +1. Introduction +System identification is a core task where the model of dynamical systems is estimated based on +observed inputs and states (Ljung (1987); Pillonetto et al. (2022)). In particular, identification of lin- +ear time-invariant (LTI) systems is a well-investigated problem that has recently received renewed +attention due to lines of research in the context of data-driven control and optimization (see, for ex- +ample, the representative works in De Persis and Tesi (2019a); Coulson et al. (2019); Hewing et al. +(2020); Berberich et al. (2020); Krishnan and Pasqualetti (2021); Li et al. (2022)). +When the observation of the state is noise-free, the LTI system matrices can be readily identified +by leveraging the Willems’ Fundamental Lemma, provided that the recorded trajectory satisfies the +persistency of excitation (PE) condition as discussed in, e.g., Willems et al. (2005); De Persis and Tesi +© Y. Chen, A.M. Ospina, F. Pasqualetti & E. Dall’Anese. + +MULTI-TASK SYSTEM IDENTIFICATION +(2019b). On the other hand, when the LTI system is subject to unknown process noise or dis- +turbances, several existing works focus on the asymptotic and finite time estimation errors and +sample complexity of the least squares (LS) estimator; see, for example, the representative works +of Sarkar and Rakhlin (2019); Simchowitz et al. (2018); Faradonbeh et al. (2018); Oymak and Ozay +(2019); Zheng and Li (2020); Xin et al. (2022a) and pertinent references therein. In particular, fi- +nite time error bounds for estimating LTI systems from a single trajectory using the LS method +are provided in Sarkar and Rakhlin (2019), and a statistical analysis of the LS estimator is pre- +sented in +Simchowitz et al. (2018); Zheng and Li (2020) and Faradonbeh et al. (2018). +Addi- +tionally, regularized system identification methods are investigated in, e.g., Chen et al. (2014); +Chiuso and Pillonetto (2019); Pillonetto et al. (2022); a low-order linear system identification via +regularized regression is considered in Sun et al. (2020). These regularized identification methods +allow one to add a prior on the system matrices, and to strike a balance between LS fit and model +complexity Hastie et al. (2009, 2015). +The performance of the LS estimator hinges on the availability of a recorded trajectory that is +sufficiently rich to satisfy the PE condition and to render the LS sufficiently well conditioned. In +this paper, we are interested in cases where the PE condition may not be satisfied. In particular, we +consider the task of estimating the system matrices of N > 1 LTI systems, in cases where we do not +have sufficiently long (or sufficiently rich) recorded trajectories for at least one of the systems (or +for some of the systems). Accordingly, the question posed in this paper is as follows: is it possible +to leverage “similarities” among the N systems to obtain accurate estimates of the system matrices, +even if the PE condition is not satisfied? In particular, if the PE condition fails for the i-th system, +can one use recorded data from the other LTI systems to improve the estimation error? +In this direction, Xin et al. (2022b), consider estimating the matrices of a linear system from +samples generated by a “similar” one; in particular, in Xin et al. (2022b), a system is considered +“similar” if its matrices are perturbed versions of the true system. In this paper, we expand the +notion of “similarity” to account for additional structural properties that the N systems may have +in common. We consider cases where the N LTI systems are similar in the sense that the system +matrices share a common sparsity pattern, their norm difference is small, or some system matrices +can be expressed as a linear combination of the ones of some of the other systems. +Leveraging these similarity models, we propose a system identification framework that bridges +core tools investigated in the context of multi-task learning (Evgeniou and Pontil (2004); Sener and Koltun +(2018); Zhang and Yang (2021); Crawshaw (2020)), statistical learning (Hastie et al. (2009)), and +regularized identification methods (Pillonetto et al. (2022)); the proposed multi-task system identi- +fication is formalized as a regularized regression problem where we minimize the LS fit for each +system plus a regularization function that enforces a prior on the structural similarities of the LTI +systems. By appropriately tuning (typically via cross-validation Hastie et al. (2015)) the weight as- +signed to the regularization function, one can find a balance between fitting of the recorded data +and model complexity. More importantly, we show experimentally that the regularization function +allows one to transfer structural information and data across systems to alleviate the ill-conditioning +of the LS for systems without sufficiently rich recorded trajectories. +Contributions. Our contributions are as follows. +(c1) We formalize a multi-task system identification problem for multiple LTI systems, where we +consider the minimization of the LS fit for each system plus a regularization function that enforces a +prior on the structural similarities of the LTI systems. We provide relevant regularization functions +2 + +MULTI-TASK SYSTEM IDENTIFICATION +that are inspired by the group Lasso Yuan and Lin (2006); Huang and Zhang (2010), nuclear norm +minimization Chandrasekaran et al. (2009); Mardani et al. (2015), and ridge regression. +(c2) We provide a proximal-gradient method for solving the multi-task system identification prob- +lem, and show that the algorithm enjoys closed-form updates. We also develop a decentralized +algorithm where the N systems collaboratively solve the identification problem without exchanging +their recorded trajectories; the decentralized algorithm involves a message-passing that is similar to +federated learning architectures Yang et al. (2019); Wang et al. (2022). +(c3) We demonstrate the effectiveness of the proposed multi-task system identification method us- +ing: (i) synthetic LTI systems that feature structural similarities, and (ii) real data from the Human +Connectome Project (HCP), where blood-oxygen-level-dependent (BOLD) signals are obtained +from resting state functional magnetic resonance imaging (fMRI) Nozari et al. (2020). In partic- +ular, we show that the proposed method requires a significantly smaller number of fMRI readings +to achieve the same error of the LS by presuming that the LTI systems modeling the brain dynamics +in number of subjects feature a common sparsity pattern. We also consider the case where only a +few fMRI readings are available for one subject, showing the ability to “transfer information” from +the dynamics of the other subjects. +In this paper, ideas and merits of the proposed method are assessed numerically; the paper does +not include analytical error bounds, which are part of our ongoing research efforts. +2. Multi-Task System Identification Problem +2.1. Modeling +We consider N linear time-invariant (LTI) systems1 +xi(t + 1) = Aixi(t) + Biui(t) + wi(t), +xi(0) ∈ Rn, +i ∈ [N], +(1) +with i ∈ [N] the system index and t ∈ N the time index, and where xi(t) ∈ Rn, ui(t) ∈ Rp, and +wi(t) ∈ Rn are the state, input and process noise, respectively, and Ai ∈ Rn×n and Bi ∈ Rn×p are +the matrices of the i-th LTI system. Assume that, for each system, the input ui(t) and state xi(t) +can be measured; on the other hand, the system matrices are unknown and the disturbance wi(t) +cannot be measured. +For the i-th system, suppose that one has access to one trajectory {xi(τ), ui(τ)}Pi +τ=0, for some +Pi ∈ N, for the state and the inputs. With these measurements, the system matrices can be estimated +using the following LS criterion: +min +A∈Rn×n,B∈Rn×p +Pi +� +τ=1 +∥xi(τ + 1) − Axi(τ) − Bui(τ)∥2 +2, +(2) +1. Notation: We denote by N and R the set of natural numbers and the set of real numbers, respectively, and define +[n] = {1, 2, . . . , n}. We let ⊤ denote transposition. For a given column vector x ∈ Rn, ∥x∥2 is the Euclidean +norm and ∥x∥1 denotes the ℓ1 norm; for a matrix X ∈ Rn×m, ∥X∥F denotes the Frobenious norm and ∥X∥∗ the +nuclear norm. Moreover (X)ij refers to the entry (i, j) of the matrix X, and vec(X) is a mn × 1 vector stacking +the columns of X. Given a differentiable function f : Rn → R, ∇f(x) denotes the gradient of f at x (taken to be a +column vector). Given a closed convex set C ⊆ Rn, projC : Rn → Rn denotes the Euclidean projection of y onto +C, namely projC(y) := arg minv∈C ∥y − v∥. Given a lower-semicontinuous convex function g : Rn → R, the +proximal operator is defined as proxλg(y) := arg minx∈Rn g(x) + +1 +2λ∥x − y∥2 +2. +3 + +MULTI-TASK SYSTEM IDENTIFICATION +which is solved for each of the N systems independently. The LS estimator (2) has been extensively +studied in the literature, especially when the recorded data {xi(τ), ui(τ)}Pi +τ=0 satisfy the persistency +of excitation (PE) condition Moore (1983); Willems et al. (2005) – where the PE condition trans- +lates into the regression matrix being full column rank. In this case, several results are available +in terms of estimation error and in terms of sample complexity; see, for example, the results in +the recent works of Faradonbeh et al. (2018); Simchowitz et al. (2018); Oymak and Ozay (2019); +Sarkar and Rakhlin (2019); Sun et al. (2020), as well as pertinent references therein. Of course, the +data for solving (2) can also be collected from multiple trajectories; see, for example, Zheng and Li +(2020). +In this paper, we are interested in cases where the PE condition is not satisfied for some of the +N LTI systems (leading to ill-conditioning of the LS for those systems where the PE fails). In this +case, the question we pose in this paper pertains to whether it is possible to leverage “similarities” +among the N systems to obtain accurate estimates of the system matrices, even if the PE condition +is not satisfied for one or more systems. Key towards answering this question is to define the notion +of “similarity” for the system matrices. A first effort in this direction was made in Xin et al. (2022b), +where the matrices {Ai}i∈[N] and {Bi}i∈[N] are given by perturbations of given common matrices +¯A ∈ Rn×n and ¯B ∈ Rn×n, respectively. In this paper, we expand this first concept of “similar +matrices” to account for the following models: +(s1) Small distance: For any pair Ai, Aj, i, j ∈ [N], there exists ǫ > 0 such ∥Ai − Aj∥2 +F ≤ ǫ. +(s2) Common sparsity: The matrices A1, . . . AN have the same sparsity pattern; i.e., (A1)ij = +(A2)ij = . . . = (AN)ij = 0 for some entries (i, j). +(s3) Linear combinations: For the subset of systems i ∈ C, C ⊆ [N], there exists {αi,j ∈ R} +such that Ai = �N +j=1,j̸=i αijAj. +Similarity (s1) models the case where the norm of the matrix difference Ai − Aj is small; this is the +case, for example, for the model considered by Xin et al. (2022b). On the other hand, (s2) captures a +prior on the structural properties of the N systems; general examples include dynamics on network +systems. As a concrete example, when (1) represents the dynamics of brain networks, (s2) naturally +emerges from a similar functional or structural connectivity of the brain across different individuals +(see, e.g., Srivastava et al. (2020); Nozari et al. (2020)). Finally, (s3) models the case where the +matrix Ai of the i-th system can be expressed as a linear combination of some of the other matrices +{Aj}N +j=1,j̸=i; as an example, this model may be applicable to traffic flows and mobility-on-demand +services (see, e.g., Turan and Alizadeh (2021)), where the LTI systems (1) model the evolution of +the density of vehicles in given geographical areas over given periods of the day. We note that, while +the list above focuses on {Ai}i∈[N], similar arguments may apply to the system matrices {Bi}i∈[N]. +In the next section, we will present appropriate reformulations of (2) that leverage the similarity +models (s1)–(s3). For notational simplicity, hereafter we assume that the matrices {Bi}i∈[N] are +known and focus on the estimation of {Ai}i∈[N] from data. However, the proposed methodology +extends directly to the case where both {Ai}i∈[N] and {Bi}i∈[N] are unknown. +2.2. Multi-task system identification problem +Assume that one can observe the states and inputs {xi(τ), ui(τ)}Pi +τ=0 for each system i ∈ [N] (as +mentioned above, Bi is known). Let Li(Ai) := �Pi +τ=1 ∥xi(τ + 1) − Aixi(τ) − Biui(τ)∥2 +2 be the +4 + +MULTI-TASK SYSTEM IDENTIFICATION +LS fit for the i-th system as in (2). In the spirit of regularized LS methods (Hastie et al. (2009)), we +consider estimating the matrices {Ai}i∈[N] by solving the following optimization problem: +{ ˆAi}i∈[N] ∈ arg min +{Ai}N +i=1 +N +� +i=1 +Li(Ai) + λR(A1, . . . , AN), +(3) +where the first term is the LS fit for the N systems, (A1, . . . , AN) �→ R(A1, . . . , AN) is a lower- +semicontinuous convex function that promotes the prior specified by (s1)–(s3), and λ > 0 is a tuning +parameter. In particular, for the priors (s1)–(s3), the following regularization functions can be used: +(r1) For (s1), one can use the function R(A1, . . . , AN) = �N +i=1 +�N +j=i+1 ∥Ai − Aj∥2 +F to penalize +large deviations between the estimated matrices (Hastie et al. (2009)). +(r2) Common sparsity patterns can be promoted by leveraging group sparsity regularization func- +tions Yuan and Lin (2006); Huang and Zhang (2010). For instance, +R(A1, . . . , AN) = +N +� +i=1 +N +� +j=1 +∥[(A1)ij, (A2)ij, . . . , (AN)ij]⊤∥2 . +(4) +(r3) For the model in (s3), when q ≪ N of the matrices {Ai}i∈[N] are such that the remaining +N − q can be represented as a linear combination of these q matrices, the n2 × N matrix +[vec(A1), vec(A2), . . . , vec(AN)] has rank q ≪ N. In this case, the regularization function +can be taken to be (see, e.g., Chandrasekaran et al. (2009); Mardani et al. (2015)): +R(A1, A2, · · · , AN) = ∥[vec(A1), vec(A2), . . . , vec(AN)]∥∗ . +(5) +In the formulation (3), the role of the regularization function λR(A1, A2, · · · , AN) is twofold: +(i) similarly to classical regularized LS criteria, the parameter λ in (3) strikes a balance between +the LS fit (in our case, the LS fit for individual LTI systems) and the complexity of the mod- +els Hastie et al. (2015); for example, for (r2), higher values of λ promote a more parsimonious set of +entries in the system matrices that best represents the data. (ii) In our specific case, λR(A1, A2, · · · , AN) +allows us to fit the data of individual systems less closely – especially for the systems where the PE +condition fails – and bypass the ill-conditioning of the LS by using the a priori information (s1)– +(s2). We note that cross-validation procedures are typically utilized to find the value of λ such that +the estimated matrices yield the lowest error on test data Hastie et al. (2015, 2009). We also note +that by varying λ we can identify whether the prior one postulates on the system matrices is true; +experimentally, if the error on test data is small for λ → 0+, then the systems may not be similar. +In the next section, we provide two low-complexity solution methods for (3). +3. Centralized and Federated Solutions +3.1. Proximal-gradient method +We note that the problem (3) is convex; when the functions (r2) and (r3) are utilized, (3) involves +a composite cost where the regularization function is not differentiable. Accordingly, we consider +a proximal-gradient method (with line search) for solving (3) (see, e.g., Beck and Teboulle (2009); +Combettes and Pesquet (2011); Parikh et al. (2014)). The algorithm is tabulated as Algorithm 1. +5 + +MULTI-TASK SYSTEM IDENTIFICATION +Algorithm 1 Proximal gradient method with line search for solving (3) +Given: ˆA(0) +1 , · · · , ˆA(0) +N , η(0), and β ∈ (0, 1). +Repeat: m = 0, 1, 2, . . . until convergence +[S1] α ← η(m). +[S2] Proximal-gradient with line search: +[S2.1] Zi = ˆA(m) +i +− α∇Li( ˆA(m) +i +), i ∈ [N] +[S2.2] {Yi}i∈[N] = proxαλR({Zi}i∈[N]) +[S2.3] Break if: �N +i=1 Li(Yi) ≤ �N +i=1 +� +Li( ˆA(m) +i +) + ∇Li( ˆA(m) +i +)⊤(Yi − ˆA(m) +i +) + +1 +2λ∥Yi − ˆA(m) +i +∥2 +F +� +[S2.4] Update α ← βα. +[S3] η(m+1) ← α, ˆA(m+1) +i +← Yi, i ∈ [N]. +We first note that the convergence to optimal solutions of (3) of Algorithm 1 is guaranteed as +shown in (Beck and Teboulle, 2009, Chapter 2). Moreover, Algorithm 1 can be converted into a +classical proximal-gradient method if the line search is not performed Parikh et al. (2014). Impor- +tantly, the proximal step [S2.2] enjoys a closed-form update when the regularization functions in +(r1)–(r3) are utilized. In particular: +(r1) Consider the function R(A1, . . . , AN) = �N +i=1 +�N +j=i+1 ∥Ai − Aj∥2 +F . For notational sim- +plicity, let zij := [(Z1)ij, (Z2)ij, . . . , (ZN)ij]⊤. Then, [S2.2] boils down to n2 parallel steps +given by: +yij = (zij + 2αλsij[1, 1, · · · , 1])/(2αλN + 1), +i, j ∈ [N], +where sij = �N +p=1(Zp)ij. +(r2) Consider (4) and let zij := [(Z1)ij, (Z2)ij, . . . , (ZN)ij]⊤. Then, [S2.2] boils down to n2 +parallel steps given by: +yij = +zij +∥zij∥2 +max(∥zij∥2 − αλ, 0), +i, j ∈ [N]. +The entries of the matrices {Yℓ}ℓ∈[N] are then filled as (Yℓ)ij = (yij)ℓ. +(r3) Consider (5) and let ¯Z = [vec(A1), vec(A2), . . . , vec(AN)]. Then, [S2.2] is given by ¯Y = +Udiag({max{σi−αλ, 0}})V ∗, where the singular value decomposition of ¯Z is Udiag({σi})V . +The matrices {Yℓ}ℓ∈[N] are then extracted from the columns of ¯Y . +Before proceeding, a couple of remarks are in order. +Remark 1 The multi-task system identification problem (3) can be extended to cases where the +system matrices {Ai}i∈[N] are similar according to more than one of the priors (s1)–(s2). For +example, if the matrices have a common sparsity patters and the differences in the non-zero en- +tries are small, one can utilize the composite regularization function λ1 +�N +i=1 +�N +j=i+1 ∥Ai − Aj∥2 +F ++λ2 +�N +i=1 +�N +j=1 ∥[(A1)ij, (A2)ij, . . . , (AN)ij]⊤∥2, where λ1, λ2 ≥ 0 are tuning parameters. +□ +Remark 2 The proximal-gradient method outlined in Algorithm 1 without line search is amenable +to an online implementation Dall’Anese et al. (2020); Chang and Shahrampour (2022). An online +proximal-gradient method is suitable for cases where the estimates of the systems matrices are +updated at each time t ∈ N after receiving a new measurement xi(t), ui(t) for at least one of the N +LTI systems. +□ +6 + +MULTI-TASK SYSTEM IDENTIFICATION +3.2. Federated Case +In this subsection, we consider a decentralized algorithm where the N systems collaboratively solve +the identification problem (3) without exchanging their recorded trajectories. We consider a message +passing strategy similar to existing federated learning architectures (see, e.g., Yang et al. (2019); +Wang et al. (2022)), where each system updates locally its own matrix ˆAi and where a central node +provides global support to the estimation process by enforcing the similarities across systems. +To this end, we consider N auxiliary optimization variables {Ki ∈ Rn×n}i∈[N], and reformu- +late (3) in the following equivalent manner: +min +{Ai,Ki}N +i=1 +N +� +i=1 +Li(Ai) + λR(K1, . . . , KN) +s.t.: Ai = Ki, +i = 1, 2, · · · , N . +(6) +The structure of the N equality constraints in (6) naturally leads to a decentralized solution approach +with a star communication strategy when primal-dual-type algorithms or the alternating direction +method of multipliers (ADMM) are utilized. Focusing on the ADMM (see, e.g., Boyd et al. (2011); +Giannakis et al. (2016)), we obtain the following updates (where m is the ADMM iteration index): +A(m+1) +i += arg min +Ai Li(Ai) + γ +2∥Ai − K(m) +i ++ γ−1Λ(n) +i +∥2 +F +i = 1, 2, · · · , N +(7a) +{K(m+1) +i +}N +i=1 = arg min +{Ki}N +i=1 +λR(K1, . . . KN) + +N +� +i=1 +γ +2∥Ki − A(m+1) +i +− γ−1Λ(m) +i +∥2 +F +(7b) +Λ(m+1) +i += Λ(m) +i ++ γ +� +A(m+1) +i +− K(m+1) +i +� +, +i = 1, 2, · · · , N +(7c) +where Λi ∈ Rn×n are the dual multipliers associated with the i-th equality constraint in (6), and +γ > 0 is a given parameter. Convergence of the ADMM (7) to solutions of (6) is well investigated +(see, e.g., Boyd et al. (2011); Giannakis et al. (2016)). +Importantly, we note that the steps (7) can be implemented in a decentralized manner where: +(i) step (7a) is implemented locally at each of the N systems; (ii) step (7b) is performed by a central +node to promote similarities across the system matrices; and (iii) copies of the multiplier matrices +can be stored and updated at both the systems and the central node. At each iteration, each of the N +systems exchange with the central note the current iterates A(m) +i +and K(m) +i +. +We note that the updates (7a) and (7b) admit closed-form expressions; these closed-form ex- +pressions are omitted from the paper because of space limitations. +4. Numerical Simulations +4.1. Experiments on brain networks +We test the proposed method for the problem of estimating the dynamics of brain networks, us- +ing data corresponding to the resting state functional magnetic resonance imaging (fMRI) from the +Human Connectome Project (HCP)2 Nozari et al. (2020); Srivastava et al. (2020); Gu et al. (2015). +Here, xi(t) is an 116-dimensional blood-oxygen-level-dependent (BOLD) time series for 116 par- +cellations of the brain of the i-th subject. Our goal here is to estimate N = 5 dynamical systems of +2. Data available at https://wiki.humanconnectome.org/ +7 + +MULTI-TASK SYSTEM IDENTIFICATION +the form xi(t+1) = Aixi(t)+wi(t), that model the evolution of BOLD signal when the individual +is in a resting state, with wi(t) capturing process noise (the model does not contain external inputs +ui due to the resting state condition). +Since the matrices {Ai}i∈[5] are unknown, we consider the following error for each system: +E(A) := 1 +n +n +� +k=1 +�p +i=1(xi(k) − [Axi](k))2 +�p +i=1(xi(k) − ¯x(k))2 +, +where n is the length of the testing vector, p is the number of testing data and ¯x(k) := 1 +p +�p +i=1 xi(k). +Note that 1 − E(A) is precisely the average R2 indicator of Nozari et al. (2020). +We consider three different methods: (i) the LS estimator (2), which is utilized per individ- +ual; (ii) the Least Absolute Shrinkage and Selection Operator (LASSO), which is again utilized per +individual as proposed in Nozari et al. (2020); and, (iii) the proposed method (3) with the group- +sparsity regularization function (4). The rationale behind the group-sparsity is that the brain dy- +namics should exhibit the same effective connectivity between parcellations, though the remaining +entries acknowledge the diversity in intensities of the interactions across individuals. We note that +the effectiveness of the LS and LASSO has been experimentally validated in Nozari et al. (2020), +where their estimation accuracy has been compared with several identification methods. Moreover, +we performed a cross-validation procedure to optimize the performance of the LASSO. +(a) +(b) +Figure 1: (a) Mean error of LS, LASSO and multi-task (MT) system identification; “Case k” means +that 100k training data points are available for each subject (k = 1, 2, · · · , 9). (b) Mean +error for subjects 2-5 and error for subject 1. “Case k” means that 25k fMRI scans are +used for subjects 1 (dashed line) while 100k (solid line) scans are used for subjects 2-5. +In Figure 1, we compare the LS, LASSO and our approach (which is labeled as “MT”) in two +cases: (a) the same amount of training data is utilized for the five subjects; and, (b) for subject 1, we +utilize only 25% of the training data points with respect to the other subjects 2-5. We use 100 test +points. In Figure 1(a) we plot the mean error across the subjects 1-5; in Figure 1(b) we plot the mean +error across the subjects 2-5 and the error for subject 1, for which fewer fMRI readings are available. +The proposed method outperforms the LS and the LASSO, on par with the number of fMRI scans +in both cases. The merits of the proposed method are particularly evident in Figure 1(b), where +the proposed method significantly outperforms the LASSO for the subject 1; on the other hand, the +LS is ill-conditioned and does not return meaningful estimates. This shows the ability to leverage +8 + +MULTI-TASK SYSTEM IDENTIFICATION +information and data (in this case, fMRI readings) from the dynamics of subjects 2-5 to assist the +estimation of the dynamics in subject 1. +(a) +(b) +(c) +Figure 2: Comparison between LS, LASSO and multi-task (MT) system identification (a) Case 1: +900 training data points for each subject. (b) Case 2: For subject 1, 75 training data +points, and 300 for subjects 2-5. (c) Case 3: For subject 1 and 3, 150 training data points, +and 600 for subjects 2, 4, and 5. In the box plots, the red center line, box limits, and +whiskers represent the median, upper and lower quartiles, and the smallest and largest +samples, respectively. Red crosses indicate outliers. +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +Figure 3: Estimated matrix ˆAi for Case 3, individuals 1, 2 and 3, for n = 116 brain parcellations. +To provide additional comparisons other than the mean error, Figure 2 shows the box plots for +the LS, the LASSO, and the proposed approach in three different scenarios. In particular, Figure +2(a) shows that proposed multi-task identification method can achieve a smaller or comparable error +9 + +MULTI-TASK SYSTEM IDENTIFICATION +(on average) than LS or LASSO when trajectories of 900 time steps are used for each subject (and +these training trajectories are sufficiently rich). Figure 2(b) considers the case where 75 training +data points are available for subject 1 and 300 for subjects 2-5. Here, the LS does not perform well +due to ill-conditioning. The performance of the LASSO is comparable with the one of the proposed +method in terms of median; however, the proposed method shows smaller upper and lower quartiles. +Finally, Figure 2(c) considers the case where fewer fMRI readings are available for subjects 1 and +3; the proposed method performs better than the LASSO in terms of quartiles and has a significantly +less error deviation across the parcellations. +Finally, a representative example of the estimated matrices ˆAi for the subjects 1-3 is provided in +Figure 3. The estimated matrices are the ones obtained in the case considered in Figure 2(c), where +subjects 1 and 3 have fewer training points. It is possible to notice that the three matrices have zeros +in many common entries. Based on this result, we will explore additional regularization methods +that will combine group sparsity with (entry-wise) sparsity. +4.2. Experiments on synthetic data +We provide additional results on synthetic data. We consider 10 systems as in (1), where {Ai}i∈[10] ∈ +R50×50, {Bi}i∈[10] ∈ R50×4, ui(t) is the vector of all ones in R4, i.e. ui(t) is constant vector and +wi(t) ∼ N(0, 0.12). We consider two different cases: common sparsity and linear combinations. +We compare the LS estimator (2) and the proposed method (3) with the group-sparsity regulariza- +tion (4) and nuclear norm regularization (5). +Figure 4 compares the LS and our approach in two cases: (a) all the 10 systems can be repre- +sented by a linear combination of 3 systems and only 25% of the training data points are accessible +for the tenth system with respect to the other systems 1-9; (b) all the 10 systems have the same +sparsity pattern and only 25% of the training data points are accessible for the tenth system with +respect to the other systems 1-9. The testing is on 60 data points. In Figure 4(a), we plot the mean +error across systems 1-9 as well as the error for system 10. The proposed method outperforms the +LS approach in both the mean error and the error for system 10, especially in the case of only a +small number of data available. In Figure 4(b), we can observe similar results. +(a) +(b) +Figure 4: Mean error curve to compare LS and multi-task (MT) system identification methods. (a) +Linear combinations. (b) Common sparsity. “Case k” means that 10k + 10 samples of +the trajectory are used for system 10 (dash line) while 40k + 40k (solid line) are used for +system 1-9, k = 1, 2, 3, 4. In “Case 5”, 75 data points are used for system 10 (dashed +line) while 300k (solid line) are used for system 1-9. +10 + +MULTI-TASK SYSTEM IDENTIFICATION +Acknowledgments +The work of Y. Chen, A. M. Ospina, and E. Dall’Anese was supported in part by the National +Science foundation through the award 1941896 and the ERC ASPIRE. The work of F. Pasqualetti +was supported in part by awards NSF-NCS-FO-1926829 and ARO-W911NF1910360. +The authors would like to thank Dr. Erfan Nozari (University of California at Riverside) for the +assistance with the data used in the simulations, and Killian Wood and Seunghyun Kim (University +of Colorado Boulder) for the discussions on this topic. +References +Amir Beck and Marc Teboulle. Gradient-based algorithms with applications to signal recovery. +Convex optimization in signal processing and communications, pages 42–88, 2009. +Julian Berberich, Johannes K¨ohler, Matthias A M¨uller, and Frank Allg¨ower. Data-driven model +predictive control with stability and robustness guarantees. +IEEE Transactions on Automatic +Control, 66(4):1702–1717, 2020. +Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed optimiza- +tion and statistical learning via the alternating direction method of multipliers. 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IEEE Control Systems Letters, 5(5):1693–1698, 2020. +14 + diff --git a/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/load_file.txt b/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..59f0a45fddd72fd58f713e1b2d552fab4e0c97cb --- /dev/null +++ b/yNAzT4oBgHgl3EQfd_yt/content/tmp_files/load_file.txt @@ -0,0 +1,566 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf,len=565 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='01430v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='SY] 4 Jan 2023 1–14 Multi-Task System Identification of Similar Linear Time-Invariant Dynamical Systems Yiting Chen YITING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='CHEN-1@COLORADO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='EDU Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder Ana M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Ospina ANA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='OSPINA@COLORADO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='EDU Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder Fabio Pasqualetti FABIOPAS@ENGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='UCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='EDU Department of Mechanical Engineering , University of California at Riverside Emiliano Dall’Anese EMILIANO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='DALLANESE@COLORADO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='EDU Department of Electrical, Computer, and Energy Engineering, University of Colorado Boulder Abstract Existing works on identification of the dynamics of linear time-invariant (LTI) systems primarily focus on the least squares (LS) method when the recorded trajectories are rich and satisfy conditions such as the persistency of excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, we consider the case where the recorded states and inputs are not sufficiently rich, and present a system identification framework – inspired by multi-task learning – that estimates the matrices of a given number of LTI systems jointly, by leveraging structural similarities across the LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' By regularizing the LS fit for each system with a function that enforces common structural properties, the proposed method alleviates the ill-conditioning of the LS when the recorded trajectories are not sufficiently rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider priors where, for example, the LTI systems are similar in the sense that the system matrices share a common sparsity pattern, some matrices are linear combinations of others, or their norm difference is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We outline a proximal-gradient method to solve the multi-task identification problem, and we propose a decentralized algorithm in the spirit of existing federated learning architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We provide empirical evidence of the effectiveness of the proposed method by considering a synthetic dataset, and by applying our method to the problem of estimating the dynamics of brain networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For the latter, the proposed method requires a significantly smaller number of fMRI readings to achieve similar error levels of the LS when estimating the brain dynamics across subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Keywords: System identification, multi-task learning, regularized regression, LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Introduction System identification is a core task where the model of dynamical systems is estimated based on observed inputs and states (Ljung (1987);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Pillonetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, identification of lin- ear time-invariant (LTI) systems is a well-investigated problem that has recently received renewed attention due to lines of research in the context of data-driven control and optimization (see, for ex- ample, the representative works in De Persis and Tesi (2019a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Coulson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Hewing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Berberich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Krishnan and Pasqualetti (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' When the observation of the state is noise-free, the LTI system matrices can be readily identified by leveraging the Willems’ Fundamental Lemma, provided that the recorded trajectory satisfies the persistency of excitation (PE) condition as discussed in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Willems et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' De Persis and Tesi © Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Ospina, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Pasqualetti & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Dall’Anese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' MULTI-TASK SYSTEM IDENTIFICATION (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' On the other hand, when the LTI system is subject to unknown process noise or dis- turbances, several existing works focus on the asymptotic and finite time estimation errors and sample complexity of the least squares (LS) estimator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' see, for example, the representative works of Sarkar and Rakhlin (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Simchowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Faradonbeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Oymak and Ozay (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Zheng and Li (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022a) and pertinent references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, fi- nite time error bounds for estimating LTI systems from a single trajectory using the LS method are provided in Sarkar and Rakhlin (2019), and a statistical analysis of the LS estimator is pre- sented in Simchowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Zheng and Li (2020) and Faradonbeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Addi- tionally, regularized system identification methods are investigated in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Chiuso and Pillonetto (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Pillonetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' a low-order linear system identification via regularized regression is considered in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' These regularized identification methods allow one to add a prior on the system matrices, and to strike a balance between LS fit and model complexity Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The performance of the LS estimator hinges on the availability of a recorded trajectory that is sufficiently rich to satisfy the PE condition and to render the LS sufficiently well conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, we are interested in cases where the PE condition may not be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, we consider the task of estimating the system matrices of N > 1 LTI systems, in cases where we do not have sufficiently long (or sufficiently rich) recorded trajectories for at least one of the systems (or for some of the systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Accordingly, the question posed in this paper is as follows: is it possible to leverage “similarities” among the N systems to obtain accurate estimates of the system matrices, even if the PE condition is not satisfied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, if the PE condition fails for the i-th system, can one use recorded data from the other LTI systems to improve the estimation error?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this direction, Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022b), consider estimating the matrices of a linear system from samples generated by a “similar” one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' in particular, in Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022b), a system is considered “similar” if its matrices are perturbed versions of the true system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, we expand the notion of “similarity” to account for additional structural properties that the N systems may have in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider cases where the N LTI systems are similar in the sense that the system matrices share a common sparsity pattern, their norm difference is small, or some system matrices can be expressed as a linear combination of the ones of some of the other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Leveraging these similarity models, we propose a system identification framework that bridges core tools investigated in the context of multi-task learning (Evgeniou and Pontil (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Sener and Koltun (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Zhang and Yang (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Crawshaw (2020)), statistical learning (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009)), and regularized identification methods (Pillonetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' the proposed multi-task system identi- fication is formalized as a regularized regression problem where we minimize the LS fit for each system plus a regularization function that enforces a prior on the structural similarities of the LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' By appropriately tuning (typically via cross-validation Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015)) the weight as- signed to the regularization function, one can find a balance between fitting of the recorded data and model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' More importantly, we show experimentally that the regularization function allows one to transfer structural information and data across systems to alleviate the ill-conditioning of the LS for systems without sufficiently rich recorded trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Our contributions are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (c1) We formalize a multi-task system identification problem for multiple LTI systems, where we consider the minimization of the LS fit for each system plus a regularization function that enforces a prior on the structural similarities of the LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We provide relevant regularization functions 2 MULTI-TASK SYSTEM IDENTIFICATION that are inspired by the group Lasso Yuan and Lin (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Huang and Zhang (2010), nuclear norm minimization Chandrasekaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Mardani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015), and ridge regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (c2) We provide a proximal-gradient method for solving the multi-task system identification prob- lem, and show that the algorithm enjoys closed-form updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We also develop a decentralized algorithm where the N systems collaboratively solve the identification problem without exchanging their recorded trajectories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' the decentralized algorithm involves a message-passing that is similar to federated learning architectures Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (c3) We demonstrate the effectiveness of the proposed multi-task system identification method us- ing: (i) synthetic LTI systems that feature structural similarities, and (ii) real data from the Human Connectome Project (HCP), where blood-oxygen-level-dependent (BOLD) signals are obtained from resting state functional magnetic resonance imaging (fMRI) Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In partic- ular, we show that the proposed method requires a significantly smaller number of fMRI readings to achieve the same error of the LS by presuming that the LTI systems modeling the brain dynamics in number of subjects feature a common sparsity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We also consider the case where only a few fMRI readings are available for one subject, showing the ability to “transfer information” from the dynamics of the other subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, ideas and merits of the proposed method are assessed numerically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' the paper does not include analytical error bounds, which are part of our ongoing research efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Multi-Task System Identification Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Modeling We consider N linear time-invariant (LTI) systems1 xi(t + 1) = Aixi(t) + Biui(t) + wi(t), xi(0) ∈ Rn, i ∈ [N], (1) with i ∈ [N] the system index and t ∈ N the time index, and where xi(t) ∈ Rn, ui(t) ∈ Rp, and wi(t) ∈ Rn are the state, input and process noise, respectively, and Ai ∈ Rn×n and Bi ∈ Rn×p are the matrices of the i-th LTI system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Assume that, for each system, the input ui(t) and state xi(t) can be measured;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' on the other hand, the system matrices are unknown and the disturbance wi(t) cannot be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For the i-th system, suppose that one has access to one trajectory {xi(τ), ui(τ)}Pi τ=0, for some Pi ∈ N, for the state and the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' With these measurements, the system matrices can be estimated using the following LS criterion: min A∈Rn×n,B∈Rn×p Pi � τ=1 ∥xi(τ + 1) − Axi(τ) − Bui(τ)∥2 2, (2) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Notation: We denote by N and R the set of natural numbers and the set of real numbers, respectively, and define [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We let ⊤ denote transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For a given column vector x ∈ Rn, ∥x∥2 is the Euclidean norm and ∥x∥1 denotes the ℓ1 norm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' for a matrix X ∈ Rn×m, ∥X∥F denotes the Frobenious norm and ∥X∥∗ the nuclear norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Moreover (X)ij refers to the entry (i, j) of the matrix X, and vec(X) is a mn × 1 vector stacking the columns of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Given a differentiable function f : Rn → R, ∇f(x) denotes the gradient of f at x (taken to be a column vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Given a closed convex set C ⊆ Rn, projC : Rn → Rn denotes the Euclidean projection of y onto C, namely projC(y) := arg minv∈C ∥y − v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Given a lower-semicontinuous convex function g : Rn → R, the proximal operator is defined as proxλg(y) := arg minx∈Rn g(x) + 1 2λ∥x − y∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 3 MULTI-TASK SYSTEM IDENTIFICATION which is solved for each of the N systems independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The LS estimator (2) has been extensively studied in the literature, especially when the recorded data {xi(τ), ui(τ)}Pi τ=0 satisfy the persistency of excitation (PE) condition Moore (1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Willems et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2005) – where the PE condition trans- lates into the regression matrix being full column rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this case, several results are available in terms of estimation error and in terms of sample complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' see, for example, the results in the recent works of Faradonbeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Simchowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Oymak and Ozay (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Sarkar and Rakhlin (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020), as well as pertinent references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Of course, the data for solving (2) can also be collected from multiple trajectories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' see, for example, Zheng and Li (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, we are interested in cases where the PE condition is not satisfied for some of the N LTI systems (leading to ill-conditioning of the LS for those systems where the PE fails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this case, the question we pose in this paper pertains to whether it is possible to leverage “similarities” among the N systems to obtain accurate estimates of the system matrices, even if the PE condition is not satisfied for one or more systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Key towards answering this question is to define the notion of “similarity” for the system matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' A first effort in this direction was made in Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022b), where the matrices {Ai}i∈[N] and {Bi}i∈[N] are given by perturbations of given common matrices ¯A ∈ Rn×n and ¯B ∈ Rn×n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this paper, we expand this first concept of “similar matrices” to account for the following models: (s1) Small distance: For any pair Ai, Aj, i, j ∈ [N], there exists ǫ > 0 such ∥Ai − Aj∥2 F ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (s2) Common sparsity: The matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' AN have the same sparsity pattern;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', (A1)ij = (A2)ij = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' = (AN)ij = 0 for some entries (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (s3) Linear combinations: For the subset of systems i ∈ C, C ⊆ [N], there exists {αi,j ∈ R} such that Ai = �N j=1,j̸=i αijAj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Similarity (s1) models the case where the norm of the matrix difference Ai − Aj is small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' this is the case, for example, for the model considered by Xin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' On the other hand, (s2) captures a prior on the structural properties of the N systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' general examples include dynamics on network systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' As a concrete example, when (1) represents the dynamics of brain networks, (s2) naturally emerges from a similar functional or structural connectivity of the brain across different individuals (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Finally, (s3) models the case where the matrix Ai of the i-th system can be expressed as a linear combination of some of the other matrices {Aj}N j=1,j̸=i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' as an example, this model may be applicable to traffic flows and mobility-on-demand services (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Turan and Alizadeh (2021)), where the LTI systems (1) model the evolution of the density of vehicles in given geographical areas over given periods of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We note that, while the list above focuses on {Ai}i∈[N], similar arguments may apply to the system matrices {Bi}i∈[N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In the next section, we will present appropriate reformulations of (2) that leverage the similarity models (s1)–(s3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For notational simplicity, hereafter we assume that the matrices {Bi}i∈[N] are known and focus on the estimation of {Ai}i∈[N] from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' However, the proposed methodology extends directly to the case where both {Ai}i∈[N] and {Bi}i∈[N] are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Multi-task system identification problem Assume that one can observe the states and inputs {xi(τ), ui(τ)}Pi τ=0 for each system i ∈ [N] (as mentioned above, Bi is known).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Let Li(Ai) := �Pi τ=1 ∥xi(τ + 1) − Aixi(τ) − Biui(τ)∥2 2 be the 4 MULTI-TASK SYSTEM IDENTIFICATION LS fit for the i-th system as in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In the spirit of regularized LS methods (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009)), we consider estimating the matrices {Ai}i∈[N] by solving the following optimization problem: { ˆAi}i∈[N] ∈ arg min {Ai}N i=1 N � i=1 Li(Ai) + λR(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN), (3) where the first term is the LS fit for the N systems, (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN) �→ R(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN) is a lower- semicontinuous convex function that promotes the prior specified by (s1)–(s3), and λ > 0 is a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, for the priors (s1)–(s3), the following regularization functions can be used: (r1) For (s1), one can use the function R(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN) = �N i=1 �N j=i+1 ∥Ai − Aj∥2 F to penalize large deviations between the estimated matrices (Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (r2) Common sparsity patterns can be promoted by leveraging group sparsity regularization func- tions Yuan and Lin (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Huang and Zhang (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For instance, R(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN) = N � i=1 N � j=1 ∥[(A1)ij, (A2)ij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , (AN)ij]⊤∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (4) (r3) For the model in (s3), when q ≪ N of the matrices {Ai}i∈[N] are such that the remaining N − q can be represented as a linear combination of these q matrices, the n2 × N matrix [vec(A1), vec(A2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , vec(AN)] has rank q ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In this case, the regularization function can be taken to be (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Chandrasekaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Mardani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015)): R(A1, A2, · · · , AN) = ∥[vec(A1), vec(A2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , vec(AN)]∥∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (5) In the formulation (3), the role of the regularization function λR(A1, A2, · · · , AN) is twofold: (i) similarly to classical regularized LS criteria, the parameter λ in (3) strikes a balance between the LS fit (in our case, the LS fit for individual LTI systems) and the complexity of the mod- els Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' for example, for (r2), higher values of λ promote a more parsimonious set of entries in the system matrices that best represents the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (ii) In our specific case, λR(A1, A2, · · · , AN) allows us to fit the data of individual systems less closely – especially for the systems where the PE condition fails – and bypass the ill-conditioning of the LS by using the a priori information (s1)– (s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We note that cross-validation procedures are typically utilized to find the value of λ such that the estimated matrices yield the lowest error on test data Hastie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We also note that by varying λ we can identify whether the prior one postulates on the system matrices is true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' experimentally, if the error on test data is small for λ → 0+, then the systems may not be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In the next section, we provide two low-complexity solution methods for (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Centralized and Federated Solutions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Proximal-gradient method We note that the problem (3) is convex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' when the functions (r2) and (r3) are utilized, (3) involves a composite cost where the regularization function is not differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Accordingly, we consider a proximal-gradient method (with line search) for solving (3) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Beck and Teboulle (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Combettes and Pesquet (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2014)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The algorithm is tabulated as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 5 MULTI-TASK SYSTEM IDENTIFICATION Algorithm 1 Proximal gradient method with line search for solving (3) Given: ˆA(0) 1 , · · · , ˆA(0) N , η(0), and β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Repeat: m = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' until convergence [S1] α ← η(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' [S2] Proximal-gradient with line search: [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='1] Zi = ˆA(m) i − α∇Li( ˆA(m) i ), i ∈ [N] [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2] {Yi}i∈[N] = proxαλR({Zi}i∈[N]) [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='3] Break if: �N i=1 Li(Yi) ≤ �N i=1 � Li( ˆA(m) i ) + ∇Li( ˆA(m) i )⊤(Yi − ˆA(m) i ) + 1 2λ∥Yi − ˆA(m) i ∥2 F � [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='4] Update α ← βα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' [S3] η(m+1) ← α, ˆA(m+1) i ← Yi, i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We first note that the convergence to optimal solutions of (3) of Algorithm 1 is guaranteed as shown in (Beck and Teboulle, 2009, Chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Moreover, Algorithm 1 can be converted into a classical proximal-gradient method if the line search is not performed Parikh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Impor- tantly, the proximal step [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2] enjoys a closed-form update when the regularization functions in (r1)–(r3) are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular: (r1) Consider the function R(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , AN) = �N i=1 �N j=i+1 ∥Ai − Aj∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For notational sim- plicity, let zij := [(Z1)ij, (Z2)ij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , (ZN)ij]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Then, [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2] boils down to n2 parallel steps given by: yij = (zij + 2αλsij[1, 1, · · · , 1])/(2αλN + 1), i, j ∈ [N], where sij = �N p=1(Zp)ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (r2) Consider (4) and let zij := [(Z1)ij, (Z2)ij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , (ZN)ij]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Then, [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2] boils down to n2 parallel steps given by: yij = zij ∥zij∥2 max(∥zij∥2 − αλ, 0), i, j ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The entries of the matrices {Yℓ}ℓ∈[N] are then filled as (Yℓ)ij = (yij)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (r3) Consider (5) and let ¯Z = [vec(A1), vec(A2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , vec(AN)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Then, [S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2] is given by ¯Y = Udiag({max{σi−αλ, 0}})V ∗, where the singular value decomposition of ¯Z is Udiag({σi})V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The matrices {Yℓ}ℓ∈[N] are then extracted from the columns of ¯Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Before proceeding, a couple of remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Remark 1 The multi-task system identification problem (3) can be extended to cases where the system matrices {Ai}i∈[N] are similar according to more than one of the priors (s1)–(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' For example, if the matrices have a common sparsity patters and the differences in the non-zero en- tries are small, one can utilize the composite regularization function λ1 �N i=1 �N j=i+1 ∥Ai − Aj∥2 F +λ2 �N i=1 �N j=1 ∥[(A1)ij, (A2)ij, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , (AN)ij]⊤∥2, where λ1, λ2 ≥ 0 are tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' □ Remark 2 The proximal-gradient method outlined in Algorithm 1 without line search is amenable to an online implementation Dall’Anese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Chang and Shahrampour (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' An online proximal-gradient method is suitable for cases where the estimates of the systems matrices are updated at each time t ∈ N after receiving a new measurement xi(t), ui(t) for at least one of the N LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' □ 6 MULTI-TASK SYSTEM IDENTIFICATION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Federated Case In this subsection, we consider a decentralized algorithm where the N systems collaboratively solve the identification problem (3) without exchanging their recorded trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider a message passing strategy similar to existing federated learning architectures (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2022)), where each system updates locally its own matrix ˆAi and where a central node provides global support to the estimation process by enforcing the similarities across systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' To this end, we consider N auxiliary optimization variables {Ki ∈ Rn×n}i∈[N], and reformu- late (3) in the following equivalent manner: min {Ai,Ki}N i=1 N � i=1 Li(Ai) + λR(K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' , KN) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' : Ai = Ki, i = 1, 2, · · · , N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (6) The structure of the N equality constraints in (6) naturally leads to a decentralized solution approach with a star communication strategy when primal-dual-type algorithms or the alternating direction method of multipliers (ADMM) are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Focusing on the ADMM (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Giannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2016)), we obtain the following updates (where m is the ADMM iteration index): A(m+1) i = arg min Ai Li(Ai) + γ 2∥Ai − K(m) i + γ−1Λ(n) i ∥2 F i = 1, 2, · · · , N (7a) {K(m+1) i }N i=1 = arg min {Ki}N i=1 λR(K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' KN) + N � i=1 γ 2∥Ki − A(m+1) i − γ−1Λ(m) i ∥2 F (7b) Λ(m+1) i = Λ(m) i + γ � A(m+1) i − K(m+1) i � , i = 1, 2, · · · , N (7c) where Λi ∈ Rn×n are the dual multipliers associated with the i-th equality constraint in (6), and γ > 0 is a given parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Convergence of the ADMM (7) to solutions of (6) is well investigated (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=', Boyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Giannakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Importantly, we note that the steps (7) can be implemented in a decentralized manner where: (i) step (7a) is implemented locally at each of the N systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (ii) step (7b) is performed by a central node to promote similarities across the system matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' and (iii) copies of the multiplier matrices can be stored and updated at both the systems and the central node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' At each iteration, each of the N systems exchange with the central note the current iterates A(m) i and K(m) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We note that the updates (7a) and (7b) admit closed-form expressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' these closed-form ex- pressions are omitted from the paper because of space limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Numerical Simulations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Experiments on brain networks We test the proposed method for the problem of estimating the dynamics of brain networks, us- ing data corresponding to the resting state functional magnetic resonance imaging (fMRI) from the Human Connectome Project (HCP)2 Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Here, xi(t) is an 116-dimensional blood-oxygen-level-dependent (BOLD) time series for 116 par- cellations of the brain of the i-th subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Our goal here is to estimate N = 5 dynamical systems of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Data available at https://wiki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='humanconnectome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='org/ 7 MULTI-TASK SYSTEM IDENTIFICATION the form xi(t+1) = Aixi(t)+wi(t), that model the evolution of BOLD signal when the individual is in a resting state, with wi(t) capturing process noise (the model does not contain external inputs ui due to the resting state condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Since the matrices {Ai}i∈[5] are unknown, we consider the following error for each system: E(A) := 1 n n � k=1 �p i=1(xi(k) − [Axi](k))2 �p i=1(xi(k) − ¯x(k))2 , where n is the length of the testing vector, p is the number of testing data and ¯x(k) := 1 p �p i=1 xi(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Note that 1 − E(A) is precisely the average R2 indicator of Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider three different methods: (i) the LS estimator (2), which is utilized per individ- ual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (ii) the Least Absolute Shrinkage and Selection Operator (LASSO), which is again utilized per individual as proposed in Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' and, (iii) the proposed method (3) with the group- sparsity regularization function (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The rationale behind the group-sparsity is that the brain dy- namics should exhibit the same effective connectivity between parcellations, though the remaining entries acknowledge the diversity in intensities of the interactions across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We note that the effectiveness of the LS and LASSO has been experimentally validated in Nozari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (2020), where their estimation accuracy has been compared with several identification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Moreover, we performed a cross-validation procedure to optimize the performance of the LASSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (a) (b) Figure 1: (a) Mean error of LS, LASSO and multi-task (MT) system identification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' “Case k” means that 100k training data points are available for each subject (k = 1, 2, · · · , 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (b) Mean error for subjects 2-5 and error for subject 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' “Case k” means that 25k fMRI scans are used for subjects 1 (dashed line) while 100k (solid line) scans are used for subjects 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In Figure 1, we compare the LS, LASSO and our approach (which is labeled as “MT”) in two cases: (a) the same amount of training data is utilized for the five subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' and, (b) for subject 1, we utilize only 25% of the training data points with respect to the other subjects 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We use 100 test points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In Figure 1(a) we plot the mean error across the subjects 1-5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' in Figure 1(b) we plot the mean error across the subjects 2-5 and the error for subject 1, for which fewer fMRI readings are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The proposed method outperforms the LS and the LASSO, on par with the number of fMRI scans in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The merits of the proposed method are particularly evident in Figure 1(b), where the proposed method significantly outperforms the LASSO for the subject 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' on the other hand, the LS is ill-conditioned and does not return meaningful estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' This shows the ability to leverage 8 MULTI-TASK SYSTEM IDENTIFICATION information and data (in this case, fMRI readings) from the dynamics of subjects 2-5 to assist the estimation of the dynamics in subject 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (a) (b) (c) Figure 2: Comparison between LS, LASSO and multi-task (MT) system identification (a) Case 1: 900 training data points for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (b) Case 2: For subject 1, 75 training data points, and 300 for subjects 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (c) Case 3: For subject 1 and 3, 150 training data points, and 600 for subjects 2, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In the box plots, the red center line, box limits, and whiskers represent the median, upper and lower quartiles, and the smallest and largest samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Red crosses indicate outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='8 Figure 3: Estimated matrix ˆAi for Case 3, individuals 1, 2 and 3, for n = 116 brain parcellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' To provide additional comparisons other than the mean error, Figure 2 shows the box plots for the LS, the LASSO, and the proposed approach in three different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In particular, Figure 2(a) shows that proposed multi-task identification method can achieve a smaller or comparable error 9 MULTI-TASK SYSTEM IDENTIFICATION (on average) than LS or LASSO when trajectories of 900 time steps are used for each subject (and these training trajectories are sufficiently rich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Figure 2(b) considers the case where 75 training data points are available for subject 1 and 300 for subjects 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Here, the LS does not perform well due to ill-conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The performance of the LASSO is comparable with the one of the proposed method in terms of median;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' however, the proposed method shows smaller upper and lower quartiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Finally, Figure 2(c) considers the case where fewer fMRI readings are available for subjects 1 and 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' the proposed method performs better than the LASSO in terms of quartiles and has a significantly less error deviation across the parcellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Finally, a representative example of the estimated matrices ˆAi for the subjects 1-3 is provided in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The estimated matrices are the ones obtained in the case considered in Figure 2(c), where subjects 1 and 3 have fewer training points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' It is possible to notice that the three matrices have zeros in many common entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Based on this result, we will explore additional regularization methods that will combine group sparsity with (entry-wise) sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Experiments on synthetic data We provide additional results on synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider 10 systems as in (1), where {Ai}i∈[10] ∈ R50×50, {Bi}i∈[10] ∈ R50×4, ui(t) is the vector of all ones in R4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' ui(t) is constant vector and wi(t) ∼ N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We consider two different cases: common sparsity and linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' We compare the LS estimator (2) and the proposed method (3) with the group-sparsity regulariza- tion (4) and nuclear norm regularization (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Figure 4 compares the LS and our approach in two cases: (a) all the 10 systems can be repre- sented by a linear combination of 3 systems and only 25% of the training data points are accessible for the tenth system with respect to the other systems 1-9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (b) all the 10 systems have the same sparsity pattern and only 25% of the training data points are accessible for the tenth system with respect to the other systems 1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The testing is on 60 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In Figure 4(a), we plot the mean error across systems 1-9 as well as the error for system 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The proposed method outperforms the LS approach in both the mean error and the error for system 10, especially in the case of only a small number of data available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In Figure 4(b), we can observe similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (a) (b) Figure 4: Mean error curve to compare LS and multi-task (MT) system identification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (a) Linear combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' (b) Common sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' “Case k” means that 10k + 10 samples of the trajectory are used for system 10 (dash line) while 40k + 40k (solid line) are used for system 1-9, k = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' In “Case 5”, 75 data points are used for system 10 (dashed line) while 300k (solid line) are used for system 1-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' 10 MULTI-TASK SYSTEM IDENTIFICATION Acknowledgments The work of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Ospina, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Dall’Anese was supported in part by the National Science foundation through the award 1941896 and the ERC ASPIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The work of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Pasqualetti was supported in part by awards NSF-NCS-FO-1926829 and ARO-W911NF1910360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' The authors would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Erfan Nozari (University of California at Riverside) for the assistance with the data used in the simulations, and Killian Wood and Seunghyun Kim (University of Colorado Boulder) for the discussions on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' References Amir Beck and Marc Teboulle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Gradient-based algorithms with applications to signal recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Convex optimization in signal processing and communications, pages 42–88, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNAzT4oBgHgl3EQfd_yt/content/2301.01430v1.pdf'} +page_content=' Julian Berberich, Johannes K¨ohler, Matthias A M¨uller, and Frank Allg¨ower.' metadata={'source': 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University of Sydney, Sydney, Australia ABSTRACT Semantic-driven 3D shape generation aims to generate 3D objects conditioned on text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Previous works face problems with single-category generation, low-frequency 3D details, and requiring a large number of paired datasets for train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' To tackle these challenges, we propose a multi-category conditional diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Specifically, 1) to alleviate the problem of lack of large-scale paired data, we bridge the text, 2D image and 3D shape based on the pre-trained CLIP model, and 2) to obtain the multi-category 3D shape feature, we apply the conditional flow model to generate 3D shape vector conditioned on CLIP embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3) to generate multi- category 3D shape, we employ the hidden-layer diffusion model conditioned on the multi-category shape vector, which greatly reduces the training time and memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Index Terms— Conditional Diffusion, Text-to-Shape, Multi-modal, Latent Vector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' INTRODUCTION As the core element in the Metaverse world [1], 3D objects play a vital role in enhancing people’s interactive experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' With the rapid development of AIGC technology [2, 3, 4], people can easily create images, audio, video, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' through text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' But 3D objects are currently designed by man- ually modeling software like Blender and Maya3D, which re- quires a great deal of time and expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Therefore, how to generate high-quality 3D objects through semantic informa- tion becomes a practical task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3D shape generation [17] is a challenging task, unlike 2D images which can be viewed as arrays of pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3D ob- jects have diverse and complex representations, such as vox- els, point clouds, grids, and implicit representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Each representation has its own advantage and limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Differ- ent representations require different processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Text-to-shape generation is also challenging [18, 19, 20] since it is hard to jointly understand 3D shape and text at the same time, resulting in it being difficult to represent them in a common space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' At the same time, unlike text-to-image gener- ation, where paired data is abundant, text-to-shape generation lacks large-scale paired text and shape data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Recently, much work has been done on 3D shape gener- ation [5, 6, 7, 14, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' DreamFusion [6] transforms the dif- fusion and denoising process in the pixel space into the op- erations in the NeRF parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Since the supervision signal in DreamFusion operates on very low-resolution im- ages (64 × 64), therefore it cannot synthesize high-frequency 3D geometric and texture details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' DPM [14] trains an en- coder to generate a shape vector representing the point cloud shape, which is then used to train a flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' After that, the pre-trained flow model can turn noise into the shape vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Subsequently, the diffusion model part utilizes this shape vector as a condition for 3D shape generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' DPM [14] is trained on the specific category, therefore it can only generate point cloud data of one type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' To tackle these challenges, we first pre-train a CLIP model which establishes a superior correspondence between text and 2D image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' At the same time, we can get a large number of high-resolution 2D images corresponding to 3D objects through the blender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Therefore, the CLIP model bridges text, 2D images, and 3D objects, thus alleviating the problem of lack of large-scale paired text-3D objects data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Thereafter, we apply a condition flow model to generate the specific category shape vector conditioned on CLIP embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Subsequently we employ a condition diffusion model to generate 3D object conditioned on the shape vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Specifically, during train- ing, the CLIP model is used to encode the 2D image as the condition, so the corresponding relationship between the 2D image and the 3D shape can be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' During inference, the CLIP model is used to encode the semantic information as the condition, thus the 3D shape corresponding to the seman- tic information can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' At the same time, in view of the high time and memory consumption problems of the dif- fusion model itself, we implement the diffusion and denoising operations on the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' To summarize, our main contributions are as follows: Considering the superior correspondence between images and texts in the CLIP model, we use images as the interme- diary to generate 3D objects with semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We propose a conditional diffusion model based on hidden layers and then use the model to generate multiple cate- gories of point cloud data, which greatly reduces training time and memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='13591v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='CV] 31 Jan 2023 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' BACKGROUND 3D object generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3D-GAN [12] uses a three-dimensional convolutional neural network to gradually map a high- dimensional hidden vector into a 3D object represented by a voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' However, due to the uncertainty of Generative Ad- versarial Networks, the results are not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' PointFlow [13] introduces a flow model to generate the shape distribution of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' It uses the hidden vector representing the shape distribution as a conditional to guide the point cloud gener- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Since point clouds are usually distributed on a two- dimensional manifold, it is difficult to obtain better results through a flow model assuming that the point cloud obeys a three-dimensional prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' In the 3D domain, DPM [14] and PVD [15] use diffusion models to generate point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Although they can generate satisfactory results, they are all trained in a specific category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Semantics-Driven 3D Object Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Text2Shape [21] proposes an end-to-end association learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' It encodes text and 3D shapes separately into the same latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' However, large-scale 3D-text datasets are still diffi- cult to obtain, so ClipForge [22] bypasses this problem with the aid of the CLIP model on text-image matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CLIP- Mesh [23] also uses the CLIP model to measure the match- ing degree between the image rendered by the grid model and the text, so as to optimize the entire model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Dreamfields [5], DreamFusion [6] and Magic3D [7] all use NeRF [24] as an implicit representation of 3D objects, and render images through differentiable renderers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' They utilize the matching degree between images and text to optimize the entire network and finally adopt the optimized implicit neural field representation to extract the 3D mesh model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' METHOD The schematic overview of the proposed architecture is il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The photo on the left is the training architecture of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' It mainly consists of four com- ponents: shape encoder, CLIP model, condition flow model, and condition diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We use the DPM model [14] as our backbone model, which samples noise data from Gaus- sian distribution and generates point cloud data through the denoising process under the guidance of the shape vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Specifically, we separate our model into two tasks during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' First, we render the 3D objects to obtain high- resolution 2D images, and then the 2D rendered images are used as the pre-trained CLIP model input, thereafter the con- ditional flow model is trained to establish the relationship between the CLIP model output and the shape vector s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Next, we adopt the shape vector as the condition to guide the 3D shape generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' During inference, the text is used as the CLIP model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Based on the bi-directionality of the flow model, we can obtain the shape vector s guided by the CLIP model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Subsequently the shape vector s guides the diffusion model to generate multi-category point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Shape Encoder: It maps the point cloud data to a distri- bution of shape vectors, namely the shape mean and shape variance, and then samples a shape vector from the shape mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The overall network includes the feature extraction layer and distribution map layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' For the feature ex- traction layer, the attribute values of point cloud data are only 3D coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' First, we use a series of 1D convolutional layers to increase the dimension of the point cloud data, and then select the maximum value of each dimension feature to perform feature dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Then for the distribu- tion mapping layer, the data after feature dimension reduction is mapped to the shape mean and variance respectively with the fully connected layer to represent the distribution of the point cloud shape vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' After obtaining the representation of the shape mean and variance, randomly generate an offset value ε to sample a shape vector s defined as equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' z = µ + ϵ ∗ exp � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='5 ∗ log � σ2�� (1) CLIP Model: It encodes text and images into the same latent space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' matching images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Therefore, based on the CLIP model, we learn the correspondence between 3D point clouds and text using images as an intermediary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The CLIP model is based on VisualTransformer [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We match images to 16*16 text vectors using the ViT-B/32 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Im- ages and texts are passed through corresponding CLIP en- coders to obtain a one-dimensional vector with a length of 256, which is normalized and input into the conditional flow model as a condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Conditional Flow Model: Traditional VAE encodes data into a standard normal distribution, while the flow model can learn a more flexible and variable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The shape vec- tor is fed into the conditional flow model to learn the transfor- mation from the Gaussian noise distribution to the distribution of s, where the CLIP encoded vector is as the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Dur- ing inference, the data is directly sampled from the Gaussian distribution, and the corresponding shape vector is obtained through the inverse transformation of the flow model, which is then input into the diffusion model as a condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We use the affine transformation layer in the RealNVP network archi- tecture [25] to build the flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The affine transformation layer divides the input into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The first part keeps the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' For the second part, the scale scaling coeffi- cient and the offset coefficient are used to transform the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Point Cloud Autoencoder : A point cloud autoencoder consists of an encoder and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The encoder is mainly based on the PointNet network architecture [9] and the graph- based max pooling layer [10] to extract point cloud features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The decoder is mainly based on the FoldingNet [26], which transforms point cloud features into raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Figure 2 shows the network architecture of the autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Similar to LDM [8], a point cloud autoencoder is first trained, and the encoded hidden vector is used as the input of the diffusion model for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' During inference, the output of the diffusion model 𝑫 𝑿(𝟎) Shape encoder Condition flow CLIP 𝑿(𝒕) 𝑿(𝑻) ℇ S Point cloud image … … Gaussian prior 𝑿(𝟎) 𝑿(𝒕) 𝑿(𝑻) … … Point cloud A boeing 747 CLIP Condition flow Gaussian prior Text S Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' An overview of our proposed model after the inverse diffusion process is a vector in the hidden space, which is decoded into point cloud data by the point cloud decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Conditional Diffusion Model: We transform noisy data into point cloud data using a diffusion model whose condi- tion is the shape vector s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The diffusion model is comprised of the diffusion process and the denoised process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The dif- fusion process of the point cloud gradually adds noise to the point cloud hidden vector, thereby converting a point cloud distribution of a specific shape into a random noise distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The diffusion process can be expressed as follows: q � x(t) i | x(t−1) i � = N � x(t) | � 1 − βtx(t−1), βtI � (2) q � x1:T i | x(0) i � = T � t=1 q � x(t) i | x(t−1) i � (3) where β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='βT are hyperparameters at each time step that controls the noise addition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The denoised process is to recover the original point cloud hidden vector from the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' First, the point cloud hid- den vector is sampled from the noise distribution, and then through the reverse Markov chain, the noise is gradually sub- tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Under the condition of shape vector s, the denoised diffusion process can be expressed as follows: pθ � x(t−1) | x(t), s � = N � x(t−1) | µθ � x(t), t, s � , βtI � (4) pθ � x(0:T ) | s � = p � x(T )� T � t=1 pθ � x(t−1) | x(t), s � (5) Among them, µθ is a mean value estimated by the neural network, s is the shape vector, and the initial data of inverse diffusion obeys the standard normal distribution N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The training objective is to maximize the likelihood func- tion of the generated point cloud data E � log pθ � X(0)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Similar to the VAE model, the specific optimization goal is still to maximize its variational lower bound (ELBO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' E[log pθ(X(0))] ≥ E � log pθ(X(0:T ), s) q(X(1:T ), s|X(0)) � = E � log p(XT ) + T � t=1 log pθ(X(t−1)|X(t), s) q(X(t)|X(t−1)) − log qφ(s|X(0)) p(s|c) � (6) Where c is the condition of the flow model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=', the vector encoded by the CLIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' s is the condition of the dif- fusion model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=', the shape vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' To simplify the above variational bound, [14] propose training on pairs of (xt, x0) to learn to parameterize this process with a simple squared L2 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The following objective is simpler to train, resem- bles denoising score matching and was found to yield higher- quality samples: L(θ) = ���ϵ − ϵθ � x(t) i , t, s ���� 2 , ϵ ∼ N(0, I) (7) where t is sampled uniformly between 1 and, and ϵθ is the learned diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' MLP Graph Layer Max Pool MLP Fold Fold Chamfer loss Decoder Encoder Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Point Cloud Autoencoder Network Architecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Dataset We use the ShapeNet (v2) dataset [27], which contains 13 cat- egories of data, and a single sample contains point cloud data and the corresponding rendered images of each 3D object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Evaluation Metrics 1-NNA: It uses the nearest neighbor classifier to test the gen- erated data separately, similar to the discriminator in GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' If it classifies the generated data close to random guessing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=', the accuracy rate is close to 50%, and the quality of the generated data is considered to be relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CLIP R-precision: It [5] can evaluate the generation ef- fect with the composite text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CLIP R-precision ranks retrieval results between generated model renderings and text to mea- sure the visual-semantic similarity between textual descrip- tions and generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The higher the ranking of the real text, the higher the quality of the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We generate point cloud data instead of images, so we need to convert point cloud data into images first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' We use the pre-trained SAP model to convert the point cloud data into a grid model, and then render the grid model to obtain an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Results The point cloud data generated using the words corresponding to the category as text is shown in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' The point cloud data generated using composite text is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' (a) airplane (b) car (c) chair (d) lamp (e) table (f) display Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Point cloud generated from the corresponding word 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CONCLUSION In this paper, we combine the CLIP model and the flow model to propose a zero-shot learning method to establish the rela- Boeing 747 Triangular plane Fighter plane Square chair Round chair Sofa chair Thin table Rectangular table Square table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Point cloud generated from composite text Airplane chair Car Method CD EMD CD EMD CD EMD r-GAN 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='25 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='21 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='24 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='57 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='68 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='65 PointFlow 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='24 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='12 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} 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+page_content='87 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='95 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='52 baseline 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='12 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='21 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='82 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='23 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='32 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='74 ours 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='27 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='68 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='82 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='35 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='07 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='93 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' 1-NNA metrics tested on three categories of Air- plane, Chair, and Car tionship between 3D shape and text through the intermediary of 2D images, which can generate multi-category point cloud and alleviate large-scale Insufficient sample data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' Due to the diffusion and denoising on the hidden layer, the training speed and memory usage are greatly optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CLIP R-precision Method CLIP B/32 CLIP B/16 CLIP L/14 DreamFusion 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='5 ours 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='8 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='1 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content='5 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} +page_content=' CLIP R-precision results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNFRT4oBgHgl3EQfjjf7/content/2301.13591v1.pdf'} 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